Classiﬁcation and equivalences of noncommutative tori and ... · 1.2 Basics of noncommutative...

Classification and equivalences ofnoncommutative tori and quantum lens spaces

ISBN: 978-90-393-5829-0Gedrukt door Wöhrmann Print Service, ZutphenIllustratie omslag: tegelpatroon op een muur van het Alhambra in Granada, Spanje.

Classification and equivalences ofnoncommutative tori and quantum lens spaces

Classificatie en equivalentiesvan niet-commutatieve tori en kwantum-lensruimtes

(met een samenvatting in het Nederlands)

Proefschrift

ter verkrijging van de graad van doctor aan de Universiteit Utrecht opgezag van de rector magnificus, prof. dr. G.J. van der Zwaan, ingevolgehet besluit van het college voor promoties in het openbaar te verdedigenop dinsdag 28 augustus 2012 des middags te 2.30 uur

door

Joannes Jitse Venselaar

geboren op 21 december 1981, te Bandung, Indonesië

Promotor: Prof. dr. G.L.M. Cornelissen

This research has been financially supported by the Netherlands Organisation for Scien-tific research (NWO) under the VICI “ From Arithmetic Geometry to NoncommutativeRiemannian Geometry, and back”.

Contents v

Contents

1 Introduction 11.1 Spin geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Basics of noncommutative geometry . . . . . . . . . . . . . . . . . . . . 31.3 This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Spectral triples and their symmetries 112.1 Definition of a real spectral triple . . . . . . . . . . . . . . . . . . . . . . 112.2 Irreducible spectral triples . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Equivariant spectral triples . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 A table of analogies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Spin structures on the torus 173.1 Outline of the classification . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Hilbert space and algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Reality operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.5 Grading and Hochschild homology . . . . . . . . . . . . . . . . . . . . . 263.6 Dimension, finiteness and regularity . . . . . . . . . . . . . . . . . . . . 293.7 Description of the real spectral triples . . . . . . . . . . . . . . . . . . . 32

4 Unitary equivalences of tori 374.1 Unitary equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2 Automorphisms of the noncommutative torus algebra . . . . . . . . . . . 384.3 Unitary equivalences of noncommutative tori . . . . . . . . . . . . . . . 40

5 Morita equivalences of tori 455.1 Morita equivalences of spectral triples . . . . . . . . . . . . . . . . . . . 465.2 Morita equivalences of noncommutative tori . . . . . . . . . . . . . . . . 495.3 Torus equivariant spectral triples . . . . . . . . . . . . . . . . . . . . . . 53

6 Quantum lens spaces 576.1 The equivariant spectral triple on SUq.2/ . . . . . . . . . . . . . . . . . 586.2 Topological quantum lens spaces . . . . . . . . . . . . . . . . . . . . . . 626.3 Geometry of quantum lens spaces . . . . . . . . . . . . . . . . . . . . . 646.4 Unitary equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

vi Contents

Samenvatting 75

Dankwoord 84

Curriculum Vitae 87

Bibliography 89

Index 95

C H A P T E R 1

Introduction

The study of topological spaces through their algebra of (continuous) functions goes backto the results of Gelfand and Naimark [36] from 1943, which state (in modern terminology)that the category of compact Hausdorff spaces is dual to the category of commutativeunital C �-algebras, and there is a a concrete procedure of reconstructing the underlyingspace from a C �-algebra. In the 1980’s, several people started studying noncommutativeC �-algebras as if they were algebras of functions on “noncommutative spaces”, such as the“irrational rotation” algebra of Rieffel [65] and the quantum groups of Woronowicz [78].Together with these examples, a first attempt at formulating a general theory was givenby Connes [12], circulated several years before publication, motivated among other thingsfrom index theory for foliations. In the 1990’s, the subject exploded, when a general theoryof noncommutative Riemannian spin manifolds (also known as “spectral triples”) wasformulated by Connes. In this introduction, we first summarize classical spin geometry,then introduce its noncommutative analog, and formulate the main results of this thesis.

1

2 Introduction

1.1 Spin geometry

The main object of study of this thesis is the noncommutative version of spin geometry.Spin geometry is a refinement of Riemannian geometry, in which an extra structure is addedto describe features not available in ordinary Riemannian geometry. We give a very briefintroduction to this subject, to motivate the definitions of several objects in noncommutativegeometry which are closely modeled on objects in spin geometry. A thorough introductionto this field can be found in the book of Lawson and Michelsohn [48].

Given an oriented Riemannian manifold M of dimension n with Riemannian metric g,the fiber bundle of orthonormal bases of the tangent space TM is a principal fiber bundleFM with structure group SO.n/, called the frame bundle. The group SO.n/ is not simplyconnected, it has fundamental group Z if nD 2 and Z=2Z if n > 2. There exists a doublecover of SO.n/, called Spin.n/, which is simply connected if n > 2. A spin structure ona manifold M as above is a non-trivial double covering of the frame bundle by a principalSpin.n/-bundle PSpin.M/. A manifold is called spin if there exists such a spin structure.Not every manifold is spin: a necessary and sufficient condition is that the exact sequence

1! Z=2Z! �1.FM/! �1.M/! 1

is split exact. If there exists a spin structure, it need not be unique. Spin structures areclassified by the elements of Hom.�1.M/;Z=2Z/.

The Spin.n/ group can be viewed as a subgroup of the Clifford algebra Cln;0. TheClifford algebra is defined as the quotient of the tensor algebra T .Rn/D

L1

kD0 .Rn/˝k

of Rn by the ideal generated by elements of the form v˝vCq.v/id with v 2 Rn and qa non-degenerate positive quadratic form. It can also be viewed as the algebra generated byRn � T .Rn/, subject to the relations

v �v D�q.v/id;

or equivalentlyv �wCw �v D�2q.v;w/;

where the bilinear form q.v;w/ is defined as q.v;w/ WD q.vCw/� q.v/� q.w/, forv;w 2 Rn. The dimension of this algebra as a vector space over R is 2

�n2

˘. Inside the

Clifford algebra lies the Clifford group, the group generated by the image of an orthonormalbasis of Rn, with respect to the bilinear form, together with �1.

There is an order two automorphism ˛ of the Clifford algebra, which is the extensionof the map v 7! �v on the vector space Rn. The C1 eigenspace of this automorphism iscalled the even part of the Clifford algebra, denoted Cl0n;0 and the �1 eigenspace is calledthe odd part, and denoted by Cl1n;0. The Spin.n/ group is then defined as the group ofinvertible elements in Cl0n;0, generated by elements v 2Rn such that q.v/¤ 0. A secondautomorphism . /t is the one defined by changing the order of the vectors in the tensorproduct:

.v1˝v2 � � �˝vk/tD vk˝�� v2˝v1:

This automorphism preserves the ideal given by v � v D �q.v/id, and hence gives anautomorphism of the Clifford algebra Cln;0. We can combine ˛ and . /t , to get an ordertwo automorphism v 7! ˛.vt / called charge conjugation.

Basics of noncommutative geometry 3

Suppose we have a complex spinor representation �C of Spin.n/ in the unitary groupU.N/. Take the embedding z WU.1/ ,!U.n/. Then the homomorphism�C�z W Spin.n/�U.1/! U.N/ has the element .�1;�1/ in its kernel. We can divide out by this element,to get the Spinc group: Spinc

n WD Spin.n/�Z=2ZU.1/. It fits in the exact sequence:

1! Z=2Z! Spincn! SO.n/�U.1/! 1:

We can then define a spinc-structure as a principal Spinc fiber bundle such that it isa nontrivial cover of the product of the frame bundle times a principal U.1/-bundle. Allspin structures can be turned into a spinc-structure in the obvious way, but the conditionof being spinc is strictly weaker. In particular, every orientable 4-manifold can be givena spinc-structure, while not every 4-manifold is spin.

Charge conjugation can be extended to the complexified Clifford algebra by adding anorder-two automorphism, complex conjugation �: v 7! .˛.vt //

�.If n is even, we can take the generators of the Clifford group and multiply them, to get

a grading operator �:� D .�i/n=2e1 � e2 � � �en:

This operator has eigenvalues ˙1, and anticommutes with the generators e1; : : : ; en of theClifford group.

Given a left Cln;0-module E, we can associate a vector bundle to the principal Spin.n/bundle PSpin.M/ on a manifold M . Let � be a representation of the Clifford algebra on E.The spinor bundle is the associated bundle defined as

S.M/D PSpin.M/��E:

This is also a left Cln;0-module.Given a spin manifold .M;g/ with a given spin structure, there exists a unique lift-

ing of the Levi-Civita connection r to the principal Spin.n/-bundle PSpin.M/, which istorsion-free and compatible with the metric. Given a spinor bundle S over M , there existsa canonical first-order differential operator D acting on sections of S , defined locally as:

Ds D

nXj D1

ej �rejs; (1.1)

where ej is the j -th orthonormal basis vector of TM in the local trivialization, r denotesthe lifted Levi-Civita connection acting on the sections of S , and � denotes the action ofRn on the Clifford module S as elements of the Clifford algebra Cln;0. This operator D iscalled the Dirac operator.

1.2 Basics of noncommutative geometry

We take as the central object of study in this thesis the spectral triple, first defined asK-cycles by Connes [14] and later as spectral triples in [15]. Several accounts of thebasic definitions and motivations of noncommutative geometry exist, starting with Connes’

4 Introduction

book [14]. Other, more introductory texts are [37], [74] and [44]. A full explanation ofthe definitions used in this thesis is given in Chapter 2. We will very briefly describe thecomponents here.

As already mentioned, the basis for noncommutative geometry are the results byGelfand and Naimark, which state that to any compact Hausdorff space X we can associatea commutative unital C �-algebra C.X/ and vice-versa. Even more, if there is continuousmap from X to Y , this gives a C �-homomorphism from C.Y / to C.X/. Every point in Xis given by a character on C.X/, with the topology defined through the weak-�-topologyon the space of characters.

Also by the results of Gelfand and Naimark, every C �-algebra, commutative or not, isisometric and �-isomorphic to a �- and norm-closed subalgebra of bounded operators ona Hilbert space H . This gives us the tools to do topology, but we need extra structure todo differential geometry. This extra structure comes in the form of a Dirac operator, anunbounded self-adjoint operatorD acting on the Hilbert space H . This operator is modeledon the Dirac operator from spin geometry.

From this Dirac operator, one can reconstruct the original metric on X by the usingthe following formula for the distance between point p and q, originally discovered byConnes [13, Proposition 1]:

d.p;q/D supfj. Op� Oq/.a/j W a 2 C.X/;kŒD;a�k � 1g ;

where we denote by Op and Oq the characters corresponding to the points p and q throughthe Gelfand-Naimark theorem. A very basic example of this theorem is the fact that thedistance of points p and q on R is given by the supremum of the difference in value ofC 1-functions with derivative bounded by 1 at the points, see Figure 1.1.

x

f .x/

p

f .p/

q

f .q/

Figure 1.1: Distance of two points, as the difference in value of a function with derivativebounded by 1.

To have a spectral triple that correctly models a differentiable manifold, we take as ouralgebra A not the full C �-algebra C.X/, but a subalgebra of the algebra of continuousfunctions. For elements a of this subalgebra, we require that ŒD;a� is bounded for all

This thesis 5

a 2A (corresponding to Lipschitz continuous) and furthermore iterated commutators with

jDj DpD2 should be bounded as well. The algebra is then not a C �-algebra anymore,

but still a pre-C �-algebra.The resulting triple of objects .A;H ;D/ is called a spectral triple, if it satisfies certain

conditions, further explained in Chapter 2. One could hope that, given suitable conditionson a spectral triple, translated from the Riemannian geometry point of view to this spectralgeometry point of view, there exists a reconstruction theorem, in the sense that givena spectral triple with a commutative algebra, there exists, up to isometries, a uniqueRiemannian manifold, and vice-versa. This turns out to be a rather delicate problem, andwas only fully resolved by Connes in 2008 [18], incorporating several extra conditions notpresent in earlier definitions [15]. It turns out that the natural object which corresponds toa commutative spectral triple is a spinc-manifold.

For noncommutative spectral triples, one can also consider the analogue of a spinmanifold, a so-called real spectral triple. To correctly encode the extra data of being spin,we need to add an antilinear isometry J of the Hilbert space, called the reality operator.This operator J can be thought of as the implementor of the charge conjugation defined inSection 1.1, together with complex conjugation, in the sense that

Jc.a/J�1D c.˛. Na/t /;

where c.�/ is the representation of a Clifford algebra on a vector space, and ˛, N� and .�/t

are the automorphisms of the Clifford algebra defined in Section 1.1, together forming thecharge conjugation. One important feature of this operator J is that it maps an elementa of the algebra A to that of the opposite algebra Ao, which commutes with the originalalgebra A as operators on H , and has the opposite order of multiplication. This condition,while somewhat trivial in the commutative case, has big consequences for the theory ofnoncommutative spectral triples. A real spectral triple consists of the objects .A;H ;D;J /.

Finally, one extra operator sometimes plays a role in the definition of spectral triples. Ifthe dimension of the manifold is even, there exists a grading operator � , the analogue ofthe grading operator defined in Section 1.1. If there is such an operator, we call the spectraltriple even. The most important feature of this operator is that it anticommutes with theDirac operator, and commutes with the representation of the algebra. The grading will notplay a big role in this thesis however.

Noncommutative geometry has found applications in physics (notably string theory,the fractional quantum Hall effect, the standard model coupled to gravity, and cosmology,cf. [19], [6], [9], and [50]) and classical differential geometry (e.g. [17] and [23]).

1.3 This thesis

In this thesis we will study several examples of noncommutative manifolds, given as spectraltriples. Two aspects we are particularly interested in is classification and the equivalencesof real spectral triples.

6 Introduction

We will classify two type of real spectral triples, noncommutative tori and quantum lensspaces. These real spectral triples have in common that they are equivariant with respect toa Hopf algebra. An equivariant spectral triple is the analogue of a homogeneous space incommutative geometry, see Section 2.3 for details. Furthermore, we restrict to irreduciblespectral triples, which are the analogue of connected manifolds, as explained in Section 2.2.

Noncommutative tori

The noncommutative n-torus A.T n�/, or irrational rotation algebra in dimension n, is

one of the first nontrivial examples of a noncommutative topological manifold, given asa deformation of the usual commutative torus, as described for example in [11] and [65].The parameter � by which the algebra is deformed is a number for a noncommutative2-torus, and an antisymmetric n�n matrix for higher dimensional tori.

The analog of putting a spin structure and a metric on this algebra is to enhance itinto a real spectral triple. This introduces a set of extra parameters

˚�in

iD1, with each

�i 2Rn, which are the analogue of the “size” of the torus. The noncommutative n-torus,both topologically and with spin structure, has found many applications in mathematicalphysics, see for example [4], [6] and [47]. A noncommutative spin structure can certainly beconstructed by deforming a spin structure on the commutative n-torus [24], so the questionbecomes whether such deformations give all possible spin structures on the noncommutativen-torus.

In dimension 2, this problem was solved by Paschke and Sitarz [59, Theorem 2.5],who showed that a noncommutative 2-torus admits exactly 4 different real spectral triples(which are deformations of spin structures on the commutative torus). This result can bereformulated as follows: any real spectral triple which is equivariant with respect to a 2-torusaction in the sense of [69] (see Section 2.3 of this thesis), is an isospectral deformation ofa spin structure on a commutative 2-torus.

Our first result is that the theorem of Paschke and Sitarz holds true in arbitrary dimen-sion:

Theorem 3.6.9 (Theorem A). All irreducible real spectral triples with an equivariantn-torus actions are isospectral deformations of spin structures on an n-torus.

The proof of [59, Theorem 2.5] does not generalize readily to higher dimensions. Ratherthan working with the grading operator as in [59], which only gives nontrivial conditions inthe even-dimensional case, we use the reality operator first. Then we establish that out ofseveral possible candidate structures, only one satisfies the growth condition and compactresolvent condition on the Dirac operator. Also, our proof uses at a crucial point Connes’reconstruction theorem [18, Theorem 11.5]. We describe the spin structures explicitly inSection 3.7 of this thesis.

In a celebrated paper [1] (see also [2, Theorem 1]), Adams used the classificationof independent vector fields on spheres to deduce elementary results on Radon-Hurwitznumbers of certain classes of matrices. Similarly, our Theorem 3.6.9 can be used to provethe following elementary result on Hermitian matrices, for which we do not know anelementary proof:

This thesis 7

Corollary 3.6.8 (Corollary B). A set of 2b �2b Hermitian matrices fAigniD1, where nD

2bC1, such that the equation

det

Xi

xiAi

!D 0;

only has the zero solution .xi D 0/niD1 in Rn, generate a Clifford algebra if and only ifX

�2Sn

sign.�/nYi

A�.i/ D �Idk ;

for some nonzero � 2R.

Isomorphisms of spectral triples

There are different views of what it means to be an isomorphism of spectral triples. In thisthesis we study two different versions: namely, unitary equivalence and Morita equivalence(a notion of “correspondence” was recently defined by Mesland [54], but we will notconsider that here). We are also interested in the difference between these two notions.

The first notion is that of unitary equivalence. A unitary equivalence between twospectral triples .A;H ;D/ and .A;H 0;D0/, is given by a unitary map U WH !H 0 thatintertwines the representations � ,� 0 of A and the operator D and D0 on the Hilbertspaces, i.e. U�.�.a//U�1 D � 0.a/ and UDU�1 D D0 with � an automorphism of theC �-algebraA such that the pre-C �-algebra A is mapped to itself. If A is the (commutative)algebra of C1 functions on a manifold X , automorphisms of the algebra correspond todiffeomorphisms of the manifold via the Gelfand-Naimark theorem.

For a commutative n-torus, the diffeomorphisms of a torus act by affine transformationson the set of spin structures (identified with the vector space .Z=2Z/n), as shown in [26].In particular, for the commutative 2-torus, there are two orbits, one consisting of oneelement, and the other consisting of three elements. In the case of the noncommutative torusa description of the full “diffeomorphism” group is not known when n > 2. Restricting toinner automorphisms of the algebra, we can show the following.

Theorem 4.3.1 (Theorem C). Except for a set of � of measure 0, the different spin structuresof the smooth noncommutative n-torus A.T n

�/ as in Theorem 3.6.9 cannot be unitarily

equivalent by an inner automorphism of the algebra.

We also compute the action of unitary transformations induced by some outer automor-phisms on the spin structures. These are the action of SL.2;Z/ on the noncommutative2-torus and the flip automorphism (see Section 4.2) on the noncommutative n-torus forn> 2. Here the results for nD 2match the commutative case, but for n> 2 no equivalencesare found.

The set of � ’s of measure 0 in Theorem 4.3.1 is determined by some Diophantineapproximation conditions given in [7], and includes � with only rational entries.

The second notion of isomorphisms that we study are Morita equivalences. Theapproach here is to study not isomorphisms of the algebras themselves, but equivalences oftheir representation theory. Two rings R and S are precisely Morita equivalent when theircategories of left (or right) representations are equivalent.

8 Introduction

The results of Gelfand and Naimark imply that points of a topological space corre-spond to one-dimensional representations of the corresponding (commutative) algebraof functions. A noncommutative algebra however might not have many interesting one-dimensional representations. This is why the entire representation theory needs to be takeninto account, and this is precisely what Morita equivalence does. For commutative algebras,Morita equivalence implies isomorphism.

A Morita equivalence between two rings R and S is implemented by two bimodules,a left R- and right S -module RPS , and a right R- and left S -module SQR, such thatSQR˝R RPS ' S and RPS ˝S SQR 'R.

In the theory of C �-algebras, a stricter notion due to Rieffel [64], called strong Moritaequivalence, turns out to be more useful. For two (pre-)C �-algebras A and B to be stronglyMorita equivalent, we demand that there exists a Hilbert C �-module BEA, with innerproducts valued in B and E , which is a left B-module and a right A-module, such thatB D EndA.E/, hE;EiB is dense in B and hE;EiA dense in A. See Chapter 5 for a fullexplanation.

This definition of equivalence of (pre-)C �-algebras can be extended to the whole (real)spectral triple .A;H ;D;J /. If A and A0 are strongly Morita equivalent algebras, theHilbert space H and reality operator J are readily transported to a new Morita equivalentspectral triple .A0;H 0;D0;J 0/. The Dirac operator needs a slight adjustment, in the formof a connection, a map from E to E˝�1

D , where �1D is modeled on the space of 1-forms,

see Section 5.1.There is however a problem with this definition of Morita “equivalence” of spectral

triples: it is not symmetric. For spectral triples where the algebra is finite dimensional,ADMn.C/, there exists a Morita equivalence to a spectral triple with Dirac operator 0.The space of 1-forms is trivial in this case, hence all connections are trivial, and there is no“equivalence” the other way around. This was first observed in general by Krajewski [46].See also Section 5.1 of this thesis.

However, when we restrict our attention to torus equivariant spectral triples, we havethe following result in this thesis:

Theorem 5.3.6 (Theorem D). For � in a dense set which is not the countable unionof nowhere dens sets, if the algebras A.T n

�/ and A.T n

� 0/ are strongly Morita equiv-alent, and if there exists a Morita equivalence .E;rD/ of equivariant spectral triplesfrom .A.T n

�/;H ;D;J / to .A.T n

� 0/;H0;D0;J 0/, then there exists a Morita equivalence

. NE;rD0/ from .A.T n� 0/;H

0;D0;J 0/ to .A.T n�/;H ;D;J / such that the composition of the

Morita equivalences gives a spectral triple unitary equivalent to the original spectral triple.A.T n

�/;H ;D;J /. Thus Morita equivalence of equivariant torus spectral triples is an

equivalence relation for a dense set of � .

This appears to be the first non-trivial example of a provably symmetric Morita equiva-lence of spectral triples.

This thesis 9

Quantum lens spaces

Finally, this thesis looks at another example of noncommutative spaces, quotients ofa noncommutative space modeled on the quantum group SUq.2/. It was shown that itis impossible to describe this quantum group in terms of a real spectral triple, but thework of Dabrowski, Landi, Sitarz, Van Suijlekom and Várilly [25] showed that a slightrelaxation of the conditions for a real spectral triple, made it possible to describe SUq.2/

as a 3-dimensional spectral geometry. Instead of strictly adhering to all commutationconditions, they put forward a new definition, where the commutation conditions holdsonly up to compact operators on the Hilbert space.

Based on this work, we study quotients of this spectral triple by finite cyclic groups,which we call quantum lens spaces. In the commutative case, lens spaces are defined asquotients of odd-dimensional spheres by free actions of finite cyclic groups. Lens spacesare interesting because they have many peculiar properties. For example, there exist lensspaces which are homotopy and homology equivalent, but not homeomorphic, as shown forone specific case by Alexander [3], and more generally by Milnor [57] and Olum [58].

The algebra of SUq.2/ is generated by two noncommuting generators a and b, andtheir complex conjugates a� and b�. The action � of a generator g the finite cyclic groupZ=pZ is given by

�.g/aD e2�i

p a �.g/b D e2�ir

p b;

with r an integer coprime to p. We denote the quantum lens space given by such anaction by Lq.p;r/. If we view the spectral triple as an equivariant spectral triple underthe action of Z=pZ, we can determine the corresponding action on the Hilbert space.There exist p different eigensubspaces HK for this action, with eigenvalues e

2�iKp for

K D 0;1; : : : ;p� 1. In principle, for each different value of K, the eigensubspace givesrise to a spectral triple, but we require irreducibility and get the following classification:

Theorem 6.3.3 (Theorem E). The quantum lens space Lq.p;r/ admits one irreduciblespectral geometry if p is odd, and two if p is even. The spectral geometries are given in thetwo cases by

� p odd: .Lq.p;r/;H0;D;J /.

� p even: .Lq.p;r/;H0;D;J / and .Lq.p;r/;Hp=2;D;J /.

with D the Dirac operator as described in (6.20) and J the operator given in (6.21).

This precisely mirrors the situation for spin structures on commutative lens spaces,which were classified by Franc [34].

Due to results by Franz [35] and Reidemeister [62], we know that two commutative lensspaces L.p;r/ and L.p0; r 0/ are isometric if and only if p0 D p and r �˙r 0˙1 mod p.Also, by the results of Ikeda and Yamamoto [41], we have that a lens space is completelycharacterized up to isometries by its Dirac spectrum. For quantum lens spaces, we have thefollowing:

Theorem 6.4.1 (Theorem F). When q 2 .0;1/, the quantum lens spaces given by .Lq.p;r/;

HK ;D;J / and .Lq.p;r0/;HK ;D;J /, with K D 0 and if p is even, K D p=2, are unitary

equivalent if r D˙r 0 mod p.

C H A P T E R 2

Spectral triples and their symmetries

In this chapter we give the definition of a real spectral triple, and define what we meanby an equivariant action of a Hopf algebra on a spectral triple. We also discuss differentnotions of irreducibility.

2.1 Definition of a real spectral triple

Since there are various, slightly different, definitions of real spectral triples, cf. [16, Pages159–162], [74, Chapter 3], [37, Chapter 10], and we need to refer to the axioms unam-biguously, we will explicitly state the definition of real spectral triple that we will use. Asmentioned in the introduction, a real spectral triple is the analogue of a spin structure innoncommutative geometry of a spin structure.

For a real spectral triple we have the following data:

� A unital, Fréchet �-algebra A, �-isomorphic to a dense subalgebra of a C �-algebraA. A is taken to be stable with respect to the holomorphic functional calculus of A,see [37, Definition 3.25],

� A separable Hilbert space H with a faithful representation � of A on H as boundedoperators,

� An unbounded self-adjoint operator D on H , called Dirac operator,

� An antilinear isometry J of H onto itself, called reality operator,

� An integer n� 0, called dimension,

� If n is even, a self-adjoint unitary operator � of H onto itself, such that �2 D Id,called grading operator. We call the spectral triple even in this case.

These objects should satisfy Conditions 1 to 11 below in order for them to be called a realspectral triple of dimension n.

Condition 1 (Compact resolvent). The Dirac operator D has compact resolvent, that is,D has finite dimensional kernel and D�1 (defined on the orthogonal complement of thekernel) is a compact operator.

The compact resolvent condition can also be written as: .D��/�1 is compact 8� …R.

11

12 Spectral triples and their symmetries

Condition 2 (Lipschitz continuity). For all a 2A, ŒD;�.a/� is a bounded operator.

This condition is the analogue of demanding that our functions be Lipschitz continu-ous [14, Lemma 1, Chapter 6.1].

Condition 3 (Grading operator). If n is even, the Z2 grading operator � splits the Hilbertspace H as H DH C˚H �, where H ˙ is the .˙1/-eigenspace of � . The operator D isodd with respect to this operator, i.e.

D� D��D; (2.1)

and the representation � of A on H is even, that is

�.a/D

�C.a/ 0

0 ��.a/

!:

with �˙ a faithful representation of A on H ˙.

Condition 4. The operators J , D and � satisfy the commutation relations from Table 2.1,and the operator J is unitary: J � D J�1.

n mod 8 0 1 2 3 4 5 6 7J 2 D ˙Id (�J ) C C � � � � C C

JD D ˙DJ (�D) C � C C C � C C

J� D ˙�J (�� ) C � C �

Table 2.1: Signs of the spectral triple

Recall that f .k/D O.kp/ means that there exists a positive constant M , and k0 > 0

such that jf .k/j �M jkpj for all k > k0.

Condition 5 (Dimension). The eigenvalues �k of jDj�1, arranged in decreasing order,grow asymptotically as

�k DO.k�n/;

for an integer n. The biggest n for which this growth condition holds is called the dimension.

Condition 6 (Opposite algebra). The reality operator J maps to a commuting algebra, i.e.for all a;b 2A we have:

Œ�.a/;J�.b�/J ��D 0: (2.2)

We will write �o.b/ D J�.b�/J �. The above formulas establish that the oppositealgebra:

AoD fao

WD J �a�J ja 2Ag; (2.3)

lies within the commutator of A. We see that the order of multiplication is reversed, whencethe name. The existence of a commuting representation with opposite order is guaranteedby Tomita theory [70], if A00, the von Neumann algebra given by the bicommutant of A

within the bounded operators B.H /, has a cyclic and separating vector.Furthermore, we also have:

Definition of a real spectral triple 13

Condition 7 (First order condition). For all a;b 2 A the following commutation rela-tion holds:

ŒŒD;�.a/�;J�.b�/J ��D 0: (2.4)

Recall that a Hochschild k-chain is defined as an element c of

Ck.A;M/DM ˝A˝k ;

with M a bimodule over A. A boundary map b W Ck! Ck�1 is defined as

b D

kXiD0

.�1/idi ;

with

�d0.m˝a1˝�� ˝ak/ Dma1˝a2 � � �˝ak ;

di .m˝a1˝�� ˝ak/ Dm˝a1˝�� ˝aiaiC1˝�� ˝ak ;

dk.m˝a1˝�� ˝ak/ D akm˝a1˝�� ˝ak�1;

for m 2M and ai 2A. Since b2 D 0, as can be easily shown, this makes Ck.A;M/ intoa chain complex. The cycles are the elements a 2 Ck.A;M/ such that b..a//D 0.

Condition 6 gives a representation of Hochschild k-chains Ck.A;A˝Ao/ on H by

�D ..a˝bo/˝a1˝�� ˝ak/D ab

oŒD;a1� : : : ŒD;ak �: (2.5)

Condition 8 (Orientability). There is a Hochschild cycle c 2 Zn.A;A˝Ao/ such that�D.c/D � when n is even, and �D.c/D Id when n is odd.

Condition 9 (Regularity). For all a 2 A, both a and ŒD;a� belong to the domain ofsmoothness

1\kD1

Dom�ık�;

where the derivation ı is given by ı.T /D ŒjDj;T �, with jDj DpD2.

Condition 10 (Finiteness). The space of smooth vectors H 1 DT1

kD1 Dom�Dk

�is

a finitely generated projective left A module. Also, there is a Hermitian pairing .�j�/on this module, satisfying

�

Z.�j�/jDj�n

D h�;�i;

where �Z

is the Dixmier trace, or noncommutative integral (defined for example in [74,

Chapter 5]), and h�;�i is the Hilbert space inner product of H .

Condition 11 (Poincaré duality). The Fredholm index of the operator D yields a nonde-generate intersection form on the K-theory ring of the algebra A˝Ao.


2.2 Irreducible spectral triples

In this thesis, we will restrict our attention to irreducible spectral triples, that is, a spectraltriple for which the only projections commuting with the action of the algebra A and Dare 1 and 0. In case A is commutative, this is equivalent to demanding that the manifold isconnected, see [16, Remark 6 on page 163]: if a manifoldX is not connected, say consistingof two disjoint parts X1 and X2, we can define the projection pX1

as projecting onto thesubspace of functions supported onX1. Since C.X/ acts on H by pointwise multiplication,it commutes with the action of the algebra, and because D is locally defined on trivializedbundles over patches of X , it also commutes with D.

Remark 2.2.1. In the literature, there are several other definitions of irreducible spectraltriples. One alternative definition is given in [37, Definition 11.2]. In this definition, wedemand that there are no non-trivial projectors commuting with the action of the algebra,the Dirac operator and the reality operator J . However, if we take this definition, thereexist commutative irreducible spectral triples which are not connected as manifolds. Takefor example two copies of the same connected manifold with identical spin structure, butinstead of the usual J operator, we let J be a map from the spinor bundle over one copy tothe other. All conditions of Section 2.1 are still satisfied, and the spectral triple is irreduciblein the sense of [37, Definition 11.2]. We find in Chapter 6 that this definition gives realspectral triples not corresponding to classical spin structures, and with this definition,counterexamples to the classification in Chapter 3 also exist.

A third definition for irreducibility is the one introduced in [42, Definition 2.1]. There,a degenerate spectral triple is defined to be one where the kernel of D has an invariantsubspace under the action of the algebra. A reducible spectral triple in this sense is onewhere there exists a proper A-invariant subspace H0 �H , such that .A;H0;DjH0

/ isnon-degenerate. For a real spectral triple, H0 should also be invariant under J . Sincethe kernel of D is taken to be finite-dimensional by the compact resolvent condition,a degenerate subspace can only exist if there is a finite-dimensional subspace invariantunder the action of the algebra, for example when the algebra is a direct sum containinga finite-dimensional component. In the cases we study in Chapters 3 and 6, this is notthe case, and then the definition reduces to that of [37, Definition 11.2], discussed in theprevious paragraph.

2.3 Equivariant spectral triples

There are different candidates for the notion of “symmetries” of noncommutative geome-tries. One obvious candidate is the group of automorphisms of the algebra A. The problemis that while lots of new interesting automorphisms, like inner automorphisms, are availablefor noncommutative algebras, if we take deformations of commutative algebras, such asnoncommutative tori, almost no outer automorphisms are known, see Chapter 4.

Equivariant spectral triples 15

To enlarge the group of symmetries in an interesting way, Sitarz [69] introduced thenotion of equivariant spectral triples. One describes symmetries of spectral triples in theform of Hopf algebras. An irreducible spectral triple which is equivariant with respect toa Hopf algebra is the analog of a connected compact homogeneous space in commutativegeometry.

An introduction to Hopf algebras is given for example in [43, Chapter 3]. We willdescribe some basic notions here, to fix the notation. A Hopf algebra is an associativeand coassociative bialgebra over C together with an antipode S . The bialgebra structure isgiven by the following maps on a Hopf algebra H :

� A multiplication � WH ˝H !H ,

� A unit � W k!H ,

� A comultiplication � WH !H ˝H ,

� A counit � WH ! k,

� The antipode S WH !H .

The antipode S has to make the following diagram commute:

H ˝H H ˝H

H k H

H ˝H H ˝H

S ˝ Id

��

�

�

�

Id ˝S

�

Any group can be made into a Hopf algebra, in several ways. One of them is thefollowing procedure, which will be used in Chapter 6. Let CG be the group algebra overC. Set�.g/D g˝g, �.g/D 1 and SgD g�1 for all g 2G, and let the multiplication andunit be the obvious ones. It can easily be checked that this determines a Hopf algebra.

We shall use Sweedler’s notation for the coproduct:

�hDX

i

hi ˝h0i WD h.1/˝h.2/:

We say that H is a �-algebra if there exists and antilinear involution � on H such that

�.h�/D .�h/�˝� �.h�/D �.h/ .S ı�/2 D Id;

where N is the complex conjugation in C. A Hopf algebra is called cocommutative ifh.1/˝h.2/ D h.2/˝h.1/ for all h 2H .

An H -module algebra is an algebra A with a complex linear representation � of H onA such that A, and � respects the algebra structure:

�.h/.a1a2/D��.h.1//a1

��.h.2//a2

�;

�.h/1D �.h/1;


for all h 2 H;a1;a2 2 A. When A is an H -module algebra, we define an equivariantleft-A-module to be a left-A-module M such that

�M .h/.am/D��A.h.1//a

��M .h.2//m

�;

for all h 2H;a 2 A;m 2M . We define equivariant right-A-modules in a similar way. Anequivariant real spectral triple is a real spectral triple .A;H ;D;J / together with a �-Hopfalgebra H over C, such that the objects of the equivariant real spectral triple transform ina compatible way under the action of the Hopf algebra:

� The algebra A is an H -module algebra;

� There is a dense subspace V �H such that V is an equivariant left A-module, andthe intersection of V and the domain D is dense;

� For every h 2H , the Dirac operator is equivariant, ŒD;h�D 0 on the dense intersec-tion of the domain of D and V ;

� The action of H op is well defined on the opposite algebra Ao via the equalityJ�1hJ D .Sh/�;

� If the spectral triple is even, Œ�;h�D 0.

2.4 A table of analogies

To summarize, we have the following analogies between various objects in noncommutativegeometry, and their counterparts in classical, commutative geometry.

Commutative geometry Noncommutative geometry

Topological (Hausdorff) space C �-algebra

Spinc manifold Spectral triple .A;H ;D/

Spin manifold Real spectral triple .A;H ;D;J /

Connected manifold Irreducible spectral triple

Homogeneous space for a group Equivariant spectral triple for a Hopf algebra

C H A P T E R 3

Classification of spin structures on thenoncommutative torus

The different possible spin structures on a differentiable manifold were classified in thework of Milnor [56]; for example, on a (commutative) n-torus, there exist 2n inequivalentspin structures. No such general classification of spin structures is currently known innoncommutative geometry — this amounts to classifying the possible real spectral triplestructures on a C �-algebra. In this chapter we prove that there exist precisely 2n differentreal spectral triples on a noncommutative n-torus, and that these structures are isospectraldeformations of spin structures on the commutative n-torus.

This chapter is based on the first part of the article “Classification of spin structures on the noncommutativen-torus” [75]

17

18 Spin structures on the torus

3.1 Outline of the classification

We first outline the proof of the classification, Theorem 3.6.9 (see Section 1.3). First, inSection 3.2 we determine the only possible irreducible action of the algebra on the Hilbertspace, such that the equivariance condition is met. This action is already well-known, butwe derive it to show that there are no other possibilities. In Section 3.3 we move to realspectral triples, and determine possible forms of the reality operator J , by considering theanti-isomorphism A.T n

�/ 7!A.T n

�/o and the equivariance condition (3.1e) below. We find

several possible families of real spectral triples, only one of which consists of isospectraldeformations of spin structures on the commutative torus. In Section 3.4, we determinethe classes of possible Dirac operators for each candidate family of real spectral triples,using equivariance of the Dirac operator and the first-order condition, and show that onlythe isospectral deformation family is compatible with the compact resolvent condition.

In Section 3.5 we determine that the parameters f�j gnj D1 �j 2Rn, in the Dirac operator

(cf. Lemma 3.4.3), must be linearly independent vectors spanning Rn, using the Hochschildhomology condition and earlier results on the Hochschild homology on noncommutativetori. The last step in the classification is done in Section 3.6, where we use Connes’reconstruction theorem and irreducibility to show that the Dirac operator is in fact a Diracoperator in the sense of spin geometry.

After the classification, we give in Section 3.7 an explicit description of all equivariantreal spectral triples on the noncommutative torus.

3.2 Hilbert space and algebra

We look for possible equivariant representations of the algebra of functions of the smoothnoncommutative torus, and give a basis of the Hilbert space H for which equivariance isobvious. We only use that the action of the algebra is equivariant and irreducible, and thatthe Hilbert space is separable.

We denote the noncommutative torus, or more precisely, the algebra of continuousfunctions on the noncommutative n-torus, by A.T n

�/, where � is an antisymmetric real

n�n matrix.In the case of equivariance with respect to a torus action, we take as Hopf algebra

the universal enveloping algebra U.tn/ of the familiar Lie algebra tn of the n-torus. Thedefinition given in Section 2.3 then translates to the following set of conditions: we havea basis of n elements ıj , generating the algebra, with a representation � on H such that fora 2A.T n

�/, v 2H :

ıj ık D ıkıj ; (3.1a)�.ıj /D ıj ˝ IdC Id˝ ıj ; (3.1b)

�.ıj /�.a/v D��.ıja/C�.a/�.ıj /

�v; (3.1c)

These relations define a commutative and cocommutative Hopf algebra, together withan equivariant representation � of the algebra A.T n

�/ on H .

Hilbert space and algebra 19

In Sections 3.3 and 3.4, we will use the following equivariance conditions on the Diracoperator and reality operator:

�.ıj /Dv DD�.ıj /v; (3.1d)�.ıj /J v D�J�.ıj /

�v: (3.1e)

Remark 3.2.1. Note that an equivariant action of n-torus is different from an n-torus actionas in [20], the former is a condition on the spectral triple, the latter is an action along whichthe algebra is deformed.

The algebra of functions on the noncommutative torus is generated by unitary elementsUe1

; : : : ;Uensuch that

UekUelD exp.2�i�kl /Uel

Uek;

with �kl the component of the matrix at position .k; l/. As a short-hand notation, wewill write

e.�/D exp.2�i �/:

The Hopf algebra action of our basis elements on the unitary generators is:

ıjUekD

(Uej

if j D k,0 if j ¤ k.

If we interpret the ej as the j -th basis vector of Zn, we can write more general unitaryelements Ux of the algebra as:

Ux D e

0@12

Xk>j

xj �jkxk

1A�Ue1

�x1�Ue2

�x2� � �.Uen

/xn ;

for xDPxj ej . The factor in front is needed to ensure the definition is independent of

the ordering of the basic unitaries Uejin the definition. The Fréchet algebra of smooth

functions on the noncommutative torus, defined later in Lemma 3.6.3, is a dense subalgebraof this algebra.

Just as for the algebra, we will write ıx for the derivation given by

ıx D ıx1ıx2� � �ıxn

:

From these definitions, it is immediate that

ıxUy D .x �y/Uy; (3.2)

where .x �y/ is the standard inner product on Zn.Now we look for a minimal Hilbert space H0 which is an equivariant leftA.T n

�/-module.

As a basis of H0 we choose mutual eigenvectors e� of the derivations:

�.ıj /e� D �j e�;

with � 2Rn.


In order for the spectral triple to be a noncommutative torus, we demand that the repre-sentation � of the algebra is equivariant with respect to a torus action, as in equation (3.1c).Written out for the unitary generators Ux, we see:

�0.Ux/e� D ux;�e�Cx;

with ux;� 2C to be determined. Thus for the minimal irreducible equivariant representationthe � will lie in a translate of a lattice, i.e. �D �0Cm:

H0 D

Mm2Zn

H�0Cm; (3.3)

where each H�0CmŠC. There are no restrictions yet on �0, these will be determined later.Since the Ux should be unitary, we have that

he�;ux;�e�Cxi D hu�x;�e��x; e�i

) ux;�ı�;�Cx D u�x;�ı��x;�; (3.4)

where ı�;�Cx is the Kronecker delta, equal to 1 if � D �Cx and 0 otherwise. Finally thedefinition of Ux in terms of Uej

gives the relations

uxCy;� D e

�1

2x ��y

�ux;�Cyuy;�: (3.5)

Lemma 3.2.2. Up to unitary transformations of H0 any unitary equivariant representationof A.T n

�/ on H0 is given by

�A0 .Ux/e� D e

�1

2x �AxCx �A�

�e�Cx; (3.6)

with A any n�n matrix such that A�At D � .

Since the representations �A0 , given by different matrices A such that A�At D � , are

equivalent, we drop the A from the notation, and just write �0 for a representation definedby equation (3.6).

Proof. Clearly, for any matrix A the representation given above satisfies the relations (3.4)and (3.5). Given two representations � and � 0 satisfying the equivariance condition (3.1c)we can write any element w 2H0 as a unique sum

w DX

x2Zn

�x�.Ux/e0;

with .�x/x2Zn 2 `2.Zn/. In other words, e0 is a cyclic vector with respect to the algebraaction. Now construct an operator T WH0!H0 by setting Te0 WD e0 and extending as

Tw WDX

x2Zn

�x�0.Ux/e0:

Reality operator 21

This is well defined since � and � 0 are representations of the same algebra A.T n�/, they

satisfy the same algebra relation (3.5), and it is an invertible map on H0, because both�.Ux/e0 and � 0.Ux/e0 span the Hilbert space if we take all x 2 Zn. By constructionT �1� 0.Ux/T D �.Ux/, and it is a unitary transformation, because we can calculate:

hT v;T wi DX

x;y2Zn

N�x�yh�0.Ux/e0;�

0.Uy/e0i

D

Xx�yD0

N�x�yh�0.Ux/e0;�

0.Uy/e0i

D

Xx2Zn

N�x��x

D hv;wi;

since hex; eyi D 0 if x ¤ y, and for all Ux, the representations �.Ux/ and � 0.Ux/ areunitary.

There might be more unitary equivalences of the algebra, a question which we explorein Section 4.1, but for now we were only interested in possible representations of thealgebra.

3.3 Reality operator

In this section, we derive conditions on the equivariant reality operator J , to give us a realspectral triple. The Hilbert space H will be a direct sum of copies of the minimal Hilbertspace H0. We will only consider conditions following from relations between A, H and Jand the equivariance condition (3.1e). No use is made of the Dirac operator in this section.

The equivariance condition for J is given in (3.1e):

�.l/J v D�J�.l�/v; (3.7)

for v 2H , and l an element of the Hopf algebra of symmetries of the noncommutativen-torus. For our basic derivations ıj , we have ı�

j D ıj , so this just means

ıjJe� D��jJe�; (3.8)

for all j . Since we are working with the representation given in (3.3), we see that J mustmap an element e� of the basis to e��.

From equation (2.2), we also see that J must map A to an algebra which commuteswith A, but with opposite order of multiplication. The Tomita involution [70] J0.a/D a

�

maps A to a a commuting representation, but it does not satisfy the condition J 2 D �J Idfrom Table 2.1, so for this to be possible, we will have to enlarge our Hilbert space. Define

H WDMj 2I

Hj ; (3.9)


with each Hj as in (3.3) and I an index set. Every nondegenerate representation of aninvolutive Banach algebra is a direct sum of cyclic representations [71, Proposition I.9.17],and because of the equivariance condition (3.1c), we get that the only cyclic representationswe can consider are the ones given by Lemma 3.2.2. We write basis vectors of this Hilbertspace as e�0;j C�;j with � 2 Zn and j 2 I .

For different j 2 I , lattices spanned by � could a priori be shifted by a different amount,satisfying (3.8), but we will see in Section 3.4 that this cannot be the case if the spectraltriple is irreducible.

We look for an antilinear operator J such that J 2 D �J Id, with signs as in Table 2.1,and so that for every a 2 A.T n

�/, Ja�J�1 commutes with all b 2 A.T n

�/, satisfying

equation (2.2). The image of A.T n�/ under the isomorphism a 7! Ja�J�1 is denoted by

A.T n�/o. We write U o

x for the image of Ux.Firstly, equation (3.7) has as a consequence that J acts as follows on elements of the

basis e�;j :Je�;j D

Xakj .�/e��;k : (3.10)

We can thus write Je�;j D ƒ.��/J0e�;j , with ƒ.��/ some unitary or skew unitarybounded linear functional, and J0 an antilinear diagonal operator:

J0ae�;j D a�e��;j :

If we apply J twice, we should get �J Id, which can be written as

J �Je�;j D Jƒ.��/e��;j Dƒ.�/ƒ�.��/e��;j ;

and so we see thatƒ.�/ƒ.��/� D �J Id: (3.11)

By applying J on a unitary generator Ux of A.T n�/ we get the following condition

on ƒ.�/:

Lemma 3.3.1. The map a 7! Ja�J � is an isomorphism into the commutant, if and only iffor all x;y 2 Zn:

ƒ.xCy/D e.x �AyCy �Ax/ƒ.x/ƒ.0/�ƒ.y/; (3.12a)

where Je�;j Dƒ.��/J0e�;j , with J0 the Tomita involution, and

ƒ.x/ƒ.x/� D Id; (3.12b)ƒ.x/ƒ.�x/� D �J Id: (3.12c)

As a consequence

U ox e�;j D e.� �AxC

1

2x �Ax/ƒ0.x/ƒ.0/�e�Cx;j ; (3.13)

whereƒ.�/D e.� �A�/ƒ0.�/; and ƒ0.�/jk D cjke.�jk.�//;

with cjk 2C and �jk WRn!R are maps such thatX

k2I

cjke.�jk.��//c�kle.��jk.�//D �J ıjl :

Reality operator 23

Proof. First we calculate U ox D JU

�x J

� using equation (3.10):

U ox e�;j D JU

�x J

�e�;j

D JU �x ƒ.�/

te��;j

D Je

�x �A�C

1

2x �Ax

�ƒ.�/te��x;j

Dƒ.�Cx/e��x �A��

1

2x �Ax

�ƒ.�/�e�Cx;j :

We compute the commutator ŒUy;Uox �:

UyUox e�;j De

�y �A.�CxC

1

2y/�x �A.�C

1

2x/�

ƒ.�Cx/ƒ�.�/e�CxCy;j

U ox Uye�;j De

��x �A.�CyC

1

2x/Cy �A.�C

1

2y/�

ƒ.�CxCy/ƒ�.�Cy/e�CxCy;j :

If the commutator vanishes, we see that by canceling common terms we must have:

e .y �Ax/ƒ.�Cx/ƒ�.�/D e .�x �Ay/ƒ.�CxCy/ƒ�.�Cy/:

This has as a consequence that ƒ.x/jk consists of e.fjk.x// with fjk a function ofthe form

fjk.x/D x �BjkxC�jk.x/C�jk :

We see that the quadratic part must be the same for each component, and that Bjk D A.The constant part �jk can be absorbed into by unitary transformation. We can thus write

ƒ.x/D e.x �Ax/ƒ0.x/; (3.14)

whereƒ0.x/ consists of functions cjke.�jk.x// such thatƒ0.x/D �Jƒ0.�x/

t and for eachx, we have ƒ0.x/ƒ0.x/� D Id. If we then calculate U o

x :

U ox e� Dƒ.�Cx/e

��x �A��

1

2x �Ax

�ƒ.�/�e�Cx;j

D e

�.�Cx/ �A.�Cx/�x �A��

1

2x �Ax�� A�

�ƒ0.�Cx/ƒ0.�/

�e�Cx;j

D e

�� AxC

1

2x �Ax

�ƒ0.�Cx/ƒ0.�/

�e�Cx;j

D e

�� AxC

1

2x �Ax

�ƒ0.x/ƒ0.0/

�e�Cx;j :

By definition (3.14) of ƒ0, we have ƒ0.0/Dƒ.0/, so this is exactly equation (3.13).


3.4 Dirac operator

The central piece of the real spectral triple is the Dirac operator. In this section, using theresults from the previous section, the equivariance condition (3.1d), and Conditions 1, 2and 7, we derive conditions for the Dirac operator D. Using irreducibility, we show thatall the �0;j must be the same. Then, by using that J is an isometry and (3.7), we showthat �0;j must be vector consisting of elements which are either 0 or 1

2. Also, we show

that for different forms of ƒ as given in Lemma 3.3.1, only the form ƒ0.x/2 D Id can leadto isospectral deformations of spin structures on the commutative n-torus. By applyingthe compact resolvent condition of Condition 1, we show that this is the only possibilitycompatible with the definition of a real spectral triple.

An equivariant Dirac operator D commutes with the basic derivations ıj as describedin (3.1d), and can thus be written as:

De�;j D

Xk2I

dk�;j e�;k :

Since D should be self-adjoint, we have that dk�;j D d

j �

�;k. We will write

De�;j DD.�/e�;j : (3.15)

This means that D.�/ is the operator D restricted to the eigenspace of derivations witheigenvalue �. This is well defined due to the equivariance condition (3.1d).

Lemma 3.4.1. All �0;j must be equal to each other, and consist of elements either 0 or 12

.

Since all �0;j are the same, we define � WD �0;j .

Proof. Given an element �DmC�0;j with m 2 Zn, we define the Hilbert space H� asthe span of eigenvectors vk of the basic derivations ıj such that ıj vk D �j vk for each j .

Between two Hilbert spaces H� and H� we have an isometry if �� 2 Zn, given bythe unitary element U�� of the algebra. Now consider the projector P� that is Id on H�

for which �� 2 Zn, and 0 otherwise. This projection clearly commutes with the algebra,and, because of (3.15), also with the Dirac operator. If the spectral triple is irreducible, onlyscalars may commute with both the algebra and the Dirac operator, so P� is the identity onthe whole Hilbert space H , hence all lattices in the different subspaces are shifted by thesame vector �0;j .

Since ıjJvD�J ıj v for all vectors v 2H by (3.7), we can conclude by the above thatfor each �D�0;j Cm we have �� .��/ 2Zn. Since �� .��/D 2�D 2�0;j C2m andm 2 Zn, we have 2�0;j 2 Zn. Hence the shift �0;j must consist of a vector with elementseither 0 or 1

2.

From the first order condition in equation (2.4), we deduce:

Lemma 3.4.2. An equivariant Dirac operator D that satisfies the first order conditionmust satisfy

D.xCy/D�ƒ0.x/ƒ.0/�

�2

.D.y/�D.0//CD.x/; (3.16)

for all x;y 2 Zn.

Dirac operator 25

Proof. To check the first order condition, it is sufficient to check it only for the unitarygenerators of A.T n

�/:

ŒŒD;Ux�;Uoy �DDUxU

0y �UxDU

oy �U

oy DUxCU

oy UxD D 0;

for all x;y 2 Zn. Using Lemma 3.3.1, we write out the first order condition:

DUxU0y e�;j D a.x;y;�/D.xCyC�/ƒ0.y/ƒ0.0/

�e�CxCy;j

UxDU0y e�;j D a.x;y;�/D.yC�/ƒ0.y/ƒ0.0/

�e�CxCy;j

U 0y DUxe�;j D a.x;y;�/ƒ0.y/ƒ0.0/

�D.xC�/e�CxCy;j

U 0y UxDe�;j D a.x;y;�/ƒ0.y/ƒ0.0/

�D.�/e�CxCy;j ;

where a.x;y;�/ is the common factor

a.x;y;�/D e�

x �A�C� �AyCx �AyC1

2.x �AxCy �Ay/

�:

This gives the relation

.D.xCyC�/�D.yC�//ƒ0.y/ƒ0.0/�Dƒ0.y/ƒ0.0/

�.D.xC�/�D.�// :

Since D is self-adjoint and ƒ0.x/ unitary, we can rewrite this as

.D.xCyC�/�D.yC�//D�ƒ0.y/ƒ0.0/

��2

.D.xC�/�D.�// :

For yD x and �D 0, the solution to the defining equation (3.16) is

D.x/DX

j

��j �x

�Aj C

�ƒ0.x/ƒ.0/�

�2

BCC; (3.17)

where the Aj , B and C are bounded operators such that Ce�;k DP

l ckle�;l and similarly

for B and Aj , and ker�P

j �j �xAj

�contains at least the x 2 Zn such that

�ƒ0.x/ƒ.0/�

�2

¤ Id:

This is the unique solution, since we see from equation (3.16) that D is fully determinedafter we choose suitable D.ej / for 1� j � n.

If .ƒ0.x/ƒ.0/�/2D Id, this gives a linear Dirac operator, familiar from commutative

geometry. However, at first glance it seems that there might be other spin structures, notcorresponding to commutative spin geometries. These other candidate geometries, where.ƒ0.x/ƒ.0/�/

2¤ Id, will however drop out because they are incompatible with the compact

resolvent condition on D.


Lemma 3.4.3. A necessary condition for the equivariant Dirac operator to have a compactresolvent is that .ƒ0.x/ƒ.0/�/

2D Id. Hence D is of the form:

De�;k D

0@Xj

��j ��

�Aj CC

1Ae�;k : (3.18)

As a corollary, we have that J is of the form:

Je�;j D e .� �A�/ƒe��;j ; (3.19)

with ƒ a constant unitary on H such that ƒ2 D �J Id.

Proof. By [61, Theorem XIII.64], we see that an unbounded self-adjoint operator Dbounded away from 0 has a compact resolvent if and only if the set

Fb D f 2 Dom.D/ W jj jj � 1I jjD jj � bg;

is compact for all b 2R. However, if .ƒ0.x/ƒ.0/�/2¤ Id for some direction x, we know

that x 2 ker�P

j �j �xAj

�and then it follows from (3.17) that the norm ofDe�x is bounded

by BCC for all � 2 Z. When we take b > jjBCC jj, Fb contains at least e�x for all� 2 Z¤ 0, so Fb cannot be compact.

3.5 Grading and Hochschild homology

In this section, we investigate what extra conditions on the spectral triple of the noncommu-tative torus come from the grading operator and Hochschild conditions, Conditions 3 and 8.We find that the parameters �i introduced in Section 3.4 must be linearly independentvectors spanning Rn.

We start by investigating the Hochschild cycle condition, which states that there isa Hochschild cycle c 2Zn.A;A˝Ao/ whose representative on H is � when n is even,or Id if n is odd. The �-operator is an isometry, with eigenvalues 1 and �1, and so byCondition 3 and the diagonal action of the algebra on the Hilbert space, we see that

� D

�C 0

0 ��

!;

where �C and �� are unitary self-adjoint operators which have only eigenvalues C1 and�1 respectively. Using a unitary transformation, we can assume these operators to bediagonal, thus

� D

IdC 0

0 �Id�

!;

where Id and Id is the identity operators on the positive and negative eigenspace of � .

Grading and Hochschild homology 27

An obvious candidate for the Hochschild cycle inZn.A;A˝Ao/ is the straightforwardgeneralization of the unique such cycle for the noncommutative 2-torus [16, Page 167],which is

c2 D U�.1;0/U

�.0;1/˝U.0;1/˝U.1;0/�U

�.0;1/U

�.1;0/˝U.1;0/˝U.0;1/:

A candidate generator of the n-th Hochschild homology of the noncommutative n-torus isgiven by

cn D

X�2Sn

0@sign.�/

0@ nYj D1

Ue�.j /

1A�nO

j D1

�Ue�.j /

�1A ; (3.20)

with ej an orthonormal basis of Zn. It is known [37, Lemma 12.15] that this a Hochschildcycle. Due to [77, Theorem 1.1], the n-th Hochschild homology of the n-torus is 1-dimen-sional. Together with Lemmas 3.5.1 and 3.5.2 below, this means (3.20) generates the n-thHochschild homology.

Lemma 3.5.1. For the noncommutative n-torus, only nontrivial cycles can be mapped by�D to � when n is even, and to Id when n is odd.

Proof. Since the Hochschild cycle consists of polynomial expressions, it is enough to provethe result for individual homogeneous polynomials, since any cycle can be written as thesum of homogeneous polynomials. Define

c0D Ux0˝U o

y ˝Ux1˝�� ˝Uxn 2ZnC1.A;A˝Ao/;

with xi ;y 2 Zn. As in [74, Chapter 3.5], we see that

�D.bc0/D .�1/n��o�Uy�� .Ux0/ ŒD;� .Ux1/� : : : ŒD;� .Uxn/�;� .UxnC1/

�: (3.21)

Since ŒD;Ux�D CxUx with Cx some operator depending on x, �D.bc0/ is proportional to

Cx1;:::;xn� .Ux0/�o�Uy�� .Ux1/ : : :� .Uxn/� .UxnC1/ :

When n is even, � maps e�;j to ˙e�;j , the total sum yCPnC1

j D0 xj must be 0. If A.T n�/

and A.T n�/o have a trivial intersection, U o

y must have total degree 0. The totalPnC1

j D0 xj is0, and since Ux and U�x commute, the commutator (3.21) vanishes. When n is odd, theargument is similar.

If A.T n�/ and A.T n

�/o have nontrivial intersection, there are y such that UxUy D UyUx

for all x 2 Zn. In that case, by the same arguments as for the trivial intersection caseabove, the Uy must lie entirely within the intersection of A.T n

�/ and A.T n

�/o, and since

xnC1 D�y�Pn

j D0 xj , the commutator (3.21) vanishes:�U o

y Ux1� � �Uxn

;UxnC1

�D U o

y�Ux1� � �Uxn

;UxnC1

�D U o

y

1� e.xnC1 ��

nXj

xj /

!D U o

y .1� e.xnC1 ��.�y�xnC1///

D 0;

since xnC1 ��xnC1 D 0 and Uy commutes with UxnC1.


Lemma 3.5.2. A necessary condition for the the Dirac operator given in Lemma 3.4.3 tosatisfy the Hochschild condition is that the vectors �j are linearly independent. The Diracoperator is

D DX

j

��j �ı

�Aj CC; (3.22)

where the Aj are bounded operators such that

X�2Sn

sign.�/Yj

A�.j / D

(det.�1�2 : : :�n/� if n is even,det.�1�2 : : :�n/Id if n is odd,

(3.23)

where we view .�1�2 : : :�n/ as an n�n matrix with the vectors �j as the columns.

Proof. To deduce the representative of the Hochschild cycle (3.20) on the Hilbert space H ,we calculate

ŒD;Ux�e�;j DDe.x �A�C1

2x �Ax/e�Cx;j �Ux

Xk

.�k ��/Ake�;j

!

D e.x �A�C1

2x �Ax/

Xk

.�k �x/Ake�Cx;j

!:

(3.24)

We suppress the e.x �A�C 12

x �Ax/ factors, since these are canceled from the left by theU �

ejfactors, and expand (3.20):

�D.c/D �D

0@X�2Sn

0@sign.�/

0@ nYj D1

Ue�.j /

1A�nO

j D1

�Ue�.j /

�1A1AD

X�2Sn

0@sign.�/

0@ nYj D1

Ue�.j /

1A�nY

j D1

ŒD;Ue�.j /�

1AD

X�2Sn

0@sign.�/X

k

nYj D1

�k � e�.j /Ak

1A :This expression should be some constant times � or Id, depending on the dimension. Dueto [18, Proposition 4.2], we can write this as

�D.c/D det.�1�2 : : :�n/X

�2Sn

sign.�/Yj

A�.j /;

from which it is immediately clear that the vectors �j must be linearly independent.

Dimension, finiteness and regularity 29

3.6 Dimension, finiteness and regularity

Here we establish, using the conditions of dimension, regularity and finiteness (Condi-tions 5, 9 and 10), that the real spectral triple must be an isospectral deformation of a spinstructure on a noncommutative torus. The result follows from Connes’ spin manifoldtheorem [18, Theorem 11.5]. In the course of proving this, we find a proof for an elementaryfact about Hermitian matrices generating a Clifford algebra, for which we do not know anelementary proof.

We first need a technical lemma on approximation of lattice points.

Lemma 3.6.1. If the �j span Rn, for arbitrary aj 2R and all � > 0, there is a t 2R anda finite set of �j 2

QZn, withP

j �k ¤ 0, where QZn is a shifted lattice as in Section 3.3such that X

j

ˇˇtaj ��j �

Xk

�k

ˇˇ< �:

Proof. Every vector p 2 Zn can be written as a sum of at most 2 vectors �k 2QZn. By

Dirichlet’s theorem of simultaneous Diophantine approximation [68, Theorem II.1B], forall N > 1 and .a0

j / 2Rn we can find integers q;p1; : : :pn with q < N , such that

jqa0j �pj j<N

�1=n;

for 1� j � n, and aj 2R.Since the �j span Rn, there is a transformation R 2GL.R;n/ depending only on the

�j such that .Rp/j D �j �p. Set a0j D .R

�1a/j . Call the eigenvalue of R with the maximalabsolute value �max, then clearly we have

jqaj ��j �

Xk

�kj D jq.Ra0/j � .Rp/j j � j�maxjjqa0j �pj j< j�maxjN

�1=n:

Choosing N such that nj�maxjN�1=n < �, we get the result.

Lemma 3.6.2. A necessary condtion for D to satisfy the compact resolvent condition ofCondition 1, is that the operator X

j

xjAj

is invertible for all xj 2 R except when all xj vanish. In particular, all Aj should beinvertible operators.

The dimension condition of Condition 5 is automatically satisfied when this happens,as the arguments of [74, Chapter 4.4] are easily seen to carry over to this case.

Proof. IfP

j xjAj is not invertible at a point where xj D �j �� for all j , then clearly thisis also the case for �0 D �� with � 2 Z. Thus the kernel cannot be finite dimensional inthis case.

IfP

j xjAj is not invertible for some non-trivial xj , but xj ¤ �j �� for all � 2 Zn, wehave the following.


Since the �j span Rn by Lemma 3.5.2, by Lemma 3.6.1 we have for every � > 0 a setof vectors �j 2 Zn such that

Pj jP

k �j ��k �xj j< �. Take an element y in the kernel ofPj xjAj . Then

jj.Xj;k

�j ��kAj / �yjj< �jjyjj:

Hence there is at least one eigenvalue ofP

j;k �j ��kAj smaller or equal to �. But thismeans that the spectrum of jDj�1 is unbounded, hence D�1 cannot be compact.

We have so far assumed nothing about the size of the Hilbert space H compared tothe basic irreducible representation on H0 of the algebra, as defined in equation (3.3). Byconstruction, H is a left A.T n

�/-module. According to Condition 10, a certain submodule

H 1 of H should be a finitely generated projective left A.T n�/-module. This has the

following consequence:

Lemma 3.6.3. A necessary condition for the spectral triple .A;H ;D/ to satisfy the finite-ness condition of Condition 10 is that the Hilbert space H is a finite direct sum of copies ofH0. If we assume the algebra A to be closed under the collection of seminorms jjık.�/jj

defined in Condition 9, it is given by

A.T n� /D

(Xx2Zn

a.x/Ux j a.x/ 2 S.Zn/

); (3.25)

where S.Zn/ is the set of Schwartz functions over Zn:

S.Zn/D

�a W Zn

!C j supx2Zn

.1Cjjxjj2/kja.x/j2 <1 for all k 2N

�:

Proof. By the arguments in [14, Lemma III.6.˛.2], we know that the intersection of thedomain of the ık is stable under the holomorphic functional calculus. The collection of semi-norms jjık.�/jj is equivalent to the collection of seminorms qk D supx2 QZn .1Cjxj2/

kja.x/j

because of the following argument.We can write

jjık.a/jj2 DX

j

Xx2Zn

jjAj jj2kjj�j �xjj2k

ja.x/j2:

Because of Lemmas 3.5.2 and 3.6.2, we know:

c1;kjjxjj2k�

Xj

jjAj jj2kjj�j �xjj2k

� c2;kjjxjj2k ;

for constants c1;k ; c2;k > 0. Thus the family of seminorms jjık.a/jj is equivalent to thefamily of seminorms q0

k.a/D

Px2Zn jjxjj2k

ja.x/j2.Now if an element a of the C �-algebra A has a finite norm in all q0

k, then clearly

supx2Zn .1Cjjxjj2/kja.x/j2 <1 for all k 2N. Conversely, if a 2A.T n�/, we can write

q0k.a/D

Xx2Zn

jjxjj2kja.x/j2

D

Xx2Zn

.1Cjjxjj2/p.1Cjjxjj2/�pjjxjj2l

ja.x/j2

Dimension, finiteness and regularity 31

and this last sum converges if p is big enough, since .1Cjjxjj2/pja.x/j2 is by assumption

less than some finite constant cp andXx2Zn

cpjjxjj2k.1Cjjxjj/�p

converges when p > n=2Ck.By Condition 10, we have that H 1 is a finitely generated projective left A.T n

�/module.

We already knew that H must be a direct sum of copies of H0 due to the discussionfollowing (3.9). The finiteness condition then ensures that the sum must be finite. Allconditions stated in Conditions 9 and 10 are then easily seen to be fulfilled.

Remark 3.6.4. The choice of smooth structure as chosen in Lemma 3.6.3 is not unique evenin the commutative case, given just the algebra C �-algebra A of the noncommutative torus.See for example [40]. However, given the equivariance condition of the Dirac operator, andthe assumption that the smooth algebra consists of all elements such that jjık.�/jj<1 forall k 2N, the smooth algebra A.T n

�/ is uniquely determined by the above argument.

The number of generators of H in terms of H0 is still undetermined, but a lower boundis given by [2, Theorem 1]. This theorem states that for complex Hermitian k�k matrices,with k D .2aC1/2b , there exists at most 2bC1 matrices satisfying the non-invertibilityproperty of Lemma 3.6.2. So in order to have at least n such matrices, k should be at least2bn=2c. Hermitian matrices generating an irreducible representation of a Clifford algebraCln;0 are an example of a set of matrices attaining this lower bound.

Remark 3.6.5. It does not follow from Lemma 3.6.2 that the algebra generated by thematrices Aj is a Clifford algebra. What remains to be shown is that .

Pj xjAj /

2 lies in thecenter of the bounded operators on H for all x 2Rn. The condition of Lemma 3.6.2 is notenough to show this.

Consider a set of self-adjoint matrices fBj g generating a Clifford algebra. These satisfythe invertibility condition of Lemma 3.6.2, and the Hochschild condition (3.23). Sincethey are self-adjoint, we can diagonalize B1 as B1 D UMU � with M a diagonal matrixwith real elements. If we rescale the elements of M each by a different nonzero amountto M 0, the set fUM 0U �g[fBj gj �2 still has the invertibility property, but not necessarilythe Hochschild property, as a calculation for any case n � 4 will show. If nD 2;3, theinvertibility property does imply the Hochschild property however.

However, a weaker form of Connes’ reconstruction theorem [18, Theorem 11.5], impliesthe following result:

Lemma 3.6.6. A necessary condtion for the candidate structure .A.T n�/;H ;D;J / to

satisfy both the Hochschild condition of Condition 8 and the dimension condition ofCondition 5, is that the matrices Aj of Lemma 3.4.3 must generate a Clifford algebra.

Proof. If we look at our conditions on A.T n�/, H , D and J , we see that none of them

depend on the antisymmetric matrix � . Also, the action of the Dirac operator on the Hilbertspace is independent of � . This means that we can just set � D 0, where we have the realspectral triple .A0;H ;D;J / of smooth functions on the n-torus. Due to the results of


Connes’ spin manifold theorem (see for example [37, Lemma 11.6], [18, Remark 5.12]),we see that ŒD;h�2 lies in the center of EndA.H

1/ when hD h� 2A. This implies thatthe Aj generate a Clifford algebra.

Remark 3.6.7. Note that for this result we do not need the strong regularity, assumed forthe full reconstruction theorem.

Because the size of the maximal set of matrices which satisfy the invertibility conditionof Lemma 3.6.2 is odd, due to [2, Theorem 1], we have as a corollary:

Corollary 3.6.8 (Corollary B). A set of 2b �2b Hermitian matrices fAj gnj D1, where nD

2bC1, such that the equation

det

0@Xj

xjAj

1AD 0;has only the zero solution .xj D 0/

nj D1 in Rn, generate a Clifford algebra if and only if

X�2Sn

sign.�/nYj

A�.j / D �Idk ;

for some nonzero � 2R.

If we assume the spectral triple to be irreducible, we need that the matrices generate anirreducible representation of the Clifford algebra. This restricts the size of the matrices tobe exactly 2bn=2c.

From the results in Sections 3.2, 3.3 and 3.4 it now follows that the remaining conditions,the Poincaré duality of Condition 11 and the Hermitian pairing of Condition 10 are alsosatisfied, since they are satisfied by isospectral deformations.

This completes the proof of Theorem 3.6.9:

Theorem 3.6.9 (Theorem A). All irreducible real spectral triples with an equivariantn-torus actions are isospectral deformations of spin structures on an n-torus.

3.7 Description of the real spectral triples

Finally, we show that given a Dirac operatorD that satisfies all conditions so far, the realityoperator J is uniquely determined, and list all the ingredients that constitute all real spectraltriples of the noncommutative n-torus. Also, we exhibit in some low dimensional caseswhat freedom there still exists for the Dirac operator.

The Clifford algebra Cln;0 and group were defined in Section 1.1.

Lemma 3.7.1. IfD is given byP

j

��j �ı

�Aj with theAj 2

bn=2c�2bn=2c matrices generat-ing the Clifford algebra Cln;0, there is a unique J operator for each n, up to multiplicationwith a complex number of norm 1.

Description of the real spectral triples 33

Proof. Since the matrices Aj generate a Clifford algebra, we can write them as linearcombinations of matrices j such that j lC l j D 2ıjl Idk , with kD 2bn=2c, the canonicalgenerators of the Clifford algebra Cln;0. Also the matrices j can be written as linearcombinations of the Aj . We only need to check wat freedom there exists for the constantunitary ƒ in (3.19). This operator must have the correct commutation relations with Das given in Table 2.1. By the same arguments as in Lemma 3.6.2, we see that it needs tosatisfy the commutation relations with all linear combinations of the Aj , so also for the j .Since the j are “orthogonal” in the sense that j kC k j D 2ıjkIdn, it is sufficient tocheck it for these matrices.

If d D 1;2;3 or 4 this can be proven for example by calculation, see Remark 3.7.5 below.We proceed by induction. First we prove existence. Recall that there are isomorphisms [48,Theorem I.4.1]:

ClnC2;0 ' Cl0;n˝Cl2;0; Cl0;nC2 ' Cln;0˝Cl0;2:

Let d > 4, and assume it has been proven for n� 4. The operator Jn D Jn�4˝ J4,acting on Cln;0 ' Cln�4;0˝Cl4;0 has precisely the right commutation relations, except forn� 1 mod 4, as can easily be calculated by looking at Table 2.1, and taking into accountthe periodicity mod 4 of the table, except for the first row, where we use J 2

4 D�1. In casen� 1 mod 4, we can achieve the same by setting Jn D Jn�4˝�4J4.

Now we prove the uniqueness. Write in for the representation of the i -th basis vector of

Rn in Cln;0. An explicit isomorphism Cln;0 ' Cln�4;0˝Cl4;0 can be chosen, for example

in D Idn�4˝

i4 for i � 4, i

n D i�4n�4˝

14

24

34

44 for i > 4.

The operators 1n

2n and 3

n 4n commute with i

n for i > 4 and anticommute with 1n ;

2n

and 3n ;

4n respectively. They square to �1, so the operator

PCWD

1

4

�1C i 1

n 2n

��1C i 3

n 4n

�;

is a projection, which commutes with in for i > 4. The projection PC does not commute

with Jn, but does commute with 2 3Jn. Also, the projection PC projects onto a subspaceof dimension 1=4 times the dimension of the irreducible representation of Cln;0, and theoperator 2 3Jn has the same commutation relations with i

n for i > 4 as Jn, and since. 2 3Jn/

2D�J 2

n it has the right signs for an .n�4/-dimensional J operator. This meansthat PC projects onto a Hilbert space belonging to an .n�4/-dimensional spectral triple,where we have a unique J -operator by the induction hypothesis. On the complement of thePC-eigenspace, we have the unique J4-operator with the right commutation relations with j , j � 4.

Stated more elaborately, we have the following result:

Theorem 3.7.2. Any irreducible torus-equivariant real spectral triple on the smooth non-commutative n-torus A.T n

�/ belongs to the following list of 2n spectral triples:

� A Hilbert space H constructed as follows:

H D

2bn=2cMj D1

Hj Hj D

Mm2Zn

HmC�; (3.26a)


with �D .�1; : : : ; �n/ 2Rn, �j 2 f0; 12g and HmC� 'C.

� An involutive algebra A with unitary generators Ux with x 2 Zn

A.T n� / WD fAD

Xx

a.x/Ux W a 2 S.Zn/g; (3.26b)

with Ux acting on a basis vector e�;j 2Hj by

Uxe�;j D e

�1

2x �AxCx �A�

�e�Cx;j ; (3.26c)

for any matrix A such that A�At D � .

� An unbounded, densely defined, self-adjoint first order operator D

D D

nXj D1

.�j �ı/Aj CC; (3.26d)

acting on H with C a bounded self-adjoint operator commuting with the algebrasatisfying JCJ�1 D �DC , and �C D �C� if n is even, where �j are n linearlyindependent vectors in Rn, AJ are n matrices of size 2bn=2c�2bn=2c, generating anirreducible representation of the Clifford group Cln;0, and ı D .ı1; : : : ; ın/ are thederivations ıj e�;k D �j e�;k .

� If n is even, the grading operator � is given byP

�2Snsign.�/

Qj A�.j /, with Aj

the matrices given above.

� A unique (up to multiplication with a complex number of modulus 1) antilinearisometry J that acts as

Je�;j D e.� �A�/ƒe��;j : (3.26e)

with ƒ a bounded linear operator such that ƒƒ� D Id and DƒD��DƒD�.

In lower dimensional cases we can explicitly calculate what form C in (3.26d) can take.In [59, Lemma 2.3, Theorem 2.5] it is proven that C D 0 if nD 2. For nD 3 and nD 4 wehave the following:

Proposition 3.7.3. If nD 3 the constant matrix C must have the form qId where q 2 Rarbitrary. If nD 4, the matrix C must have the form

˙0 0 a b

0 0 �Nb Na

Na �b 0 0

Nb a 0 0

�

;

where a;b 2C.

Description of the real spectral triples 35

Proof. If nD 3 it follows from Theorem 3.7.2 that

J D e.� �A�/Xk2I

ajke��;k ;

with I D f1;2g. We choose a particular form of D, given by the particular representationof the Clifford group Cl3;0 known as the Pauli matrices. We see that

J D e.� �A�/

0 �1

1 0

!;

just as in the nD 2 case in [59, Theorem 2.5]. Using the appropriate values for �J and �Dfound in Table 2.1, we see that the defining equation is JCJ D C and by a calculation thisshows that C D qId where q 2R arbitrary. Similarly for nD 4, we can just check what theconditions are for C to satisfy the equations �D D�D� and JC D CJ , and this givesthe possibilities given in the proposition.

Remark 3.7.4. It is trivial to calculate similar conditions for higher dimensions. Due tothe increase in the size of matrices, and relaxation of the commutation relation with � whengoing from nD 2k to nD 2kC1, the number of parameters will increase when n grows.

Remark 3.7.5. If we choose a representation for Cl1;0, Cl2;0 and Cl0;2, all Clifford algebrasCln;0 can be constructed by the basic isomorphisms [48, Theorem I.4.1]:

Cln;0˝Cl0;2 ' Cl0;nC2;

Cl0;n˝Cl2;0 ' ClnC2;0:

If we choose a representation:

Cl1;0 D 1;

Cl2;0 D

0 i

�i 0

!;

1 0

0 �1

!;

Cl0;2 D

0 �1

1 0

!;

i 0

0 �i

!;


the (unique up to multiplication with a complex number of norm 1) matrix part of the Joperator can easily be calculated:

J2 D J3 D

0 1

�1 0

!(3.27a)

J4 D

˙0 1 0 0

�1 0 0 0

0 0 0 �1

0 0 1 0

�

(3.27b)

J5 D

˙0 0 �1 0

0 0 0 1

1 0 0 0

0 �1 0 0

�

: (3.27c)

In this representation, we also see that for all n, the matrix component of J has preciselyone nonzero element in every column or row.

C H A P T E R 4

Unitary equivalences of torus equivariantspectral triples

We know from the previous chapter that all real spectral triples on the noncommutativen-torus are isospectral deformations of spin structures on the commutative n-torus. Inthe commutative case, a diffeomorphism acting on the torus can transform one spin struc-ture into another, according to the action of the diffeomorphism group as calculated byDabrowski and Percacci for all Riemann surfaces [26]. Here we show that in the noncommu-tative case, when nD 2, these results carry over. When n > 2 the result is less conclusive,due to insufficient knowledge of the automorphism group of the C �-algebra, but we obtainsome results for inner automorphisms.

This chapter is based on the second part of the article “Classification of spin structures on the noncommutativen-torus” [75]

37

38 Unitary equivalences of tori

4.1 Unitary equivalences

The definition of a unitary equivalence between two (even) real spectral triples is as follows:

Definition (Unitary equivalence). A unitary equivalence between two real spectral triples.A;H;D;J;�/ and .A;H;D0;J 0;� 0/, with � the representation of A on H and � 0 therepresentation of A on H 0, is given by a unitary operator W acting on the Hilbert space Hsuch that

W�.a/W �1D �.�.a// 8a 2A; (4.1a)

WDW �1DD0; (4.1b)

WJW �1D J 0; (4.1c)

W �W �1D � 0; (4.1d)

where � is a �-automorphism of the (unique) completion of the pre-C �-algebra A, theC �-algebra A, such that the algebra A is mapped into itself.

4.2 Automorphisms of the noncommutative torus algebra

We first recall what is known about the automorphisms of the C �-algebra A.T n�/. We view

the space of n�n matrices as Rn2and equip it with the Lebesgue measure. The main

tool for understanding the automorphism group of a general noncommutative n-torus isa theorem of Elliott and Evans [30] for nD 2 and Boca [7, Theorem I] for n > 2, whichtells us that for � in a set which has full measure in the space of all antisymmetric matrices,the algebra A.T n

�/ is an inductive limit of direct sums of circle algebras. This results allows

one to generalize many results on noncommutative two-tori to higher dimensional tori.We first investigate some properties of automorphisms of the continuous noncommuta-

tive n-torus A.T n�/, of which the automorphisms of the smooth noncommutative n-torus

A.T n�/ form a subgroup. Denote by Inn.A.T n

�// the inner automorphisms of A.T n

�/, and

by Inn.A.T n�// its closure in the weak-� topology. We assume that � is such that A.T n

�/ is

an inductive limit of direct sums of circle algebras and simple.For a noncommutative n-torus which is an inductive limit of direct sums of circle

algebras, the automorphism group fits in the following exact sequence [32, Theorem 2.1]:

1! Inn.A.T n� //! Aut.A.T n

� //! Aut.K.A.T n� ///! 1;

where an automorphism of K.A.T n�// WDK0.A.T n

�//˚K1.A.T n

�// should preserve the

order unit Œ1A.Tn�

/� and the order structure�K0.A.T n

�//�C. Since we assume the algebra A

is simple, this is full order structure.The closure of the space of inner automorphisms Inn.A.T n

�// can be further specified

by [32, Corollary 4.6], which states that for algebras which are inductive limits of directsums of circle algebras the group Inn.A.T n

�// is pathwise connected, hence equal to the

closure of the group of inner automorphisms determined by unitaries in the connected

Automorphisms of the noncommutative torus algebra 39

component of the identity: Inn0.A.T n�//. This means that an inner automorphism of

a noncommutative torus can give a unitary equivalence of two spin structures only if theidentity automorphism gives a unitary equivalence between the two spin structures.

From the 6-term exact sequence of Pimsner and Voiculescu [60] we can deduce thatthe K-theory of the noncommutative n-torus is the same as that of the commutativen-torus, i.e. K0.A.T n

�//'K1.A.T n

�//'Z2n�1

. By [66, Theorem 6.1] the order structure.K0.A.T n

�///C consists precisely of those elements for which the normalized trace is

positive, and by [28, Theorem 3.1], the image of this trace on K0 is equal to the range ofthe exterior exponential of � :

exp^� D 1˚�˚

1

2.� ^�/˚ : : : W

even

Zn!R:

For noncommutative 2-tori with irrational � , this means that all automorphisms ofK.A.T 2�//

must be the identity on K0.A.T 2�//D ZC�Z. In this case Aut.K1.A.T 2

�///D GL.2;Z/

and a partial lifting of the automorphism group of the ordered K-theory is known [8], andgiven by the action of SL.2;Z/ on the lattice Z2 of unitary generators Ux, i.e. g 2 SL.2;Z/acts as Ux 7! Ug.x/.

Remark 4.2.1. For the smooth noncommutative 2-torus A.T 2�/, we know from [29] that

in fact the whole automorphism group of the algebra A.T 2�/ for irrational � with certain

extra Diophantine conditions is given by a semidirect product of this action, the canonicaltorus action: Uej

7! �jUej, j D 1;2, and PU.A.T 2

�//

0 the connected component of theidentity of the projectivized group of unitaries of A.T 2

�/:

Aut.A.T 2� //D PU.A.T

2� //

0 Ì .T 2 Ì SL.2;Z//:

Thus, for the smooth noncommutative 2-torus, all outer automorphisms are given by theaction of SL.2;Z/ on the lattice Z2 above.

For n > 2 the situation is less clear, since the action on the K0 group need not be trivialanymore.

Lemma 4.2.2. For the noncommutative n-torus A.T n�/ with n > 2, if � is a matrix such

that all �ij , j > i are independent over Z, an outer automorphism cannot be of the formUx 7! U�.x/ with � 2GL.n;Z/, except � D˙Idn.

Since automorphisms of A.T n�/, the smooth torus algebra, are defined as automor-

phisms of A.T n�/ such that the smooth algebra is mapped to itself, this result also holds for

A.T n�/.

Proof. An automorphism � of this form must satisfy e.�.x/ � ��.y// D e.x � �y/ for allx;y 2 Zn. If the �ij , j > i are independent over Z, we see that �.x/D ˛11xC˛12y and

�.y/D ˛21xC˛22y with

˛11 ˛12

˛21 ˛22

!2 SL.2;Z/. When n > 2 the only solution for all x

and y is �.x/D˙x with sign the same for all x.

The automorphism defined by �.x/D�x is called the flip automorphism.


4.3 Unitary equivalences of noncommutative tori

With this knowledge of the automorphism group, we can proceed to our main theorem ofthis chapter.

Theorem 4.3.1 (Theorem C). Except for a set of � of measure 0, the different spin structuresof the smooth noncommutative n-torus A.T n

�/ cannot be unitarily equivalent by an inner

automorphism of the algebra.

In case nD 2, the theorem was proven in [59, Theorem 2.5].While for the proof of Theorem 4.3.1 it is only necessary to consider inner automor-

phisms, we get a stronger result for n D 2, Corollary 4.3.2 below, if we also considerautomorphisms induced by an action of SL.2;Z/ on the algebra.

Proof. We actually prove the theorem for A.T n�/, the full C �-algebra. Since any automor-

phism (inner or outer) of A.T n�/ has to be an automorphism of A.T n

�/, this is enough to

prove the theorem for A.T n�/.

We will assume in the following that the components of � in the upper right cornerare independent over Z. The set of � all of whose components in the upper right cornerare independent over Z is of full measure, and so is the set of � for which Inn.A.T n

�//D

Inn0.A.T n�// by the discussion in Section 4.2, so their intersection also has full measure.

We label the basis of the Hilbert space as described in Theorem 3.7.2 for the differentspin structures � by the same labels m 2 Zn, so the Hilbert space H is spanned by vectorse�;j with �DmC�, and 1� j � 2bn=2c. We see that for a spin structure � the operatorJ , written as in equation (3.19), acts as

J�em;i Dƒij e ..mC�/ �A.mC�//e�mC2�;j :

We consider a unitary transformation W , induced by an automorphism � 2 SL.2;Z/ ifnD 2, or � D˙Idn if n > 2, and denote

We0;i D

Xk

Xj

wk;ij ek;j ;

for the action on the e0;i . Also we write �.x/ for the obvious action of the automorphism� (either an element of SL.2;Z/ when nD 2, or the action x 7! ˙x when n > 2) on thevector x 2 Zn.

We first show that the action of the unitary transformation W on the Hilbert space isfully determined by the action on the different e0;i . Next we show that the condition (4.1c)implies that the action of the unitary transformation on the basis vectors must be such thata basis vector e0;i is mapped to a linear combination of vectors ek;l with k D Q��.�/,where � is the original spin structure and Q� the new spin structure, as defined just belowLemma 3.4.1. If �.�/D˙�, this means that the spin structure is unchanged, since k 2 Zn.For the noncommutative 2-torus, we have � 2 SL.2;Z/, and we see that if �D 0, the spinstructure is unchanged. The spin structures � on the noncommutative 2-torus with �¤ 0,are unitarily equivalent to each other, see Corollary 4.3.2.

Unitary equivalences of noncommutative tori 41

Consider a unitary transformation W that maps a spin structure � to Q�. Since ex;i D

Uxe��

12

x �Ax�x �A��e0;i and using (4.1a) we can write

Wem;i DWUme.�1

2m �Am�m �A�/e0;i

D U�.m/e.�1

2m �Am�m �A�/

0@Xk;j

wk;ij ek;j

1AD

Xk;j

e

��.m/ �A.kC Q�/C

1

2�.m/ �A�.m/

�

� e

��1

2m �Am�m �A�

�wk;ij ekC�.m/;j :

This shows that the action of W on the Hilbert space is fully determined by the action onthe e0;i . Requirement (4.1c) gives the following equations:

WJ�em;i Dƒij e ..mC�/ �A.mC�//We�mC2�;j

D

Xk;l

ƒij e

��.�mC2�/ �A.kC Q�C

1

2�.�mC2�//

�

� e

�.mC�/ �A.mC�/� .2��m/ �A.�C

1

2.2��m//

�wk;jlekC�.�mC2�/;l

D

Xk;l

ƒij e .� �A.�3�C2m/C�.�/ �A.2. Q�Ck/C�.2��m///

� e

�m �A.3�C

1

2m/C�.m/ �A.� Q��kC�.

1

2m��//

�wk;jlekC�.�mC2�/;l

J Q�Wem;i D

Xk;j

e

��.m/ �A.kC Q�C

1

2�.m//�m �A.�C

1

2m/�wk;ij ekC�.m/;j

D

Xk;j

ƒjle ..kC�.m/C Q�/ �A.kC�.m/C Q�//

� e

��.m/ �A.kC Q�C

1

2�.m//Cm �A.�C

1

2m/�w�

k;ij e�k��.m/C2 Q�;j

D

Xk;j

ƒjle

�. Q�Ck/ �A. Q�CkC�.m//Cm �A.�C

1

2m/�

� e

�1

2�.m/ �A�.m/

�w�

k;ij e�k��.m/C2 Q�;j :

Collecting the vectors with the same indices and in the same Hilbert space Hj we see thatfor indices kC�.�mC2�/D�k0��.m/C2 Q�, or k0 D�kC2 Q��2�.�/:

e

��.�mC2�/ �A.kC Q�C

1

2�.�mC2�//� .2��m/ �A.�C

1

2.2��m//

�wk;jl

Dƒjle

�. Q�Ck/ �A. Q�CkC�.m//Cm �A.�C

1

2m/C

1

2�.m/ �A�.m/

�w�

k;ij :


Since ƒ�1 D ƒ� andP

j ƒljƒ�j i D �J ıil by (3.26e) with ıil the Kronecker delta, this

implies that

wk;ij D�J e�2� �A.3��2m/C3 Q� �A.3 Q��kC�.�2�Cm//

�� e�k �A.�3 Q�CkC�.2��m//C�.�/ �A.�4 Q�C4kC3�.2��m//

�� e��.m/ �A. Q�CkC�.�//�2m �A�

�w�

�kC2 Q��2�.�/;ij : (4.2)

Applying this same formula again for w��kC2 Q��2�.�/;ij

, we get

w��kC2 Q��2�.�/;ij D e

�� A.3��2m/� . Q�Ck/ �A. Q�CkC�.m//�m �A.4�Cm/

�� e��.�/ �A.6 Q��2k��.2��m//C�.m/ �A.3 Q��k��.�Cm//

�wk;jk :

(4.3)

Filling in the expression of (4.3) in (4.2), we see

wk;jk D e�2 Q� �A.4 Q��2kC�.m�3�//C2k �A.�.��m/�2 Q�/

�� e�2�.�/ �A.3k�5 Q�C�.4��m//C2�.m/ �A.k� Q�C�.�//

�wk;jk :

Collecting all terms which contain m, we see that these add up to

2�.m/ ��.kC�.�/� Q�/:

Since the equality above should hold for all m, this means that either wk;ij D 0, or

m ��.kC�.�/� Q�/D 0;

for all m.If all components of � in the upper right corner are independent over Z this can only

be the case if kC�.�/� Q�D 0, hence kD Q��.�/. Since k must lie in Zn, we see that if�.�/D˙�, the spin structure cannot change.

From the proof also follows:

Corollary 4.3.2. Let � 2 SL.2;Z/. The automorphism � of Z2 induces an automorphismof the noncommutative 2-torus of the form Ux 7! U�.x/.

This automorphism induces a unitary equivalence of real spectral triples which mapsa spin structure � to Q� D �.�/. In particular, real spectral triples on the noncommuta-tive 2-torus which are not isospectral deformations of the trivial spin structure on thecommutative 2-torus are unitary equivalent to each other, via the following unitary mapW :

We� D e�.�/;

W UxW�1D U�.x/;

D0D

0@Xj

��1.�j / �ı

1A˝Aj ;

J 0D J;

composed with an additional unitary map given by Lemma 3.2.2 that maps the representa-tion �A0

with A0 D � t A� to the original �A.

Unitary equivalences of noncommutative tori 43

Proof. The statement about the spin structures follows from the statements at the end of theproof of Theorem 4.3.1, and the classification of automorphisms ofA.T 2

�/ of Remark 4.2.1.

Since � 2 SL.2;Z/ is an automorphism of the lattice Z2, it cannot map an � … Z2 to one

in Z2 and vice-versa. On the other hand, set �10 D

12

0

!;�01 D

012

!;�11 D

1212

!, and

M D

1 0

�1 1

!;N D

1 �1

0 1

!in SL.2;Z/. We see

�11

�10 �01

M

NM �1

NM �1

MN �1

N �1

where the matrices M and N act by left multiplication, so there exist � 2 SL.2;Z/such that the � … Z2 are mapped to one another. Given a unitary operator We� D e�.�/,we can calculate its action on the Dirac operator by using the definition (4.1b):

D0e�;k DWDW�1e�;k

DWDe��1.�/;k

DWX

j

��j � .�

�1.�//�˝Aj e��1.�/;k

D

Xj

��1.�j / ��

�˝Aj e�;k :

Remark 4.3.3. The action of the automorphisms � 2 SL.2;Z/ on the Dirac operator when�D 0 was also determined in [74, Section 7.1], with the change in notation that our �1 is

given there by

0

Im�

!and our �2 is

1

Re�

!in [74].

C H A P T E R 5

Morita equivalences of torus equivariantspectral triples

Two algebras are Morita equivalent if their categories of representations are equivalent. Ifboth the algebras are commutative, this implies that they are isomorphic, but for noncommu-tative algebras, the notion is more general. This leads to another notion of “isomorphism”or “isometry” of spectral triples, as follows: we require the algebras to be Morita equivalent,and modify the Hilbert space and Dirac operator in such a way that all geometric data ispreserved, as proposed by Connes [16]. If the algebra is noncommutative, this gives rise tointeresting new possibilities which do not occur if the algebra is commutative.

For example, given a spectral triple where the algebra is the algebra of smooth func-tions on a four-manifold times a certain finite dimensional matrix algebra, the Moritaself-equivalences of the algebra change the Dirac operator. These changes take the formof connections, which for suitably chosen matrix algebras constitute the gauge group ofthe Standard Model of particle physics, as shown by Chamseddine and Connes [9]. Thenotion of Morita equivalence of spectral triples is a little different depending on whether ornot the spectral triple one considers is a real spectral triple, or spectral triple without a realstructure, as illustrated by the examples in Section 5.1 below.

However, there is one catch: a Morita equivalence of spectral triples is not a trueequivalence relation, as it is not symmetric in general, see for example [22, Remark 1.143].In this chapter, using the results of Rieffel and Schwarz [67] and Elliott and Li [31, Theorem1.1] on Morita equivalences of algebras of noncommutative tori, we show that if we restrictto equivariant spectral triples for the noncommutative n-torus (as classified in Chapter 3),Morita equivalences are symmetric. This appears to be the first non-trivial example wheresymmetry can be proven. The Morita equivalences also lead to new “isometries” betweenreal spectral triples which are not present in the commutative case, even though all suchspectral triples were shown to be isospectral deformations of commutative tori.

This chapter is based on the preprint “Morita “equivalences” of equivariant torus spectral triples” [76]

45

46 Morita equivalences of tori

5.1 Morita equivalences of spectral triples

Definitions: general case

The right context for strong Morita equivalences of C �-algebras are Hilbert C �-modules.For a modern introduction, see Meyer [55].

Definition. Let A be a C �-algebra. A Hilbert A-module is a right A-module E witha compatible C-vector space structure, equipped with a conjugate-bilinear “inner product”h�; �iA with values in A, such that for all e;f 2 E and a 2 A:

� he;eiA � 0,

� he;eiA D 0 if and only if e D 0,

� he;f iA D hf;ei�A,

� he;f �aiA D he;f iAa.

The map jjejj D jjhe;eiAjj12 then defines a norm on E , and the module has to be complete

in this norm.

We can also define left A-modules in a similar way. The same definition also makessense if A is a pre-C �-algebra, but then we do not require the module E to be complete,but its completion should be compatible with the completion of A. The modules we studyin this chapter are all finitely generated projective modules over their algebras.

Two pre-C � algebras A and A0 are strongly Morita equivalent if there exists a HilbertC �-bimodule A0EA for which we have A0 D EndA.E/ [63]. If A and A0 are stronglyMorita equivalent, we say that the spectral triple .A;H ;D/ is Morita equivalent to thespectral triple .A0;H 0;D0/, if H 0 D E ˝A H and the Dirac operators D and D0 arerelated by a connection. The connection is a way to transport the action of D from H toH 0 D E˝A H . This is needed, since the action ofD does not in general commute with A.

Definition (Connection). Define the space of one-forms as:

�1D D spanfaŒD;b� j a;b 2Ag: (5.1)

A connection rD on E is a linear map E! E˝A�1D that satisfies the Leibniz rule:

rD.sa/Dr.s/aC s˝ ŒD;a�; (5.2)

for all s 2 E and a 2A, and is Hermitian:

hr jrDsiA�hrDr jsiA D ŒD;hr jsiA�; (5.3)

for all r;s 2 E , where hr;siA is the inner product of the Hilbert A-module E , takingvalues in A.

Morita equivalences of spectral triples 47

If we then define the Dirac operator D0 on H 0 as:

D0.s˝ �/DrD.s/�C s˝D�; (5.4)

it has all the right properties of a Dirac operator on H 0.From (5.4), the Dirac operator D0 can be determined, up to a component commuting

with the algebra A0, as follows:

ŒD0;b�.s˝ /D ŒrD;b�s˝ Cbs˝D �bs˝D D ŒrD;b�s˝ : (5.5)

We will denote a Morita equivalence of .A;H ;D/ to .A0;H 0;D0/ given by an equivalencebimodule E and connection rD by .E;rD/.

Definition: real spectral triple

If we are considering a spectral triple with a real structure J , the Hilbert space H isan A-bimodule, with the opposite algebra Ao D JAJ�1 acting from the right. If E

is projective and finitely generated over A (so E D pAm with p a projection), we candefine the opposite equivalence bimodule NE D .Ao/mp. The Hilbert space H 0 can then beconverted to a A0-bimodule by setting H 0 D E˝A H ˝Ao NE and

D0.s˝ ˝ t /DrD.s/ ˝ tC s˝D ˝ tC s˝ .rDt /: (5.6)

The real structure J 0 is then given in the obvious way

J 0.s˝ �˝ Nt /D t˝J �˝ Ns:

We see that (5.5) still can be used to calculate the new Dirac operator in this case.

Examples

� If A is a noncommutative algebra, non-trivial examples of Morita equivalences ofspectral triples are induced by Morita equivalences of A with itself. Take as equiva-lence bimodule the algebra A itself. The algebra and Hilbert space are unchanged,but in the presence of a real structure, the Dirac operator D changes as [16]:

D0DDCAC �JJAJ �; (5.7)

where A is given by a finite sumP

j aj ŒD;bj �, aj ;bj 2A. The sign �D in the equationis taken from Table 2.1 and isC1 if the dimension n of the spectral triple is n¥ 1 mod4 and �1 is n� 1 mod 4, where the dimension n is a number describing the growthof the spectrum of the Dirac operator, as in Condition 5. For the noncommutativen-torus, the dimension is n.

The element A of�1D is called a gauge potential, and is an important part of the non-

commutative geometry approach of the Standard Model of particle physics [9],[10].

� To illustrate the difference between a Morita equivalence of real spectral triples asopposed to that of spectral triples, we compute a Morita equivalence of a commutative

spectral triple .C1.S1/;L2.S1/; i@

@t/, with the function algebra C1.S1/ generated

by unitary generators e2�ikt , k 2 Z.


We take the equivalence bimodule to be the algebra itself, C1.S1/. The Hilbertspace H thus remains unchanged. We have �1

D ' C1.S1/, and so the connection

rD is a linear map C1.S1/! C1.S1/. The Leibniz rule (5.2) in this case reducestorD.ab/DrD.a/bCaŒD;b�, hencerD.a/D ŒD;a�Cca with c 2C1.S1/. TheHermitian condition (5.3) then ensures c D c�. The new Dirac operator is then thesame up to an additive term, henceD0DDCc with c D c� 2C1.S1/. By contrast,in the real case we have as reality operator J induced by complex conjugation onC1.S1/, and since JJ � D 1 and JD D�DJ in dimension 1, we see that

JcJ �D JaŒD;b�J �

D a�ŒJDJ �;J bJ ��D�a�ŒD;b��D�c�;

hence D0 DDC c� c� DD, so that D remains unchanged. This is in fact true forall Morita equivalences of commutative real spectral triples when the Hilbert space isgiven (as usual) by L2-sections of spinors, see the discussion in [51, Section 3.4]. Ifthe Hilbert space is taken to be more general, commutative examples where D0 ¤D

can be constructed, see [72].

Equivalence relation

Morita equivalence is an equivalence relation for C �-algebras, but what about Moritaequivalences of spectral triples? For a spectral triple .A;H ;D/, by taking as bimodule thealgebra A and as connection a 7! ŒD;a�, we see that

D0.a˝ �/D ŒD;a��CaD� DDa�;

and so .A;H ;D/ is Morita equivalent with itself. Also not so hard to see is transitivity:

Lemma 5.1.1. Morita equivalence of spectral triples is a transitive relation.

Proof. Let A, B and C be pre-C � algebras, such that A is strongly Morita equivalent to B

via the bimodule BEA and B strongly Morita equivalent to C via the bimodule C FB . Onecan form the bimodule C GA D F ˝B E , and it is well known this is a Morita equivalencebimodule between A and C .

Now given a Morita equivalence of spectral triples from .A;H ;D/ to .B;H 0;D0/, anda Morita equivalence of spectral triples from .B;H 0;D0/ to .C ;H 00;D00/, we constructtheir composition, a Morita equivalence from .A;H ;D/ to .C ;H 00;D00/. Clearly H 00 D

F ˝H 0 D F ˝E˝H D G ˝H .Denote the respective connections of the Morita equivalences by rD and rD0 . We can

then define an �1D-connection rD00 on G by

rD00 WD rD0˝ IdEC IdF ˝rD : (5.8)

To show that it is connection, we remark that it satisfies the Leibniz rule for the A-action,since the action of rD0 on F trivially commutes with the action of the algebra A on E . Asa linear combination of two Hermitian connections, it is a Hermitian connection.

So we are left to show that rD00 is a map G ! G ˝�1D . For the IdF ˝rD part this

is clear.

Morita equivalences of noncommutative tori 49

TherD0˝IdE -part is a priori a map F ˝E!F ˝�1D0˝E , with�1

D0 the A0 bimodulespanned by faŒD0;b�ja;b 2 A0g. But since ŒD0;b� D ŒrD;b� acting from the left on E ,and rD is a map from E to E˝�1

D , we can view elements of F ˝�1D0˝E as lying in

F ˝E˝�1D , hence in G ˝�1

D .This connection is such that ŒrD00 ; c�D ŒD00; c� for all c 2 C , since ŒD00; c�D ŒrD0 ; c�,

and the second part of (5.8) commutes with the C -action. Also, ŒrD00 ;a�D ŒD;a� for alla 2A, since the first part of (5.8) commutes with the A-action, hence it is a well-defined�1

D-connection if rD is.

Remark 5.1.2. This construction is also compatible with a real structure, since we canjust write out (5.6) for both connections, and check that they are the same. Also, the realstructure J 00 can easily be described in terms of the real structure J , since

J 00.u˝ s˝ �˝ Nt˝ Nv/D v˝J 0.s˝ �˝ Nt /˝ NuD v˝˝tJ �˝ Ns˝ Nu;

with s; t 2 E , u;v 2 F and � 2H .

An open question is when Morita equivalences of spectral triples are proper equiva-lences, that is, not only reflexive and transitive, but also symmetric. It is well known thisdoes not hold in all cases. For example, if .A;Ck ;D;J / is a real spectral triple with a finitedimensional algebra A, by the results of [46] the Dirac operator can be written as

D D�CJ�J�1;

with � a self-adjoint operator in �1D . Together with (5.7), this implies that there is

a Morita self-equivalence with the spectral triple .A;H ;0;J /. Since �10 D f0g, there is

no connection such that there is Morita equivalence of spectral triples from .A;Ck ;0;J /

to .A;Ck ;D;J /, hence the relation cannot be symmetric. With the same arguments, ifa spectral triple can be written as a direct sum of other spectral triples, and one of thesespectral triples is finite, then the relation cannot be symmetric. In the following section,we will present the first case where the Morita “equivalence” is provably an equivalencerelation.

Remark 5.1.3. In [80], Zhang introduced a notion of Morita equivalence for spectral triples,that is symmetric [80, Theorem 5.6], by allowing for a more general type of connections,taking values in E˝B.H /, not just E˝�1

D , still satisfying the Leibniz rule (5.2) andHermitian condition (5.3). This implies in particular that for a Morita self-equivalence ofa spectral triple with a finite dimensional algebra A, a connection can take value in the setof all bounded self-adjoint operators on H . The geometric meaning of this is unclear.

5.2 Morita equivalences of noncommutative tori

From now on, we will denote the algebra A.T n�/ as A� . From [49, Theorem 1.1] we know

that two algebra A�1and A�2

of smooth noncommutative n-tori, with n > 2, are stronglyMorita equivalent if and only if their matrices �1 and �2 lie in the same SO.n;njZ/-orbit.This is the group of 2n�2n matrices which leave the quadratic form

Pi xixnCi invariant,


have determinant 1 and have integer entries. We can write elements of SO.n;njZ/ in thefollowing block form:

A B

C D

!; (5.9)

with A;B;C;D n�n matrices with integer entries.We can identify several subgroups of this group. For the group GL.n;Z/, if we take

any R 2GL.n;Z/ the matrix given by

�.R/D

R 0

0�Rt��1

!; (5.10)

is in SO.n;njZ/. Also, for the additive group Skewn.Z/ of skew-symmetric n�nmatriceswith integer entries, for any such matrix N the matrix given by

�.N /D

Idn N

0 Idn

!(5.11)

is in SO.n;njZ/. Finally we have an element �2, that acts on .xi /2niD1 by interchanging x1

with xnC1, x2 with xnC2 and leaving the rest invariant. The following is known:

Lemma 5.2.1 ([67]). The group SO.n;njZ/ is generated by the elements �.R/;R 2GL.n;Z/ and �.N /;N 2 Skewn.Z/ defined above, and the single element �2.

The action of SO.n;njZ/ on an n�n skew-symmetric matrix � is given by

g� WD .A�CB/.C�CD/�1; (5.12)

where A;B;C and D are the matrix components defined in (5.9). This action is onlydefined for the subset of the skew-symmetric matrices � for which C�CD is invertible

for all possible C;D such that

A B

C D

!2 SO.n;njZ/. We will restrict our attention to

this set of � and denote it by ‚0n. This is a dense set of the “second category” in the Baire

sense (i.e. not a countable union of nowhere dense subsets) in the set of all skew-symmetricn�n matrices, as shown in [67]. In [49] Morita equivalence is extended to skew-symmetricmatrices � such that g� is not defined for all g.

We will describe the Morita equivalences of C �-algebras induced by the generators ofthe group, following [67]. Since Morita equivalence of C �-algebras is well known to bea transitive relation, this is enough to calculate the equivalences for the entire group.

Given a skew-symmetric n�n matrix N with integer entries, the action of �.N / on thematrix � can be deduced from (5.11) as � 0 D �CN . We see immediately that the algebrasA� and A� 0 are actually isomorphic, since e.x ��y/D e.x � .�CN/y/.

For every element R 2 GL.n;Z/ we have an element �.R/ of SO.n;njZ/ given by(5.10) and we see that �.R/.�/ D R�Rt . The commutation relations become UxUy D

e.x �R�Rt y/UyUx so the algebra A�.R/� is isomorphic to A� .The last generator �2 of SO.n;njZ/ is not an isomorphism, so the equivalence bimodule

is a bit more involved. We will calculate here explicitly the effect of �2 on the action of thealgebra A� , following the description given in [67] and [66].

Morita equivalences of noncommutative tori 51

Set q D n�2. The Morita equivalence goes via the bimodule S.R�Zq/, the space ofSchwartz functions over R�Zq . Denote by

S.R/D

(f 2 C1

ˇˇ sup

x2Rjxn d

k

dxkf .x/j<18k;n 2N

);

the Schwartz space over R, and define S.Z/ as the restriction of the above to functionsover the integers. We have S.R�Zq/' S.R/�S.Z/q . Write

� D

�11 �12

�21 �22

!;

with �11 the top 2� 2 part of � . If �11 ¤ 0, this is an invertible matrix, since �11 isskew-symmetric. Then

�2.�/D

��1

11 ��111 �12

�21��111 �22��21�

�111 �12

!; (5.13)

Define

Jo D

0 1

�1 0

!; J2 D

�Jo 0 0

0 0 Idq

0 �Idq 0

�

:

Write xq for the vector in Rq consisting of the last q components of x and let T11 be any2�2 matrix such that

T11tJoT11 D��11

and T32 any q�q matrix such that

T32t�T32 D �22:

If �11 is not zero, Jo is similar to ��11, and so there always exists such a T11. Also, �22 isskew-symmetric, so there exists a suitable T32.

We now follow [67] in defining a right action of A� and left action of A� 0 with� 0 D �2.�/ on S.R�Zq/ such that their actions commute.

Lemma 5.2.2. If �11 is invertible (i.e. not zero), a right action of the generators Ux withx 2 Zn of A� on S.R�Zq/ is given by

U rx f .t;p/D e

�.T11 �x/1 � tC

�� t

12 T32

��x��p�f .tC .T11 �x/2;pCxq/; (5.14)

with .T11 �x/i the i -th component of T11 � x and xq the vector composed of the last qcomponents of x.

Proof. Define the homomorphism T of Zn into R2�Zq �Rq by the matrix action

T D

�T11 0

0 Idq

� t12 T32

�

:


We see that T maps Zn ' Z2�Zq onto a lattice in R2�Zq �Rq . Given an element.s; t;u;v/ of R�R�Zq �Rq , one has the following action on S.R�Zq/:

�.s; t;u;v/f .x;p/D hs;xihv;pif .xC t;pCu/;

with hs;xi D e2�isx and hv;pi D e2�i.v�p/. Thus hs;�i defines a character on R andhv;�;i a character on Zq .

The value of v only plays a role modulo Z, hence we can reduce T to map with imagein R�R�Zq �T q . But this means we can write T as a map from Zn into M � OM , withM D R�Zq and OM is the Pontryagin dual of M . Because T tJ2T D �� , we see thatthe generators Ux with x 2 Zn, and hence the whole algebra, give a right action of A� onS.R�Zq/.

The way to find the endomorphism algebra, hence the strongly Morita equivalent algebraA� 0 , is to find an embedding S of Zn into R�R�Zq �T q such that S.Zn/ and T .Zn/

are dual lattices, i.e. S.x/ �JT .y/ 2 Z for all x;y 2 Zn.

Lemma 5.2.3. The algebra A� 0 with � 0 D �2.�/ acts on S.R�Zq/ by

U lx f .t;p/D e

��Jo

�T11

t��1

�I2 ��12

��x�

2� tC

��0 T32

��x��p�

f .tCJo

��T11

t��1

�I2 ��12

��x�

1;pCxq/; (5.15)

and this action commutes with the action of A� on S.R�Zq/ as defined in Lemma 5.2.2.

Proof. Consider the map S W Zn!R2�Zq �Rq given by the matrix

S D

�Jo

�T11

t��1

�Jo

�T11

t��1

�12

0 Idq

0 �T32t

�

:

Since S t ıJ2T 2 Z, this gives a lattice dual to T .Zn/ in R2 �Zq �Rq , and thus givesa commuting left action.

We calculate S tJ2S :

S tJ2S D

T11

�1J to 0 0

�� t12T11

�1J to Idq �T32

!�J2 �

�Jo

�T11

t��1

�Jo

�T11

t��1

�12

0 Idq

0 �T32t

�

D

T11

�1 0 0

�� t12T11

�1 T32 Idq

!�

�Jo

�T11

t��1

�Jo

�T11

t��1

�12

0 Idq

0 �T32t

�

D

T11

�1Jo.T11t /

�1�T11

�1Jo.T11t /

�1�12

�� t12T11

�1Jo.T11t /

�1� t

12T11�1Jo.T11

t /�1�12CT32�T32

t

!;

where we used in the first step that J toJo D Id2. Since T11

�1Jo.T11t /

�1D ��1

11 andT32�T32

tD��22, this is equal to ��2.�/ as given by (5.13), hence the algebra defined

by the embedding S is A�2.�/.

Torus equivariant spectral triples 53

We thus have a Morita equivalence between the two algebras A� and A� 0 , via thebimodule S.Rp �Zq/, with the right action given by (5.14) and the commuting left actiongiven by (5.15).

5.3 Morita equivalence of torus equivariant spectral triples

We now consider Morita equivalences of equivariant real spectral triples on the noncommu-tative n-torus, as described in Chapter 3. We first determine the structure of the bimodule�1

D of one-forms, and then deduce the general form of a connection from this structure.

Lemma 5.3.1. Let A� be the smooth algebra of the noncommutative n-torus. The bimodule�1

D D spanfaŒD;b�ja;b 2A�g is a free module of rank n, i.e.

�1D 'A˚n

�:

Thus we have for E D S.R�Zq/, that E˝A��1

D ' E˚n.

Proof. We know from Theorem 3.6.9 in Chapter 3, that D can be written as

D DX

j

.�j �ı/˝Aj CC

with fAj gnj D1 a set of 2bn=2c � 2bn=2c matrices generating an irreducible representation

of the Clifford algebra Cln;0, and C a bounded operator commuting with the algebraaction. To see that the Aj generate the whole module, first consider the module �1

D0 withD0 D

Pi

�ej �ı

�Aj . The modules�1

D0 and�1D are isomorphic by the action of an element

G 2GL.n;R/ such thatP

i G.�i / �ıAj DP

j ej �ıAj for all j . Such a G exists, since the�j are n independent vectors in Rn.The bounded operator C does not play a role in thedefinition of �1

D , since it commutes with the action of the algebra.Since for all j , Œıj ;a� 2A� , and the Ai operators commute with the algebra action, we

can write any b DP

k ak ŒD;ck � 2�1D as a sum

Pj bjAj , with each bj D

Pk ak Œık ; ck �.

Also, for any element a 2A� we have a D aU �ejŒıj ;Uej

�, hence any b DP

j biAj givesrise to an element b 2�1

D .The generators Ai of this module are independent with respect to the action � of A� ,

because we have Xj

.�.aj /Aj /�X

k

�.ak/Ak D

Xj

�.a�j aj /;

where the last equality uses the fact that the Aj are generators of a Clifford algebra. Thisis a sum of positive operators a�

j aj , hence only zero if all aj are zero, thus the module isfree.

Using this decomposition, we can describe a connection rD in terms of its componentsacting on the different subspaces of E˝�1

D . In fact a more convenient description, interms of the generators of the Hopf algebra U.tn/, as described in Section 3.2, is available.


Lemma 5.3.2. The connection rD on E D S.R�Zq/ is a linear combination of n con-nections rj Cbj , for which

ŒX

k

.�j /krk ;Urx � ˝Aj v D ˝ Œ�j �ıAj ;Ux�v;

and bj 2A� 0 , with � 0 D �2.�/.

Proof. Via the isomorphism induced by the action of GL.n;R/ introduced in the proof ofLemma 5.3.1, we see that in order to deduce the connection rD , it is enough to calculatethe connections ri where ri is defined through the Leibniz rule as:

rj . a/v D .rj /avC .ıja/v;

with 2 E , v 2H , a 2A� using the right action of A� on E . This determines the rj

up to an additive term bj commuting with the action of the algebra A� on E . BecauseE is a Morita equivalence bimodule between A� and A� 0 with � 0 D �2.�/, we havebj 2A� 0 .

Let f 2 S.R�Zq/. Denote by t 2 R its first variable, and by p 2 Zq the last qvariables.

Lemma 5.3.3. The connections rj of Lemma 5.3.2 are given by:

for j D 1;2: rj D T11�1�

�t 1

2�i

@

@t

�;

for j D 3; : : :n: rj D p:

Proof. The right action of A� on S.R�Zq/ is given by (5.14). We see that

Œt;U rx �f .t;p/D .T11 �x/2U

rx f .t;p/

Œ@

@t;U r

x �f .t;p/D 2�i.T11 �x/1Urx f .t;p/

Œpi�2;Urx �f .t;p/D xiU

rx f .t;p/;

so these connections give the right commutators with generators of the algebra A� .

Remark 5.3.4. While these connections lead to an equivariant spectral triple, they them-selves do not form the Hopf algebra for an equivariant torus action, since

Œr1;r2�D .��111 /12:

We only have equivariance with respect to an equivariant action on the last q D n� 2components.

Recall that all Morita equivalences for A� are given by the group SO.n;njZ/ if � 2‚0n,

the set of all skew-symmetric matrices such that the SO.n;njZ/ action is defined for allelements of SO.n;njZ/. This is a dense set of the second category, see Section 5.2.

Torus equivariant spectral triples 55

Lemma 5.3.5. Given an equivariant noncommutative n-torus spectral triple .A� ;H ;D;J /

withD as in (3.26d), � 2‚0n, and � 0D �2.�/, then the Dirac operatorD0 in the equivariant

Morita equivalent spectral triple .A� 0 ;H 0;D0;J 0/ is given by:

D0D

Xj

�j �

��1

11 ��111 �12

0 Idq

!ı

!˝ j CC

0

D

Xj

��1

11 0

� t12�

�111 Idq

!�j �ı

!˝ j CC

0; (5.16)

with C 0 an arbitrary bounded self-adjoint operator commuting with the algebra A� 0 , andthe j matrices generating an irreducible representation of the Clifford algebra Cln;0 and�j as in (3.26d).

Proof. The Dirac operator D0 on H 0 can be calculated using (5.5):

ŒD0;U lx �.s˝ �˝ Nt /D .ŒrD;U

lx �s/�˝ Nt ;

whereU lx is defined as in (5.15). We knowrD in terms of theri calculated in Lemma 5.3.3:h

ri ;Ulx

iD T11

�1�

�Jo

�T11

t��1

�I2 �T32

t��x�

D ��111

�I2 ��12

��x:

From this data we can now compute D0 as

D0D

Xi

.�0i �ı/˝Ai CC

0; (5.17)

where Œ�0i � ı;U

lx � ˝ v D Œr

0i ;U

lx � ˝ v, and ı are the generators of the equivariant

torus action.The additional term we had for the connections, the bj 2A� 0 are seen to vanish when

we demand a Dirac operator of the form (5.17). Since an equivariant Dirac operator shouldcommute with the Hopf algebra as in (3.1d), we have that Œık ;bj �D 0 for all i;j . However,then we have that Œbj ;U l

x �D 0 for all j and x 2 Zn, and so the contribution of the bj to theDirac operator is a bounded operator commuting with the algebra, hence can be absorbedinto C 0.

Theorem 5.3.6 (Theorem D). For � 2‚0n, if the algebras A.T n

�/ and A.T n

� 0/ are stronglyMorita equivalent, and if there exists a Morita equivalence .E;rD/ of equivariant spectraltriples from .A.T n

�/;H ;D;J / to .A.T n

� 0/;H0;D0;J 0/, then there exists a Morita equiva-

lence . NE;rD0/ from .A.T n� 0/;H

0;D0;J 0/ to .A.T n�/;H ;D;J / such that the composition

of the Morita equivalences gives a spectral triple unitary equivalent to the original spectraltriple .A.T n

�/;H ;D;J /. Thus Morita equivalence of equivariant torus spectral triples is

an equivalence relation when � 2‚0n


Proof. We have for all � 2‚0n that the set of Morita equivalences is given by SO.n;njZ/,

and by Lemma 5.2.1 we know that this group is generated by GL.n;Z/, Skewn.Z/ andthe element �2. Since we know Morita equivalence of spectral triples is a transitiverelation by Lemma 5.1.1, it is enough to construct inverses for the generators of this group.If g 2 GL.n;Z/ or g 2 Skewn.Z/, the Morita equivalence of algebras was actually anisomorphism. Since the algebras are isomorphic, the corresponding transformation of theDirac operator is easily seen to be invertible. In fact, for the Skewn.Z/ case, the Diracoperator is unchanged, and if R 2GL.n;Z/, the Dirac operator changes under �.R/ as:

D0D

nXiD1

.R�1� �ı/Ai CB: (5.18)

For the �2 generator, the result follows from Lemma 5.3.5: Since �2 ı �2 D Id onmatrices, we see that the inverse of the Morita equivalence of algebras is given by applying�2 again. The corresponding Dirac operator is given by applying (5.16) twice, and using

� 0D �2.�/D

��1

11 ��111 �12

�21��111 �22��21�

�111 �12

!:

We then see:

�2.D0/D

Xj

��11 0

� t12�

�111 �11 Idq

! ��1

11 0

� t12�

�111 Idq

!�j �ı

!˝ j CC

D

Xj

��j �ı

�˝ j CC;

hence the relation is symmetric. From Lemma 5.1.1 we know that Morita equivalence ofspectral triples is a transitive relation, and reflexivity follows from the discussion precedingLemma 5.1.1, thus, Morita equivalence of equivariant spectral triples of the noncommutativen-torus is an equivalence relation.

The fact that for equivariant spectral triples the Morita equivalences are provablyinvertible suggests that these type of spectral triples are in some way special among allspectral triples. In fact, even for noncommutative n-tori with � 2 ‚0

n, the kernel ofD0 DDCACJAJ�1, with D an equivariant Dirac operator and A 2�1

D is not knownif D0 is not equivariant, see for example the discussion following [33, Corollary 5.2]. IfD0 is equivariant, D DD0. One would need to investigate if there is a canonical way topick out these preferred spectral triples, even among spectral triples which do not admit anobvious equivariant action. Also the structure of the moduli space of noncommutative tori,especially the extra equivalences due to the �2-equivalence, merits investigation.

C H A P T E R 6

Quantum lens spaces

Lens spaces, orbit spaces of free actions of cyclic groups on odd-dimensional spheres,were first introduced in 1884 by Walther Dyck [27]. Lens spaces are interesting becausethey are some of the simplest manifolds exhibiting the difference between homotopy typeand homeomorphism type. In this chapter we study spectral geometries on quantum lensspaces, as orbit spaces of free actions of cyclic groups on the spectral geometry on thequantum group SUq.2/, as constructed in [25]. These spectral geometries are given byweakening some of the conditions of a real spectral triple, just like in [25]. We classify theirreducible spectral geometries on quantum lens spaces and we study unitary equivalencesof such quantum lens spaces.

This chapter is based on joint work with Andrzej Sitarz

57

58 Quantum lens spaces

6.1 The equivariant spectral triple on SUq.2/

We recall the construction of the equivariant real spectral triple on SUq.2/ from [25]. Thisis not a real spectral triple in the sense of [15], since the opposite algebra only commuteswith the algebra up to compact operators. This was done in order to cope with certain “nogo-theorems”, which showed that it was impossible for a su.2/-equivariant spectral tosatisfy all conditions of a real spectral triple [25, Remark 6.6].

Let q denote a real number, 0 < q < 1. Let A.SUq.2// be the �-algebra generated bythe two elements a, b, satisfying the following relations:

baD qab (6.1a)b�aD qab� (6.1b)bb�D b�b (6.1c)

a�aCq2b�b D 1 (6.1d)aa�Cbb�

D 1: (6.1e)

From these relations it follows that a�bD qba�, a�b�D qb�a� and Œa;a��D .q2�1/bb�.If q D 1, we recover the generators of SU.2/ as a commutative space.

The Hopf �-algebra Uq.su.2//, for which we require the spectral triple to be equivari-ant, is generated over C by elements e;f;k;k�1, satisfying:

kk�1D 1; ek D qke; kf D qf k; k2

�k�2D .q�q�1/.fe� ef /; (6.2)

with the coproduct given by

�k D k˝k; �e D e˝kCk�1˝ e; �f D f ˝kCk�1

˝f; (6.3)

and antipode given by

Sk D k�1; Sf D�qf; Se D�q�1e: (6.4)

This algebra has irreducible finite dimensional representations �l on the 2lC1 dimensionalvector space Vl , labeled by half-integers l D 0; 1

2;1; 3

2; : : : [45]. Let jl;mi, (where m D

�l;�lC1; : : : l) be a basis for the irreducible Uq.su.2//-module Vl . The representation�l on Vl is then given by

�l .k/jl;mi D qmjl;mi;

�l .f /jl;mi DqŒl �m�qŒlCmC1�qjl;mC1i; (6.5)

�l .e/jl;mi DqŒl �mC1�qŒlCm�qjl;m�1i:

We can define two commuting actions of su.2/ on the algebra A.SUq.2//, as follows. Oneis the usual left action F, defined by

k FaD q12 a; k Fa�

D q� 12 a�; k Fb D q� 1

2 b; k Fb�D q

12 b�;

f FaD 0; f Fa�D�qb�; f Fb D a; f Fb�

D 0; (6.6)

e FaD b; e Fa�D 0; e Fb D 0; e Fb�

D�q�1a�:

The equivariant spectral triple on SUq.2/ 59

The other, commuting, action is given by applying the order 2 automorphism � of su.2/,which maps e to �f and maps k to k�1, and the inverse of the antipode to the usual rightaction. As can be checked, this gives a left representation �, commuting with (6.6):

k �aD q12 a; k �a�

D q� 12 a�; k �b D q

12 b; k �b�

D q� 12 b�;

f �aD 0; f �a�D qb�; f �b D 0; f �b�

D�a; (6.7)

e �aD�b�; e �a�D 0; e �b D q�1a�; e �b�

D 0:

There is a vector space basis eklm of A.SUq.2//, given by monomials of the form(eklm WD a

kblb�m k 2 Z;k � 0; l;m 2N

eklm WD a��kblb�m k 2 Z;k < 0; l;m 2N

(6.8)

There is a so-called Haar state, which is the unique [78, Theorem 4.2] bi-invariant linearfunctional on the C �-completion of A.SUq.2//, given by [78, Appendix A1]:(

.eklm/D 0 when k > 0 or l ¤m

.e0mm/D1�q2

1�q2mC2 :

Just as in the commutative (q! 1) case, the GNS-representation space V DL2.SUq.2/; /

with respect to the Haar state has a Peter-Weyl decomposition

V D

1M2lD0

Vl ˝Vl ;

by a suitable analogue of the Peter-Weyl theorem for compact quantum groups [79, Section6]. We abbreviate a vector jlmi˝ jlni 2 Vl ˝Vl by jlmni.

On this space we have two commuting left su.2/-actions, �;�, given on the subspaceVl ˝Vl by

� WD id˝�l ; � WD �l ˝ id: (6.9)

These two commuting actions are an extension to q ¤ 1 of the classical case where wecan identify

SU.2/D S3D Spin.4/=Spin.3/D .SU.2/�SU.2//=SU.2/;

with the action of Spin.4/ on SU.2/ given by .g;h/x D gxh�1.We recall the definition of a .�;�/-equivariant representation [25, Definition 3.2]:

Definition. Let � and � be two commuting representations of a Hopf-algebraH on a vectorspace V . A representation … of a �-algebra A on V is .�;�/-equivariant if the followingcompatibility relations hold:

.�.h/….a//v˝w D ��h.1/ �a

��h.2/

�v;

.�.h/….a//v˝w D ��h.1/ Fa

��h.2/

�v; (6.10)

for all h 2H , a 2A and v 2 V .


In our case, the two commuting actions � and � on the Hilbert space are given by (6.9).Just as in the case for noncommutative tori, see Chapter 3, in order to have a real

spectral triple, we need multiple copies of the Hilbert space. In the case of SUq.2/, weneed to consider two copies [25]:

H D V ˝C2' V ˝V 1

2' V 1

2˚

1M2j D1

�Vj C 1

2˝Vj

�˚

�Vj � 1

2˝Vj

�(6.11)

For convenience, we rename the spaces on the right hand side as

H DH"

0 ˚

M2j �1

H"

j ˚H#

j ; (6.12)

with H"

j '

�Vj C 1

2˝Vj

�and H

#

j '

�Vj � 1

2˝Vj

�.

The definition of the .�;�/ action on V in (6.9) can be amplified to H in several ways.Following [25], we define:

Definition. The representations � and � of Uq.su.2// on V are amplified to �0 and �0

on H , as

�0.h/ WD��˝� 1

2

�.�h/D �.h.1//˝� 1

2.h.2//;

�0.h/ WD �.h/˝ id: (6.13)

It is easy to check that �0 and �0 commute.We now define an explicit basis of H which is suited to .�0;�0/-equivariance, i.e. basis

vectors are eigenvectors for �0.k/ and �0.k/ and �0.f /, �0.f /, �0.e/ and �0.e/ are ladderoperators. Define Œn�D Œn�q D

qn�q�n

q�q�1 , and k˙ D k˙ 12

. Set

Cj� WD q�.j C�/=2 Œj ��

12

Œ2j �12

and Sj� WD q.j ��/=2 Œj C��

12

Œ2j �12

: (6.14)

Using the decomposition of H of (6.11), we define for j D lC 12

,�Dm� 12

,�D�j; : : : ;jand nD�j C 1

2; : : : ;j � 1

2:

jj�n #i WD Cj�jj��Cni˝ j

1

2;�1

2iCSj�jj

��ni˝ j1

2;C1

2i; (6.15)

with m˙ D m˙ 12

, and jj��Cni 2 Vj � . For j D l � 12

, � as before and n D �j �12; : : : ;j C 1

2:

jj�n "i WD �Sj�jjC�Cni˝ j

1

2;�1

2iCCj�jj

C��ni˝ j1

2;C1

2i: (6.16)

The action of the elements of Uq.su.2// can now be calculated, and it is easily shown thatthe jj�n "i and jj�n #i are joint eigenvectors for �0.k/ and �0.k/ with eigenvalues q� for�0 and qn for �0.

Summarizing, we have the following definition for the Hilbert space H :

The equivariant spectral triple on SUq.2/ 61

Definition. Let j D 0; 12;1; 3

2; : : :, and let �D �j;�j C1; : : : ;j and nD �j � 1

2;�j C

12; : : : ;j C 1

2. The Hilbert space H of the real spectral triple .A.SUq.2//;H ;D;J / is

spanned by basis vectors

jj�nii WD

jj�n "i

jj�n #i

!(6.17)

with the convention that the jj�n #i component is zero when nD˙.j C 12/ or j D 0.

An equivariant representation of the algebra A.SUq.2// on this Hilbert is given by:

Proposition 6.1.1 ([25]). The following representation … of A.SUq.2// on the Hilbertspace H with orthonormal basis jj�nii, is equivariant with respect to the .�0;�0/-actionof Uq.su.2//.

….a/ jj;�;nii D ˛C

j�njjC�CnC

iiC˛�j�njj

��CnCii (6.18a)

….b/ jj;�;nii D ˇC

j�njjC�Cn�

iiCˇ�j�njj

��Cn�ii (6.18b)

…�a��jj;�;nii D QCj�njj

C��n�iiC Q

�j�njj

��n�ii (6.18c)

…�b��jj;�;nii D QCj�njj

C��nCiiC Q

�j�njj

��nCii (6.18d)

where ˛˙j�n and ˇ˙

j�n are, up to phase factors depending only on j , the following triangular2�2 matrices:

˛Cj�n D q

.�Cn� 12 /=2Œj C�C1�

12

�q�j � 1

2

hj CnC 3

2

i 12

Œ2j C2�0

q12

hj �nC 1

2

i 12

Œ2j C1�Œ2j C2�q�j

hj CnC 1

2

i 12

Œ2j C1�

˘

(6.19a)

˛�j�n D q

.�Cn� 12 /=2Œj ��

12

�qj C1

hj �nC 1

2

i 12

Œ2j C1��q

12

hj CnC 1

2

i 12

Œ2j �Œ2j C1�

0 qj C 12

hj �n� 1

2

i 12

Œ2j �

˘

(6.19b)

ˇCj�n D q

.�Cn� 12 /=2Œj C�C1�

12

� hj �nC 3

2

i 12

Œ2j C2�0

�q�j �1

hj CnC 1

2

i 12

Œ2j C1�Œ2j C2�q� 1

2

hj �nC 1

2

i 12

Œ2j C1�

˘

(6.19c)

ˇ�j�n D q

.�Cn� 12 /=2Œj ��

12

��q� 1

2

hj CnC 1

2

i 12

Œ2j C1��qj

hj �nC 1

2

i 12

Œ2j �Œ2j C1�

0 �

hj Cn� 1

2

i 12

Œ2j �

˘

(6.19d)

and the remaining matrices are the hermitian conjugates

Q˙j�n D .˛

�

j ˙��n�/�; Q˙

j�n D .ˇ�

j ˙��nC/�:

The Dirac operator of the spectral triple constructed in [25, Section 5] is given by:

D

jj�n "i

jj 0�0n0 #i

!D

�2j C 3

2

�jj�n "i

��2j 0C

12

�jj 0�0n0 #i

!: (6.20)


The reality operator, constructed in [25, Section 6] is given by:

J

jj�n "i

jj 0�0n0 #i

!D

i2.2j C�Cn/jj;��;�n "i

i2.2j 0��0�n0/jj 0;��0;�n0 #i

!: (6.21)

Together, the algebra A.SUq.2//, with representation as given in Proposition 6.1.1,Dirac operator D given by (6.20), and reality operator J given by (6.21), satisfy mostconditions as stated in Chapter 2, with some slight modifications. These modificationsare that Œ….x/;J….y�/J �� and ŒŒD;….x/�;J….y�/J �� are not exactly 0, but lie in thetwo-sided ideal Kq in B.H / generated by the compact, positive trace class operators

Lq W Lqjj;�;nii D qjjj;�;nii: (6.22)

This means that Conditions 6 and 7 should be modified appropriately.Also, it is currently unknown whether the Hochschild cycle condition of Condition 8

and the Poincaré duality condition of Condition 11 are satisfied. For a discussion of theseconditions for SUq.2/, see Remark 6.3.2.

6.2 Topological quantum lens spaces

Commutative lens spaces are defined as the quotient of an odd-dimensional sphere by a freeaction of a finite cyclic group. We will only consider 3-dimensional lens spaces here. Letp;r1; r2 be integers such that gcd.r1;p/D gcd.r2;p/D 1.

We view S3 as the points .z1;z2/ 2C2 such that jz1j2Cjz2j

2 D 1. We can define anaction of the cyclic group Z=pZ on S3 as

.z1;z2/ 7! .e2�i

p r1z1; e2�i

p r2z2/:

The lens space L.pIr1; r2/ is then defined as the quotient of S3 by this action of the cyclicgroup Z=pZ.

For 3-dimensional lens spaces we see that L.pIr1; r2/DL.pI1;r�11 r2/ by multiplying

each component by e2�i

p r�11 (where r�1

1 is meant mod p). We will thus write L.p;r/ forthe lens space L.pIr1; r2/, with r D r�1

1 r2.Since S3 ' SU.2/, it is natural to define quantum lens spaces as quotients of SUq.2/

by free actions of finite cyclic groups (the analogy is z1$ a, and z2$ b). We want toconsider quotients of the algebra A.SUq.2// under actions of the group Z=pZ. In order todo this, we need to embed Z=pZ into Aut.SUq.2//. Since there exists only one non-trivialouter automorphism of A.SUq.2// [38, Proposition 3.1], namely the map of order twoinduced by a 7! a;b 7! b�, we can fit the automorphisms of A.SUq.2// into the followingexact sequence:

0! Inn.A.SUq.2///! Aut.A.SUq.2///! Z=2ZËU.1/! 0; (6.23)

with U.1/ acting by multiplying the a and b generators by complex numbers of modulus 1.To maximize the analogy with the commutative case, we consider actions of cyclic groupsZ=pZ embedded in U.1/ as outer automorphisms. Let g denote a generator of Z=pZ, andlet r be an integer relatively prime to p. Let �.r/D .rC1/ mod 2. We have:

Topological quantum lens spaces 63

Proposition 6.2.1. The following action of Z=pZ on the algebra A.SUq.2//:

�.g/FaD e2�i

p a (6.24a)

�.g/Fb D e2�i

p rb: (6.24b)

generates a unitary representation � of Z=pZ on H given by:

�.g/jj;�;nii D e2�i

p

�.1Cr/�C.1�r/nC 1

2 �.r/�jj;�;nii: (6.25)

The presence of �.r/ guarantees that the expression in brackets is an integer.

Proof. Since Z=pZ is a group, we demand that the action of A on H is equivariant withrespect to the group viewed as a Hopf algebra, with coproduct �.g/D g˝g. From thisit follows that �.g/jj�nii D cj�njj�nii, for some cj�n 2 C, since otherwise this wouldnot be compatible with the equivariance condition �.g/….a/vD….g.1/a/�.g.1//v, where

g.1/a is e2�i

p a, by Proposition 6.1.1. We can then calculate:

�.g/….a/jj�nii D…�e

2�ip a

��.g/jj�nii;

�.g/….b/jj�nii D…�e

2�ip rb

��.g/jj�nii:

From (6.18) we see that the action of a and a� on H leaves the difference ��n constant,and b leaves the sum �Cn constant.

We get the recurrence relations �.g/jj�niiD e2�i

p �.g/jj˙�CnCii and �.g/jj�niiDe

2�ip r�.g/jj˙�Cn�ii. Solving these recurrence relations, we see that �.g/jj�nii D

e2�i

p .r.�Cn/C.��n//Cc , where c is any constant. In order to have gp D 1, we see thatc D �.r/ plus an additional integer, which can be set to zero.

Observe that if we replace r by r�p the action on the generators does not change. If wetake p even then r is necessarily odd, and �.r/ is 0. If r is even, then p is necessarily odd,and r �p is odd, and on the Hilbert space the actions determined by .p;r/ and .p;r �p/are equivalent since:

e2�i

p ..1C.r�p//�C.1�.r�p//n/D

�e

2�ip.n��/p e�

�i�.r/p

�e

2�ip

�.1Cr/�C.1�r/nC 1

2 �.r/�

D

��e�

�i�.r/p

��.g/;

where we have used that n�� 2 12ZnZ. Since��e�

�i�.r/p

�p

D 1; (6.26)

for odd p, the actions are equivalent. For this reason, we can always take r odd, and�.r/ D 0. Let K D 0;1;2; : : : ;p� 1. We define HK as the eigensubspace of H for theaction of g with eigenvalue e

2�iKp . Further, let Lq.p;r/ be the subalgebra of A.SUq.2//

which is invariant under the action of Z=pZ.We have


Proposition 6.2.2. For eachKD 0;1;2; : : : ;p�1 and each x 2Lq.p;r/,….x/HK �HK .

Proof. From (6.8) we know that A.SUq.2// is generated as a vector space by the elementsakblb�m and blb�ma�n for k; l;m;n 2N. Since p and r are taken to be relatively prime,we see from (6.24a) and (6.24b) that L.p;r/ is generated by elements akblb�m withkC r.l �m/� 0 mod p and blb�ma�n with r.l �m/�n� 0 mod p. From (6.18), wecan calculate that the action of akblb�m on � and n is given by

� 7! �C1

2kC

1

2l �

1

2m; n 7! nC

1

2k�

1

2lC

1

2m:

Then if �.g/jj;�;nii D e2�i

p Kjj;�;nii, we have .1C r/�C .1� r/n�K mod p and it

then follows that

.1C r/.�C1

2kC

1

2l �

1

2m/C .1� r/.nC

1

2k�

1

2lC

1

2m/D .1C r/�C .1� r/nCkC r.l �m/

�K mod p:

Proposition 6.2.3. The equivariant real structure J , as given in (6.21) satisfies:

JHK DHK0 ; (6.27)

where KCK 0 � 0 mod p.

Proof. We have J jj�nii D cj�njj ��nii with cj�n a complex number, and from

.1C r/�C .1� r/nDK mod p;

it follows that.1C r/ � .��/C .1� r/ � .�n/D�K mod p:

6.3 Geometry of quantum lens spaces

We now turn to the construction of the real spectral triple of the quantum lens space. Asstated at the end of Section 6.1, we modify some conditions of a real spectral triple, exactlyas in [25], i.e. the Conditions 6 and 7 only hold up to the compact operators of positivetrace class defined in (6.22).

Furthermore, it is unknown if te Hochschild cycle condition of Condition 8, and thePoincaré duality of Condition 11 are satisfied for SUq.2/. This means that also forLq.p;r/

we do not know if they are satisfied.We call a structure, satisfying all conditions of Chapter 2, with the modification of

the first order condition, and the removal of the Hochschild cycle condition and Poincaréduality a spectral geometry.

Proposition 6.3.1. Let Lq.p;r/, q 2 .0;1/, be the quantum lens space as defined above.Then for any K D 0;1; : : :p�1, the Hilbert space HK˚HK0 , where KCK 0 � 0 mod p,the reality structure J and the Dirac operator D taken as the restrictions of J and D fromthe A.SUq.2// spectral geometry constitute a spectral geometry over the quantum lensspace Lq.p;r/.

Geometry of quantum lens spaces 65

Proof. Almost all the usual conditions as described in Chapter 2 are easily seen to carryover from the SUq.2/ case. For completeness, we list them here, and give short argumentswhy they carry over.

� The compact resolvent and dimension growth of Conditions 1 and 5 follow fromthe fact that for j big enough, there always exist � and n such that there are vectorsjj;�;nii 2HK , and thus also in HK0 , hence the dimension growth is satisfied. Thecompact resolvent condition also follows from this, and the fact that the dimensionof the kernel of D still clearly is finite dimensional.

� The commutation relations between J and D and the sign of J 2 are trivially thesame as for SUq.2/.

� The algebra satisfies the regularity condition of Condition 9: ŒD;a� is a boundedoperator on H for all a 2A and both a and ŒD;a� belong to the domain of smoothnessT1

kD1 Dom.ık/ of the derivation ı, with ı.T /D ŒjDj;T �. This condition is satisfiedsinceLq.p;r/ is a subalgebra of the algebra A.SUq.2//which satisfies this conditionby [73, Proposition 2.1].

� The finiteness condition of Condition 10 is satisfied because the space of smooth vec-tors H 1 WD

T1

kD1 Dom.Dk/ is a finitely generated projective left Lq.p;r/-module,because both are given as eigenspaces by the same group action, commuting with theDirac operator.

The opposite algebra and first order condition of Conditions 6 and 7 are weakened asdescribed in [25]: the opposite algebra condition Œa;J b�J�1� D 0 and the first ordercondition ŒŒD;a�;J b�J�1� D 0 should hold up to compact operators in Kq as definedin (6.22). We should check whether this condition still holds for the lens space. First,observe that the Lq operators are unchanged, since they only see the j component, whichdoes not play a role in the �.g/ action. In fact, due to [25, Proposition 7.2] for thegenerators a;a�;b;b�, the commutation relations are given by Œ�.x/;�o.y/�D L2

qA, withA a bounded operator depending on x and y. Since for any a;b in a ring R, we have that ifŒa;b� 2 I, with I an ideal of R, we have Œam;bn� 2 I by repeatedly applying Œa;b� 2 I, thegenerators of Lq.p;r/ as described in Proposition 6.2.2 also satisfy this condition.

Remark 6.3.2. The Hochschild cycle condition, the analogue of having a nowhere vanish-ing volume form, is problematic, because as shown in [52, Proposition 1.2], for SUq.2/

there are no nontrivial Hochschild cycles n-cycles for n > 1. This “dimension drop” phe-nomenon can be addressed by using twisted Hochschild homology as in [39, Theorem1.2], where an analogue of the volume form, dA was constructed in twisted Hochschildhomology for SUq.2/. However, what the meaning of this volume form is in terms ofspectral triples is still unclear. The Poincaré duality of Condition 11, is intimately connectedto the Hochschild homology, and so its status for spectral triples on SUq.2/ is also unclear.This “volume form” dA is invariant under the action � of Z=pZ, but it is unclear for nowwhat �D.dA/ is.


Irreducibility Though any of the above listed spectral geometries for the quantum lensspaces is admissible from the point of view of the noncommutative axiomatic approach, notall correspond to spin structures on commutative lens spaces. As discussed in Section 2.2,we demand that our spectral be irreducible as the analogue of a connected manifoldin commutative geometry. For the three different notions of irreducibility discussed inSection 2.2, only one gives the same number of spin structures on the lens space as inthe commutative case. If we use [37, Definition 11.2] or [42, Definition 2.1], all the realspectral triples above are irreducible, since the J operator interchanges the HK and HK0

spaces. If we use the definition of [16, Remark 6 on p.163], only irreducibility with respectto the algebra action and Dirac operator are demanded. None of the above real spectraltriples are then irreducible, however the cases where K DK 0 D 0 and K DK 0 D p=2 if pis even can be made irreducible by dropping one of the two copies of the Hilbert space, andsetting the real spectral triple to be .Lq.p;r/;HK ;D;J /. Since JHK �HK in this case,this is a well-defined spectral triple, and irreducible.

For even p we obtain two possible spin structures, for odd p just one, just as in thecommutative case [34].

Theorem 6.3.3 (Theorem E). The quantum lens space Lq.p;r/ admits one irreduciblespectral geometry if p is odd, and two if p is even. The spectral geometries are given in thetwo cases by

� p mod 2D 1: .Lq.p;r/;H0;D;J /.

� p D 2P : .Lq.p;r/;H0;D;J / and .Lq.p;r/;HP ;D;J /.

with D the Dirac operator as described in (6.20) and J the operator given in (6.21).

The spectrum of the Dirac operator Let us recall that the spectrum of the Dirac operatorover A.SUq.2// (with appropriate normalization) is given by:

Djj;�;n;"i D

�2j C

3

2

�jj;�;n;"i with j D 0;

1

2; : : :

with multiplicity: .2j C1/.2j C2/

Djj;�;n;#i D �

�2j C

1

2

�jj;�;n;#i with j D

1

2;1; : : :

with multiplicity: .2j C1/2j

Note that the spectrum is symmetric. Since in the construction of the spectral geometrieson quantum lens spaces we keep the Dirac operator of SUq.2/ (and only restrict the Hilbertspace), the spectrum remains unchanged, only the multiplicities differ. We shall reducethe problem of computing these multiplicities to a number theoretic problem of solvingcongruence relations.

Proposition 6.3.4. The eigenvalues of D belonging to an irreducible spectral geometryas in Theorem 6.3.3 with Hilbert space HK (K D 0 or K D 1

2p) are 2j C 3

2and �2j � 1

2,

with respective multiplicity NC.j / and N�.j /, (either of them could be 0, which means


that the value is not present in the spectrum) whereN˙.j / denotes the number of solutionsto the equation:

.1C r/�C .1� r/n�K mod p (6.28)

with �j � �� j and �.j ˙ 12/� n� j ˙ 1

2.

The exact calculation of the number of eigenvalues is a tedious task. The solutiondepends heavily on the properties of .1C r/ and .1� r/, in particular on the greatestcommon divisor of 1˙ r and p. Although in each case the explicit solutions for � and ncan be easily found, calculating the number of solutions for a given j is rather difficult foran abstract choice of r and p.

For illustration, we show here some pictures of what the spectra for small p looks like.We represent the basis vectors jj�n "i; jj�n #i of the Hilbert space H of the spectraltriple on SUq.2/ as defined in Proposition 6.1.1 as a lattice, with � on the horizontal axisand n on the vertical axis. We project the j coordinate away, since it does not play a rolefor determining whether a vector lies in HK , only in confining the possible � and n. Weillustrate this by drawing lines through the allowed values (in the form of rectangles aroundthe origin) for j D 3 and in the " part of H . We draw a circle through the origin�D 0;nD 0.

The stars ( ) and diamonds( ) represent basis vectors of the HK Hilbert space of thelens space .Lq.p;r/;HK ;D;J /. The circles ( ) represent basis vectors of the Hilbert spaceof SUq.2/ which are not part of the lens space. The stars are the allowed values for integervalued j , the diamonds are the allowed values for half-integer valued j .

Figure 6.1: H0 for p D 2;r D 1 Figure 6.2: H1 for p D 2;r D 1


Figure 6.3: H0 for p D 5;r D 1 Figure 6.4: H0 for p D 5;r D�3

Figure 6.5: H0 for p D 7;r D�5 Figure 6.6: H0 for p D 7;r D 3

Example: the L.p;p � 1/ lens space This corresponds to the choice r D p � 1 orequivalently r D�1 (then �.r/D 0).

Proposition 6.3.5. The spectrum of the Dirac operator on L.p;p�1/ is:

� for p � 1 mod 2:

�D D

†2kpC2lC 1

2; with multiplicity: 2.2kC1/.kpC l/

k D 0;1; : : : l D 0;1; : : : ;p�1

.2kC1/pC2lC 12; with multiplicity: .2kC2/..2kC1/pC2l/

k D 0;1; : : : l D 0;1; : : : ;p�1

�2kp�2l � 32; with multiplicity: 2.2kC1/.kpC2lC1/

k D 0;1; : : : l D 0;1; : : : ;p�1

�.2kC1/p�2lC 12; with multiplicity: .2kC2/..2kC1/pC2lC2/k D 0;1; : : : l D 0;1; : : : ;p�1

Note that the �D D12

eigenvalue (k D l D 0) has multiplicity 0.


� for p D 2P and K D 0 we have

�D D

„2kpC2lC 1

2with multiplicity: 2.2kC1/.kpC l/k D 0;1; : : : l D 0;1; : : : ;p�1

�2kp�2l � 32

with multiplicity: 2.2kC1/.kpC lC1/k D 0;1; : : : l D 0;1; : : : ;p�1

Again, the �D D12

eigenvalue has multiplicity 0 (k D l D 0).

� for p D 2P and and K D P :

�D D

„.2kC1/P C2lC 1

2with multiplicity: .2kC2/..2kC1/pC2l/k D 0;1; : : : l D 0;1; : : : ;P �1

�.2kC1/P �2l � 32

with multiplicity: .2kC2/..2kC1/pC2lC2/k D 0;1; : : : l D 0;1; : : : ;P �1

Remark 6.3.6. The result is the same as the result for the commutative case as describedin [5, Theorem 5], when we set T D 1.

Proof. Since r D �1, .1C r/� D 0 and so we only need to consider n in the analysisbelow. The multiplicity of the � factor for a given j is 2j C 1. When we consider thestandard spin structure K D 0, we see that 2n� 0 mod p.

Consider first the case p odd. 2n� 0 mod p means that either n is integer valued, andnD kp for some k 2Z, or n is half integer valued and nD 1

2pCkp for some k 2Z. Since

the spectrum is symmetric, we only need to consider the " part.In case n is integer valued, j must be half-integer valued. When j < p� 1, there

is only one possibility for n, namely 0. When p� 1 < j < 2p� 1, we have 3 possibil-ities for n: �p;0;p. In general, when kp� 1 < j < .kC 1/p� 1, k an integer > 0, jhalf-integer valued, we have 2kC1 possibilities for n. If n is half-integer valued, j mustbe integer-valued. The first value for j such that n D 1

2pC kp, k 2 Z, is possible, is

j D 12p� 1

2. There are two possibilities for n: 1

2p and �1

2p. We get 2 new possibilities

of n if we increase j by p. Since the eigenvalue of D on a vector jj�n "i is 2j C 32

, andj D 1

2; 3

2; : : :, or j D 1

2p� 1

2; 1

2pC 1

2; : : :, we get the asked spectrum for p � 1 mod 2,

where the first formula corresponds to j half-integer valued, and the second to j integervalued.

For p even and K D 0, we see that n half-integer valued is not a possibility, and theanalysis for the n integer-valued case is the same as for p odd.

WhenpD 2P is even andKDP , we see that ifP is even, then n is only integer-valued,and if P is odd, then n is only half-integer valued. If P is even, we then have j ishalf-integer valued, and the first solution for n arises when j D P=2� 1

2. We get two

solutions for n, P=2 and �P=2, in this case, and two new solutions for n when we increasej by P . When P is odd, we need n to be half-integer valued, with the first two possibilitiesarising when j D P=2� 1

2, and again, two new possibilities for n when we increase j by

P , hence the spectrum given above.


There is a second case which we can easily derive from the calculations above. If welook at equation (6.28), we see that in addition to the r D�1 case, the r DC1 case shouldeasily be calculable. Classically, the L.p;p�1/ and L.p;1/ lens spaces are isometric, theorientation is just reversed, so one expects the spin structures to be the same. We will showthat this is the case for the quantum lens spaces in the next section.

6.4 Unitary equivalences

It is known [62] that two commutative lens spaces L.p;r/ and L.p0; r 0/ are homeomorphicif and only if p D p0 and r 0 �˙r˙1 mod p. This can be intuitively understood by lookingat diagrams as in Section 6.3. There we see that the diagram of L.p;r/ is the same as thediagram of L.p;r 0/, r 0 D˙r˙1, up to rotation, mirroring and interchanging the stars anddiamonds. For example, 3��.�5/�1 mod 7, and we see in Figures 6.5 and 6.6 that theyare the same if we rotate Figure 6.5 one quarter clockwise and flip the stars and diamonds.The L.5;1/ and L.5;�3/ lens space diagrams of Figures 6.3 and 6.4 are very differenthowever, and not related by mirroring and rotations by quarter-turns.

In the noncommutative case when q ¤ 1, the lens spaces Lq.p;r/ and Lq.p;r0/ are

shown to be unitary equivalent when r D˙r 0, if we take care of the subtleties of equivariantrepresentations.

Theorem 6.4.1 (Theorem F). When q 2 .0;1/, the quantum lens spaces given by .Lq.p;r/;

HK ;D;J / and .Lq.p;r0/;HK ;D;J / are unitary equivalent if:

� r 0� r mod p. The unitary equivalenceU is then given byU jj�n"iD cj�n"jj�n "i

and U jj�n #i D cj�n#jj�n #i, with cj�nl complex numbers of norm 1.

� r 0 � �r mod p. The unitary equivalence is implemented by the order order-twoautomorphism �.a/D a, �.b/D�b� of A.SUq.2// and the actionU on the Hilbertspace given by U jj�n "i D �cj�njj

Cn� #i and U jj�n #i D cj�njj�n� "i, with

again cj�n a complex number of norm 1.

Proof. It is clear that the first map is a unitary equivalence. To show that the second mapis a unitary equivalence, we first study the equivalence of Hilbert spaces. If v 2HK withK D 0 or K D p=2, we have .1C r/�C .1� r/n� 0 mod p or p=2 respectively. If wetake r 0 D�r , we see that .1C r 0/�C .1� r 0/nD .1� r/�C .1C r/n and we see that ifwe interchange � and n, v 2HK is mapped to a vector v 2H 0

K in the K D 0 or K D p=2subspace of the � action for r 0 D�r .

To define a compatible action of the algebra on H 0K , we see from Proposition 6.1.1 that

we need to interchange b and�b�. Now to show that this indeed gives a unitary equivalenceon the algebra, we need to show that the U�1.…0.�b//U D….b�/, and U�1.…0.a//U D

….a/. However, since U interchanges the " and # part, …0 is not the representation asgiven in Proposition 6.1.1, since then U�1…0.b/U is not .�0;�0/-equivariant as describedin Section 6.1. We need to demand that …0 is equivariant with respect to a differentsu.2/-action .�00;�00/, such that U�1�00.g/U D �0.g/ and U�1�00.g/U D �0.g/. We willstudy the proof of [25, Proposition 4.4] to see how the representation must be modified.

Unitary equivalences 71

Write

AD

A"" A"#

A#" A##

!:

for a matrix acting on jj�nii 2H by

A

jj�n "i

jj�n #i

!D

A""jj�n "iCA#"jj�n #i

A"#jj�n "iCA##jj�n #i

!:

Subsequently ˛C

j�n;""is the part of ˛C

j�n in (6.19) which maps the "-subspace of theHilbert space to the "-subspace, etc. , and we call the "" and ## part the diagonal.

The off-diagonal parts clearly are not the same for � 0 as for � . We see immediatelythat .˛C

j�n/"#D 0, while

�U�1�.˛C

j�n/U�

"#D �.˛C

j Cn�/#"¤ 0. We go through the

determination of the equivariant algebra action on the Hilbert space in [25, Proposition 4.4].We can calculate the .�00;�00/-action on H by using �00.h/D U�0.h/U�1. Using [25,

Lemma 4.2], we get that �00.h/D �0.h/ and �00.h/D �0.h/ on the Hilbert space. By (6.6)and (6.7), we see that the relations for a and a0 used in [25, Proposition 4.4] are unchanged,and that b should be mapped to �b� and vice versa. There is one extra subtlety however: ifwe wish to get the action of ….a/ on jj�nii, ….b/ on jj�nii, etc. , we need to determine

…0.h/

jj�n� #i

jjCn� "i

!;

i.e. the action …0 on a vector of the form

jj 0n� #i

jj 00n� "i

!, with j 00 D j 0C 1. We will see

that this changes the off-diagonal parts of ˛C, ˛�, ˇC and ˇ�. Using the commutationrelations as described in (6.7), we get:

�00.k/…0.a/U jj�nii D…0.q12 a/q�U jj�nii; �00.k/…0.b/U jj�nii D…0.q

12 b/q�U jj�nii;

�00.k/…0.a/U jj�nii D…0.q12 a/qnU jj�nii; �00.k/…0.b/U jj�nii D…0.q� 1

2 b/qnU jj�nii;

from which it follows that

…0.a/

jj�n� #i

jjCnC�C "i

!D

Xj 0

Cj 0j�n#jj

�nC�C #i

Cj 0j�n"jjCn� "i

!; (6.29)

with Cj 0j�nl to be determined. Here we encounter our first difference between the rep-resentations …0 and …. Since �00.f /…0.a/D q� 1

2…0.a/�00.f /, we see that if we apply�00.f /

r to both sides of (6.29), we get that the left hand side of (6.29) vanishes in theupper part for all r such that �C r > j�, whereas the right hand side vanishes for all rsuch that �CC r > j 0, hence Cj 0j�n# D 0 for j 0 > j . Similarly, we get that Cj 0j�n" D 0

for all j 0 > j C1, and by considering �00.e/, we get that Cj 0j�n# D 0 for j 0 < j �1 andCj 0j�n" D 0 for all j 0 < j . Hence

…0.a/

jj�n� #i

jjCnC�C "i

!D ˛0C

j�n

jjnC�C #i

jj C1;nC�C "i

!C˛0�

j�n

jj �1;nC�C #i

jjnC�C "i

!:


The rest of the calculation in the proof of [25, Proposition 4.4] can be done with the abovein mind, to get the following .�00;�00/-equivariant representation of A.SUq.2// on UH :

…0.a/

�jj�n "i

jj C1;�n #i

�D ˛0C

j�n

�jjC�CnC "i

jj C 32 ;�

CnC #i

�C˛0�

j�n

�jj��CnC "i

jjC�CnC #i

�…0.b/

�jj�n "i

jj C1;�n #i

�D ˇ0C

j�n

�jjC�Cn� "i

jj C 32 ;�

Cn� #i

�Cˇ0�

j�n

�jj��Cn� "i

jjC�Cn� #i

�…0.a�/

�jj�n "i

jj C1;�n #i

�D Q0

C

j�n

�jjC��n� "i

jj C 32 ;�

�n� #i

�C Q0

�

j�n

�jj��n� "i

jjC��n� #i

�…0.b�/

�jj�n "i

jj C1;�n #i

�D Q0

C

j�n

�jjC��nC "i

jj C 32 ;�

�nC #i

�C Q0

�

j�n

�jj��nC "i

jjC��nC #i

�

With ˛0˙

j�n and ˇ0˙

j�n matrices given by:

˛0C

j�n D q.�Cn� 1

2 /=2

�q�j � 1

2Œj C�C1�

12

hj CnC 3

2

i 12

Œ2j C2��q

12

Œj ��C1�12

hj CnC 3

2

i 12

Œ2j C2�Œ2j C3�

0 q�j �1Œj C�C2�

12

hj CnC 3

2

i 12

Œ2j C3�

˘

;

˛0�


2 /=2

�qj C1

Œj ��12

hj �nC 1

2

i 12

Œ2j C1�0

q12

Œj C�C1�12

hj �nC 1

2

i 12

Œ2j C1�Œ2j C2�qj C 3

2Œj ��C1�

12

hj �nC 1

2

i 12

Œ2j C2�

˘

;

ˇ0C


2 /=2

�Œj C�C1�

12

hj �nC 3

2

i 12

Œ2j C2��qj C1

Œj ��C1�12

hj �nC 3

2

i 12


0 q� 12

Œj C�C2�12

hj �nC 3

2

i 12

Œ2j C3�

˘

;

ˇ0�


2 /=2

��q� 1

2Œj ��

12

hj CnC 1

2

i 12

Œ2j C1�0

q�j �1Œj C�C1�

12

hj CnC 1

2

i 12

Œ2j C1�Œ2j C2��

Œj ��C1�12

hj CnC 1

2

i 12

Œ2j C2�

˘

:

We can now calculate that U…0.a/U�1D….a/, U…0.a�/U�1D….a�/, U…0.b/U�1D

….�b�/ andU…0.b�/U�1D….�b/. The parts with a are easily seen to match from (6.8).We show the identity U…0.b/U�1 D….�b�/, for one the ˇC-part. That the Q� and ˇ0�

matrices are the same follows from a similar calculation. First, we derive QCj�n from

Proposition 6.1.1, suppressing the q.�Cn� 12 /=2 factor, which is clearly the same for the

representation … and …0:

QCj�n D Œj ��C1�

12

��q� 1

2

hj CnC 3

2

i 12

Œ2j C2�0

�qj C 12

hj �nC 1

2

i 12

Œ2j C1�Œ2j C2��

hj CnC 1

2

i 12

Œ2j C1�

˘

:

Unitary equivalences 73

In order to match the vectors jj�nii, we calculate Uˇ0C

j � 12 ;n�

U�1, again suppressing the

q.�Cn� 12 /=2 factor:

Uˇ0C

j � 12 ;n�

U�1

D

�0 1

�1 0

�� hj CnC 1

2

i 12

Œj ��C1�12

Œ2j C1��qj C 1

2

hj �nC 1

2

i 12

Œj ��C1�12


0 q� 12

hj CnC 3

2

i 12

Œj ��C1�12

Œ2j C2�

˘�0 �1

1 0

�

D

�q� 1

2

hj CnC 3

2

i 12

Œj ��C1�12

Œ2j C2�0

qj C 12

hj �nC 1

2

i 12

Œj ��C1�12


hj CnC 1

2

i 12

Œj ��C1�12

Œ2j C1�

˘

;

and we see that � QCj�n D Uˇ0C

j � 12 ;n�

U�1.

Remark 6.4.2. For the other type of equivalences, i.e. the r! r�1 case, we do not knowif they give rise to unitary equivalences in the q ¤ 1 case. For isomorphisms of Lq.p;r/

coming from the automorphisms of A.SUq.2//, as classified by (6.23), we can showthat they do not give rise to isomorphisms between Lq.p;r/ and Lq.p;r

�1/. Take forexample an element of the form a�blb�m

2 Lq.p;r/, with r.l �m/�1� 0 mod p, i.e.l �m� r�1 mod p. We have r�1.l �m/� .r�1/2 ¥ 1 mod p if r�1 ¥ r . This meansthat if r�1 ¥ r , the identity automorphism is not a map from Lq.p;r/ to Lq.p;r

�1/. Also,we have r�1.m� l/ � r�1 � .�r�1/ ¥ 1 mod p if r�1 ¥ �r , hence the automorphisma! a, b! b� is not a map from Lq.p;r/ to Lq.p;r

�1/. Hence the automorphisms ofA.SUq.2// do not give homomorphisms from Lq.p;r/ to Lq.p;r

�1/ if r�1 ¥˙r . Thesame argument holds for Lq.p;r/ and Lq.p;�r

�1/.We do not know that there are isomorphisms of Lq.p;r/ not compatible with automor-

phisms of A.SUq.2//.

Remark 6.4.3. If we consider instead of a q-deformation a � -deformation as in [21], theredoes exist an algebra automorphism between the algebras of the L.p;r/ and L.p;r�1/

lens spaces, as shown in [53, Proposition 5.7]. This is because the generators a and b havethe same properties, i.e. aa� D a�a and bb� D b�b.

Samenvatting

Deze samenvatting is als volgt opgedeeld: eerst geef ik een korte beschrijving van mijnvakgebied, de niet-commutatieve meetkunde, en de resultaten in dit proefschrift. Daarnazal ik beschrijven hoe resultaten uit dit proefschrift raken aan wiskunde die op dit momentgebruikt wordt voor draadloze communicatie.

Een korte introductie in de niet-commutatieve meetkunde

Veel waarnemingen van de wereld om ons heen zijn in feite spectraal: we zien, horen, ofmeten een spectrum. Wat een spectrum is, is het eenvoudigst te zien bij geluid: als webijvoorbeeld een snaar van een gitaar bespelen, gaat deze snaar trillen. Deze trillingen zijnopgebouwd uit eigentrillingen, oftewel de staande golven. Deze golven worden omgezet intrillingen van de lucht, die op hun beurt weer ons trommelvlies laten bewegen, waarna wehet geluid horen.

Wiskundig kunnen we de staande golven als volgt bekijken. De functie die de trillingbeschrijft, moet voldoen aan een bepaalde differentiaalvergelijking, een vergelijking die deafgeleide (snelheid) op een punt relateert aan de waarde (uitwijking) van de functie op datpunt. De vergelijking heet in dit geval de golfvergelijking. In deze golfvergelijking staateen zogenoemde differentiaaloperator, iets waar je een functie in stopt en een combinatievan afgeleiden van die functie teruggeeft. Een methode om deze vergelijking op te lossen,en dus te vinden hoe een snaar kan trillen, is om te zoeken naar eigenfuncties, functies dieals we ze in de differentiaaloperator stoppen, eruit komen als een veelvoud van de functiezelf. Een voorbeeld dat misschien bekend is, is de exponentiële functie x 7! ex . Als weals differentiaaloperator de afgeleide naar x nemen, dan is dit een eigenfunctie, aangeziend

dxex D ex , de functie zelf is.Als we een eigenfunctie in een differentiaaloperator stoppen, komt er een constant getal

keer de eigenfunctie uit. Dit getal heet de eigenwaarde van de eigenfunctie. De verzamelingvan alle mogelijke eigenwaarden heet het spectrum.

Voor een snaar geldt dat, met de juiste randvoorwaarden (de uiteindes van de snaarmogen niet uitwijken, die zitten vast), er voor elk geheel getal k � 0 een oplossing uk vandeze vergelijking is, en dus een trilling, waar er k punten (behalve de uiteindes) “stil” staan.Voor elk zo’n k is deze golffunctie uk een eigenfunctie van de differentiaaloperator, met een

75

76 Samenvatting

eigenwaarde lk . Wat dit getal lk precies is, hangt af van de spanning, lengte en materiaalvan de snaar. Het is wel altijd zo dat lk proportioneel met k groeit: als k twee keer zo grootwordt, dan wordt lk ook twee keer zo groot (als k groot genoeg is). Dit groeigedrag isspecifiek voor snaren: voor andere dingen, zoals trommels, zal dit anders zijn.

Snaartrillingen met nul, een en twee stilstaande punten

Andere waarnemingen van de wereld om ons heen blijken ook vaak te bestaan uitspectra: zo kunnen we bepalen welke stoffen er in sterren zitten, door te kijken naarwelke spectraallijnen het licht dat ze uitzenden bevat. Elk element zendt licht uit inbepaalde golflengtes, bepaald door eigenwaarden van een differentiaaloperator. Het elementhelium is ontdekt doordat men in het spectrum van de zon een spectraallijn vond die nietbij een bekend element hoorde. Ook de roodverschuiving, waarmee we de afstand vanverre melkwegstelsels kunnen berekenen, wordt bepaald door te kijken naar diezelfdespectraallijnen.

400nm 700nm

400nm 700nm

400nm 500nm 600nm 700nm

De spectraallijnen van de elementen waterstof (boven) en helium (onder) in het zichtbare licht: linksde korte golflengtes (paars), rechts de lange golflengtes (rood)

Ook in de deeltjesfysica, de theorie die de kleinste deeltjes waaruit wij bestaan beschrijft,zijn spectra overal te vinden. Een deeltje is zelfs niets anders dan een eigenfunctie vaneen bepaalde differentiaaloperator, in dit geval de Hamiltoniaan. Het spectrum van dezeoperator geeft dan de mogelijke energiewaardes die een deeltje kan hebben. De kleinstepositieve eigenwaarde is de massa van het deeltje.

In de niet-commutatieve meetkunde wordt een dergelijke spectrale aanpak ook voormeetkunde gebruikt. Er is bekend dat alleen het spectrum niet genoeg is om alle meet-kunde te beschrijven: er zijn objecten die meetkundig verschillend zijn, maar hetzelfdespectrum hebben. De vraag “Kunnen we de vorm van een trommel horen?”, gesteld door dewiskundige Salomon Bochner, en bekend gemaakt door Mark Kac in 1965, moet negatiefbeantwoord worden: er zijn trommels met verschillende vormen, die hetzelfde klinken. Omtoch de hele meetkunde op deze manier (dus spectraal) te beschrijven bedacht Alain Connesde spectrale drietallen, waar naast de differentiaaloperator die het spectrum geeft, in ditgeval de Dirac-operator, ook de algebra van gladde functies in staat. De Dirac-operator

Resultaten uit dit proefschrift 77

werkt niet direct op deze functies, maar allebei werken ze op de kwadratisch integreerbarefuncties, die de derde component van het drietal vormt. Een spectraal drietal .A;H ;D/,met A de algebra van functies, H de kwadratisch integreerbare functies, en D de Dirac-operator, met extra condities op de objecten van het drietal, bevat alle meetkunde van eenruimte. Dit is in 2008 door Alain Connes bewezen.

De algebra van gladde functies bepaalt de grove vorm van de ruimte: uit hoeveellosse stukken bestaat deze, zitten er gaten in, enzovoort. Als we over een klassiekeruimte praten, vormen de functies een commutatieve algebra. We kunnen deze functiesoptellen, vermenigvuldigen met getallen en vermenigvuldigen met andere functies, en dezevermenigvuldiging is commutatief, oftewel de volgorde van vermenigvuldigen maakt nietuit. De optelling en vermenigvuldiging gebeurt puntsgewijs, dat wil zeggen dat de waardevan de functie f Cg in het punt x van de ruimte wordt gegeven door de waarde van f inhet punt x, plus de waarde van g in het punt x:

.f Cg/.x/D f .x/Cg.x/;

.a �f /.x/D a �f .x/;

.f �g/.x/D f .x/ �g.x/;

.f �g/.x/D .g �f /.x/;

waar f en g functies zijn, x een punt in de ruimte, en a een getal.Het werk van Alain Connes laat zien dat we niet alleen naar concrete algebra’s van

functies op een klassieke ruimte hoeven te kijken om meetkunde te doen. Het is voldoendeom dit met een abstracte algebra te doen. De vermenigvuldiging van zo’n algebra kanniet-commutatief zijn, dat betekent dat de volgorde van vermenigvuldiging wel uitmaakt.Het blijkt dat er een sterke correlatie bestaat: voor het type algebra’s waar we naar kijken,C �-algebra’s, is een commutatieve algebra altijd een algebra van functies met de punts-gewijze optelling en vermenigvuldiging. Bij een niet-commutatieve algebra hoort geenruimte meer die bestaat uit punten. Een spectraal drietal met een niet-commutatieve algebrabeschrijft volgens deze aanpak een niet-commutatieve ruimte. Dit lijkt in eerste instantieonnodig, wat betekent meetkunde immers zonder punten, maar het blijkt van nut te zijn inuiteenlopende takken van de wiskunde en natuurkunde. Daarnaast geeft deze – spectrale –kijk op meetkunde nieuwe inzichten in de gewone – commutatieve – meetkunde.

Resultaten uit dit proefschrift

In mijn proefschrift bestudeer ik verschillende soorten niet-commutatieve ruimtes, enbeschrijf ik mogelijke equivalenties ertussen. In de eerste plaats kijk ik naar zogenoemdeniet-commutatieve tori. Een voorbeeld van een (commutatieve) torus is het oppervlak vaneen fietsband. Deze band is een tweedimensionale torus. Een torus kunnen we maken doorde tegenoverliggende randen van een vierkant of parallellogram aan elkaar te plakken, eersttot een cilinder, daarna de uiteinden van de cilinder weer aan elkaar tot een torus.

Een driedimensionale torus kunnen we op dezelfde manier maken door de randen vaneen kubus (of parallellepipedum) aan elkaar te plakken. Hogerdimensionale tori kun jekrijgen door hogerdimensionale kubussen (in vier of meer dimensies) aan elkaar te plakken.

78 Samenvatting

De opbouw van een torus: eerst plakken we de bovenste en onderste rand van het vierkant aan elkaar,daarna plakken we de uiteinden van de verkregen cilinder aan elkaar

Functies op een torus kunnen we dan beschouwen als functies op een vierkant, kubus, etc.,die met dat plakken weer een nette functie opleveren. De waardes van de functie op deuiteinden die aan elkaar geplakt worden, komen dan overeen.

Om een spectraal drietal te vormen, hebben we ook een Dirac-operator nodig. DeDirac-operator die bij de torus hoort geeft informatie over de grootte en vorm van hetparallellogram. In het commutatieve geval horen deze onlosmakelijk bij elkaar: als weweten dat de functies gegeven worden door de functies op een parallellogram dan kennenwe de algemene vorm van de Dirac-operator. In dit proefschrift heb ik laten zien dat ookvoor een niet-commutatieve algebra gebaseerd op de algebra van functies van de torus, eenzogenoemde � -deformatie, de Dirac-operator van het spectrale drietal hetzelfde moet zijnals de Dirac-operator van een commutatieve torus.

Het tweede soort ruimtes waar ik naar heb gekeken zijn lensruimtes. De commutatievelensruimtes zijn als volgt te beschrijven. Als we een bol bekijken, kijken we naar eendriedimensionale ruimte. Als we alleen naar de rand ervan bekijken noemen we dat eentwee-sfeer, het is immers op hele kleine schaal een (tweedimensionaal) vlak, zoals hetaardoppervlak ook vlak lijkt als je het van dichtbij bekijkt.

Een polder aan de Groeneweg in Oud Zuilen: het lijkt vlak, maar buigt mee met het aardoppervlak

Resultaten uit dit proefschrift 79

In vier dimensies kunnen we ook een bol beschrijven, en de rand hiervan noemen weeen drie-sfeer. Als we een gewone bol in drie dimensies ronddraaien, zullen er altijd puntenvan de twee-sfeer op zichzelf worden afgebeeld. Dit zien we bijvoorbeeld bij de aarde dieom zijn as draait, waarbij de Noord- en Zuidpool op hun plek blijven. Voor de drie-sfeer isdat niet het geval: als we om één as ronddraaien hebben we nog steeds twee vaste punten,maar als we deze om meerdere assen tegelijk draaien, bijvoorbeeld om één as een vijfdevan een hele ronde, en om een tweede as tegelijkertijd twee vijfde van een hele ronde, danwordt er geen enkel punt vastgehouden. Na vijf keer deze rotatie uit te voeren, is de sfeerweer in de oorspronkelijke positie terug. We kunnen een aaneengesloten stuk van de bolvinden, waarvoor voor elk punt op de drie-sfeer geldt dat er precies één punt van dit stukna een aantal rotaties op wordt afgebeeld. Dit stuk van de bol noemen we de lensruimteL.5I1;2/, waar de 5 staat voor hoe vaak we moeten roteren om de oorspronkelijke bolterug te krijgen, en de 1 en de 2 voor de verhoudingen van de rotatie om de twee assen: éénslag om de ene as, twee slagen om de andere as.

Voortbouwend op een eerdere constructie van een spectraal drietal op een “kwantum-sfeer” door een groep wiskundigen, worden in dit proefschrift alle spectrale drietallen opkwantum-lensruimtes geconstrueerd. De methode van classificatie wijkt af van die voorniet-commutatieve tori, aangezien er voor de kwantumsfeer strikt genomen geen spectraaldrietal bestaat: we moeten ofwel loslaten dat het spectrum hetzelfde is als de gewonesfeer, ofwel enkele condities voor een spectraal drietal loslaten. In de classificatie vankwantum-lensruimtes beginnen we met het spectrum van de sfeer als basis, dit heeft totgevolg dat er bepaalde eigenschappen van spectrale drietallen verloren gaan.

Daarnaast richtte mijn onderzoek zich op de vraag wanneer twee niet-commutatieve“ruimtes” equivalent oftewel “hetzelfde” zijn. Een equivalentierelatie heeft als eigen-schappen dat een object equivalent met zichzelf moet zijn (reflexief ), dat als een object aequivalent is met b en b equivalent is met c, a ook equivalent met c moet zijn (transitief ),en dat als a equivalent is met b, b ook equivalent moet zijn met a (symmetrisch).

Er bestaan verschillende voorstellen voor equivalenties in de niet-commutatieve meet-kunde. In het eerste voorstel dat in dit proefschrift bekeken wordt, is het van belang datde functie-algebra’s overeenkomen, isomorf zijn. Twee algebra’s A en B zijn isomorfals er een één-op-éénafbeelding h is die de structuur behoudt, in het bijzonder in de ver-menigvuldiging en optelling. Als de algebra’s commutatief zijn, hebben de ruimtes waarze bij horen dezelfde vorm, hetzelfde aantal stukken, gaten, enzovoorts, als hun functie-algebra’s isomorf zijn. Ik heb gekeken hoe de structuren op de niet-commutatieve tori enkwantum-lensruimtes die in dit proefschrift zijn beschreven, samenhangen. In het geval vande tweedimensionale torus wordt in dit proefschrift een definitief antwoord gegeven, in deandere gevallen (hogerdimensionale tori en de lensruimtes) er nog geen goede beschrijvingvan alle isomorfismes bestaat.

Daarnaast heb ik een tweede type equivalenties bestudeerd, de Morita-equivalenties,vernoemd naar de Japanse wiskundige Kiiti Morita. Deze gaat niet uit van een isomorfismevan algebra’s, maar van isomorfisme van representaties van algebra’s. Als we naar dedefinitie van een spectraal drietal kijken, een algebra A, een Dirac-operator D, die allebeiwerken op kwadratisch integreerbare functies H , zien we dat niet de algebra A zelf vanbelang is, maar hoe deze algebra op de ruimte van kwadratisch integreerbare “functies”H werkt. Dit is de representatie van de algebra. Als twee algebra’s dezelfde structuur

80 Samenvatting

van representaties hebben, dan zijn ze Morita-equivalent. Voor commutatieve algebra’sis dit niets anders dan isomorf zijn, maar voor niet-commutatieve algebra’s hoeft dit nietzo te zijn.

Er is alleen een probleem met deze Morita-equivalenties. Wat de algebra’s betreftspreken we bij Morita-equivalenties echt van equivalentierelaties, die zijn altijd én reflexief,én transitief, én symmetrisch. Als we het begrip echter uitbreiden naar het hele spectraledrietal is er een probleem: de manier waarop de Dirac-operator verandert is niet altijdsymmetrisch! Er zijn gevallen bekend van niet-commutatieve spectrale drietallen, waarbij ereen Morita-equivalentie de ene kant op is, maar niet terug de andere kant op. Een dergelijkeMorita-equivalentie is dus niet symmetrisch. In het algemeen is niet bekend of en wanneereen Morita-equivalentie van spectrale drietallen symmetrisch is. In dit proefschrift wordtvoor het specifieke geval van niet-commutatieve tori bewezen dat Morita-equivalenties welsymmetrisch zijn.

Draadloze communicatie

De schoonheid van wiskunde ligt voor een deel in de onverwachte resultaten. Uit declassificatie van niet-commutatieve tori volgt een resultaat over bepaalde types matrices.Dit resultaat hangt samen met het sturen van signalen over kanalen met ruis, zoals in dedraadloze communicatie. Een onverwachte toepassing, zonder dat ik er naar op zoek was.

We noemen een matrix A inverteerbaar als er een inverse, A�1 bestaat, een matrixzodanig dat A �A�1 D A�1 �AD Id. In het algemeen is een matrix A inverteerbaar als hetbeeld en domein van de matrix gelijke dimensie hebben en voor elke twee vectoren x;y inhet domein geldt dat Ax ¤Ay als x ¤ y. We kunnen ons nu afvragen wanneer de som vaninverteerbare matrices weer inverteerbaar is. Om de vraag iets algemener te stellen: bestaaner matrices A1; : : : ;An zodanig dat x1A1C : : :CxnAn inverteerbaar is, met x1; : : :xn 2R?Dit is een generalisatie van een bekend probleem in de lineaire algebra: bepaal of devectoren y1; : : : ;yn lineair onafhankelijk zijn, oftewel: is de combinatie x1y1C : : :Cxnyn

nooit nul? Het probleem met matrices is iets moeilijker, aangezien we “niet nul” hebbenvervangen door “inverteerbaar”.

Het eerste dat opvalt is dat als x1 D x2 D �� D xn D 0, dit niet het geval kan zijn, ditsluiten we dus uit. We kunnen ons afvragen hoe groot de verzameling van onafhankelijkematrices fA1; : : : ;Ang kan zijn (dus zodat A1x1C �� CAnxn inverteerbaar is voor allex1; : : : ;xn in R die niet allemaal tegelijk nul zijn).

In het geval van lineaire onafhankelijkheid van vectoren is de oplossing bekend: er zijnhooguit n onafhankelijke vectoren in een n-dimensionale vectorruimte. Voor het geval vanmatrices is het resultaat wat ingewikkelder, en moeten methoden van buiten de lineairealgebra worden gebruikt:

Stelling (Adams, Lax en Phillips, 1965). Schrijf nD .2aC 1/2b . Dan is het maximaalaantal complexe n�n matrices Ai waarvoor

PxiAi inverteerbaar is 2bC2.

Het bijbehorende bewijs van deze stelling is verrassend: het maakt gebruik van eenstelling uit de algebraïsche topologie, een compleet ander onderdeel van de wiskunde. Pre-ciezer: de stelling is gebaseerd op Adams’ classificatie van vectorvelden op n-dimensionale

Draadloze communicatie 81

sferen. In 1962 bewees Adams dat als n D .2aC 1/2cC4d met 0 � c < 4, dan is hetaantal lineair onafhankelijke vectorvelden op de eenheidssfeer in n dimensies 2cC 8d .Met behulp van enkele slimme afschattingen kan hieruit het resultaat van Adams, Lax enPhillips worden bewezen. Het resultaat van Adams over vectorvelden op bollen is een grotegeneralisatie van een oudere stelling van Brouwer:

Stelling (Brouwer, 1912). Ieder glad vectorveld op een sfeer in 3 dimensies is ergens nul.

Deze stelling wordt ook wel geparafraseerd als “je kunt het haar op een kokosnoot nietkammen”.

Elke manier van kammen op een bol levert een kruin op

Het resultaat van Adams, Lax en Phillips lijkt behoorlijk abstract, wat kun je nu metdie matrices? Een onverwachte toepassing kwam in de jaren ’90, met de zogenoemderuimte-tijdblokcodes. Deze codes vormen een methode om op een efficiënte manier metmeerdere antennes informatie draadloos te versturen.

De eerste van dit soort types codes werd bedacht door Alamouti, en is inmiddels onder-deel van verschillende standaarden voor draadloze netwerken, zoals de 802.11n-standaarddie moderne routers gebruiken. Het idee is om meerdere antennes tegelijk te laten zenden,dit signaal op verschillende manieren een aantal keer achter elkaar te versturen, en danaan de ontvangstzijde het signaal te decoderen. Om zo goed mogelijk aan de eisen vande consument te voldoen is het noodzakelijk om zo veel mogelijk informatie in zo weinigmogelijk tijd te versturen, met zo min mogelijk kans op fouten. Een andere eis is datde antennes om te versturen dicht bij elkaar staan (om de installatie van het systeem tereduceren tot het neerzetten van één wireless router), en dat bij ontvangst maar één antennenodig is (om laptops en tablets te ondersteunen).

De methode werkt als volgt: neem een signaal c, breek dat op in n stukken c1; c2; : : : ; cn.We versturen dit signaal over k antennes in m stappen als volgt:

antennes!

tij

d Pi ci .Ai /11 : : :

Pi ci .Ai /1k

:::: : :P

i ci .Ai /m1 : : :P

i ci .Ai /mk

82 Samenvatting

Een draadloos modem met meerdere antennes

Hierbij is fAigniD1 een verzameling van m�k-matrices. Het signaal op stap t is dan de

te rij van de somP

k ckAk plus nog de mogelijke verstoring. Met wat matrixalgebra en dekennis van matrices Ai kan het meest waarschijnlijke signaal hieruit worden herleid. Defout wordt zo klein mogelijk, als voor twee verschillende signalen c1c2 : : : ck en e1e2 : : : ek

de matrix Xi

.ci � ei /Ai ;

maximale rang heeft. Als de Ai vierkante k�k-matrices zijn, kan dit criterium ook wordenopgevat als:

Pi xiAi is inverteerbaar als tenminste een van de xi niet nul is. Hierbij nemen

we aan dat de xi reëel zijn. De resultaten van Adams, Lax en Phillips leveren dus eenbovengrens op voor efficiëntie van deze methode. Nemen we voor de Ai k�k Hermitischematrices, dus k antennes in k stappen, dan kunnen we hooguit 2bC1 codes versturen, metk D .2aC1/2b . We gaan ervan uit dat de antennes alleen reële signalen kunnen sturen,maar we coderen een complex getal xCIy als volgt:

x y

�y x

!:

Met het decoderingsmechanisme kunnen we nu met één antenne eerst xCy versturen, endaarna x�y, en dit decoderen als xC Iy. Het versturen van een complex signaal kostdus twee keer zoveel tijdstappen. De code van Alamouti is een complexe 2�2-methode,dus twee antennes in vier stappen. Als we naast Hermitische matrices ook algemenematrices toestaan krijgen we één extra mogelijke matrix uit de resultaten van Adams, Laxen Phillips, dus met k D 2b zijn er maximaal 2bC2 matrices. De methode van Alamoutiis een voorbeeld van zo’n verzameling matrices en kan dus in vier tijdstappen vier signalenversturen. Dit heeft een efficiëntie van 1 (signalen/tijdstappen). Er kan bewezen worden datdit de maximale efficiëntie is en dat dit alleen mogelijk is voor complexe 2�2-matrices.

Een methode om matrices te construeren die voor een bepaalde grootte een maximaleefficiëntie bereiken (en dus een maximale verzameling van matrices is waarvoor de somaltijd inverteerbaar is), is gebruik te maken van Clifford-algebra’s. Een Clifford-algebra

Draadloze communicatie 83

is een algebra gegenereerd door elementen xj , zodanig dat xjxkCxkxj D 2g.xj ;xk/Idwaarbij g.;/ een symmetrische vorm is (g.x;y/D g.y;x/, en g.x;y/ 2C) die lineair isin beide argumenten en positief-definiet (g.x;x/� 0 voor alle x en g.x;x/D 0 alleen alsx D 0). Een korte berekening laat zien dat als de generatoren van deze Clifford-algebraHermitisch zijn, dit een verzameling matrices oplevert waarvan de som van maximale rangis en dat de verzameling maximaal is voor de grens van Adams, Lax en Phillips. Een vraagdie dan rijst is: zijn alle voorbeelden van maximale verzamelingen matrices te reconstruerenuit Clifford-algebra’s?

Dit blijkt niet het geval te zijn, er zijn maximale verzamelingen matrices waarvan desom altijd inverteerbaar is maar die geen Clifford-algebra genereren. Een volgende vraagdie men kan stellen is dan: “Wat zijn voldoende extra condities aan maximale verzamelingenmatrices waarvan de som altijd inverteerbaar is, zodanig dat de matrices de generatorenvan een Clifford-algebra zijn?”

Deze vraag kwam op een gegeven moment ook naar voren bij mijn onderzoek naarde classificatie van niet-commutatieve tori. De Dirac-operator, die de meetkunde bepaalt,bleek onder andere opgebouwd uit matrices, die aan bepaalde eisen moesten voldoen. Hetwas nodig om te bepalen wat voor soort matrices aan bepaalde eisen voldeden. De eersteeis was dezelfde als die voor de optimale matrices voor de ruimte-tijdblokcode:X

i

xiAi inverteerbaar als tenminste een van de xi ¤ 0:

De tweede eis is de volgende:X�2Sn

teken.�/nY

iD1

A�.i/ D xId;

waarbij Sn de symmetrische groep van n elementen is, x een getal ongelijk 0 en Id deidentiteitsmatrix. De linkerzijde van deze vergelijking is een “antisymmetrische som”: voornD 2 staat hier bijvoorbeeld A1A2�A2A1, en voor nD 3:

A1A2A3�A2A1A3CA2A3A1�A3A2A1CA3A1A2�A1A3A2;

enzovoorts. Als we met een commutatieve torus van doen hebben, is bekend dat dezematrices een Clifford-algebra moeten genereren. Nu kunnen we het bewijs van de clas-sificatie tot op dit punt bekijken, en dan blijkt dat we nergens gebruik hebben gemaaktvan de (niet-)commutativiteit van de functies. Verder zijn de eisen aan de matrices het-zelfde voor de commutatieve en de niet-commutatieve torus. Nu kunnen we gebruikmakenvan de reconstructiestelling van Alain Connes (bewezen in 2008), die zegt dat we uiteen commutatieve algebra van functies, samen met een Dirac-operator, de onderliggenderuimte en bijbehorende meetkunde kunnen reconstrueren. Dat betekent in dit geval dat deDirac-operator die aan deze eisen voldoet de al bekende Dirac-operator op een torus is,en dus dat de matrices die aan deze twee eisen voldoen een Clifford-algebra genereren.We hebben dus de vraag beantwoord die hiervoor werd gesteld: de extra eis die nodig is,is de conditie op de antisymmetrische som. Ik ken hiervan geen rechtstreeks bewijs doormiddel van lineaire algebra. De classificatie van niet-commutatieve tori, een resultaat in deniet-commutatieve differentiaalmeetkunde, wordt hier dus gebruikt om een resultaat in delineaire algebra te bewijzen.

Dankwoord

This thesis would not have been possible without the help, encouragement and guidance bya lot of people.

Op de eerste plaats wil ik mijn begeleider, Gunther Cornelissen bedanken. Zonder hemzou dit proefschrift er niet zijn. Niet alleen in de zin dat er zonder begeleider geen promotiekan plaatsvinden, maar zijn invulling van die taak was ook nog eens geweldig. Naast allehulp bij de wiskunde en het schrijven, was hij ook van onschatbare waarde bij het vindenvan een baan na mijn promotie. Ik had me geen betere begeleider kunnen wensen.

The members of the thesis committee, Erik van den Ban, Alan Carey, Klaas Landsman,Giovanni Landi and Matilde Marcolli, made this thesis much more readable with manyhelpful suggestions. Andrzej Sitarz provided the basis of the last chapter of this thesisand I would also like to thank him for his hospitality during my stay in Krakow at theUniwersytet Jagiellonski.

Thankfully, in contrast to the popular bias, mathematical research is not a solitary occu-pation: I have had the fortune to have stimulating discussions with many mathematicians,of which I would especially like to thank Bram Mesland, Walter van Suijlekom and JorgePlazas Vargas, besides the others already mentioned.

Wat dichter bij huis wil ik mijn (voormalige) kamergenoten bedanken, Jaap, Bas, Arjenen Sanjay. Hoewel de gezelligheid op de korte termijn de wiskunde wel eens in de wegstond, droeg het op de lange termijn zeker veel bij aan mijn wiskundige opvoeding, mededoor alle discussies over vormen van dobbelstenen, de techniek achter 3D-engines, de“beste” manier om zaken in LATEX vorm te geven en de spelling en grammatica van hetNederlands tegenover die van het Engels.

Bart en Job zorgden niet alleen voor heerlijke koekjes bij de koffie ’s ochtends, maargaven ook uitleg over verschillende wiskundige concepten die aan de grondslag liggen vanenkele lemma’s in dit proefschrift.

Together with Janne, Jan Willem, Sebastiaan and Jakub I shared the privilege of havingGunther as an advisor, and if I could not find him, at least one of you always knew wherehe was. Our mathematical topics were very much apart, but the discussions we had werevery fruitful.

The Mathematical Institute in Utrecht was a wonderful place to work, very much duethe many friendly people that work there. I have many fond memories of discussing a vastrange of subjects during lunch, varying from a revolutionary concept for securing shoppingcarts in supermarkets, to the infamous shaving of the kiwifruit, but also the various ways ofteaching mathematics, with Arthur, Tammo-Jan, Charlene, Aleksandra, Albert-Jan, Vincent,

85

86 Dankwoord

Jantien, Sebastiaan Janssens, Jeroen, Bas Janssens, Esther, Wilfred, Wouter, KaYin, Dana,Ionut, Roy, Anshui, Lee, Boris, Maria, Timo, João, Inan, Viktor and Ori. I really enjoyedour trips together, such as the trip to Berlin and the mountainbike trip in the “Loonse enDrunense Duinen”.

Simone Baddou verzorgde voortreffelijk het ontwerp voor de omslag en de uitnodiging.Al mijn vrienden en familie: ik ga jullie missen de komende twee jaar, hopelijk komenjullie langs. Ik wil mijn ouders en Niels bedanken voor alle aandacht, en de moeite die zededen om te begrijpen waar ik mee bezig was. Hopelijk draagt dit boekje een beetje bij aanhet begrip.

En, natuurlijk, Louise. Jouw lach maakt het allemaal waard.

Curriculum Vitae

Jan Jitse Venselaar werd op 21 december 1981 geboren te Bandung in Indonesië. Hijgroeide op in Leusden en Zutphen. In 2000 haalde hij zijn VWO-diploma aan het StedelijkDaltoncollege Zutphen. Aansluitend studeerde hij Wiskunde en Natuurkunde in het TWIN-programma aan de Universiteit Utrecht. In september 2008 rondde hij beide studies af meteen scriptie over Seiberg-Witten-theorie en Donaldson-theorie door middel van equivariantelokalisatie, onder begeleiding van Luuk Hoevenaars en Eduard Looijenga. In januari 2009begon hij met een promotie aan de Universiteit Utrecht onder supervisie van GuntherCornelissen, wat uiteindelijk in augustus 2012 tot dit proefschrift leidde. Tijdens dezeperiode begeleidde hij ook een aantal werkcolleges, waarvoor hij in 2010 door de studententot “AiO van het jaar” werd gekozen.

Jan Jitse Venselaar is vanaf september 2012 werkzaam als postdoc bij het CaliforniaInstitute of Technology in de Verenigde Staten, in de onderzoeksgroep van Matilde Marcolli.

87

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Index

SUq.2/, 9, 58

charge conjugation, 2, 3, 5Clifford algebra, 2, 31Clifford group, 2compact resolvent, 11connection, 46

dimension, 11, 12Dirac operator, 3, 4, 11, 24

spin geometry, 3

equivalence bimodule, 46

flip automorphism, 39frame bundle, 2

grading, 3, 11, 12

Hilbert C�-module, 46Hochschild homology, 13Hopf algebra, 15

su.2/, 58torus, 18

Leibniz rule, 46lens space, 9, 62

quantum, 9, 62, 64spectrum, 67, 69

Morita equivalence, 7C�-algebra, 46algebra, 45circle, 47finite spectral triple, 49noncommutative torus, 49real spectral triple, 47strong, 46transitive, 48

noncommutative torus, 6, 17algebra, 19automorphisms, 39connection, 54Morita equivalence, 8, 53, 55real spectral triple, 6, 33smooth structure, 31volume form, 27

one-form, 46opposite algebra, 5, 12orientability, 13

Poincaré duality, 13pre-C�-algebra, 5

reality operator, 5, 11regularity, 13

spectral geometry, 9, 64spectral triple, 11

equivariant, 15, 16even, 12finite, 49irreducible, 14real, 5, 11, 13

spin geometry, 2spin structure, 2, 17Spin.n/, 2Spinc , 3spinor bundle, 3

unitary equivalence, 7, 38lens space, 70noncommutative torus, 7, 40

95

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