Quantumevenspheres.pdf

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arXiv:math.Q

A/0

211462v1

29No

v2002

Quantum even spheres 2n

qfrom Poisson double suspension

F.Bonechi1,2, N.Ciccoli3, M.Tarlini1,2

May 3, 2004

1 INFN Sezione di Firenze2

Dipartimento di Fisica, Universita di Firenze, Italy.3 Dipartimento di Matematica, Universita di Perugia, Italy.

e-mail: [email protected], [email protected], [email protected]

Abstract

We define even dimensional quantum spheres 2nq that generalize to higher

dimension the standard quantum two-sphere of Podles and the four-sphere

4q obtained in the quantization of the Hopf bundle. The construction relies

on an iterated Poisson double susp ension of the standard Podles two-sphere.

The Poisson spheres that we get have the same symplectic foliation consisting

of a degenerate point and a symplectic plane and, after quantization, havethe same Calgebraic completion. We investigate their K-homology and

K-theory by introducing Fredholm modules and projectors.

Math.Subj.Classification: 14D21, 19D55, 58B32, 58B34, 81R50

Keywords: Noncommutative geometry, quantum spheres, quantum suspension,Poisson geometry.

1 Introduction

In the seminal paper [13] by Podles on SUq(2)-covariant quantum two-spheres itwas introduced a family of deformations S2q,d , depending on d R and q < 1. Theydefine three distinct quantum topological spaces S2q,d : the case d = 0, the so calledstandard sphere, d > 0, the non standard sphere, and d = q2n, n = 1, 2 . . ., theexceptional case.

The geometry of these quantum spaces can be nicely interpreted by lookingat the underlying Poisson geometry and considering the sphere as a patching of


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symplectic leaves. There exists, in fact, a 1parameter family of SU(2)covariantPoisson bivectors on the sphere S2 such that S2q,d can be seen as a quantization ofsuch structures ([15]). In the case d = 0 the symplectic foliation is made of a pointand a symplectic R2; after quantization, the symplectic plane is quantized to K, the

algebra of compact operators, and the degenerate point survives as a character. TheC-algebra C(S2q,0) is then isomorphic to the minimal unitization of compacts K, andsatisfies the following exact sequence 0 K C(S2q,0) C 0. In the case d > 0symplectic foliation consists of an S1-family of degenerate points, and two symplecticdisks. After quantization C(S2q,d) satisfies 0 K K C(S

2q,d) C(S

1) 0.The exceptional spheres correspond to the symplectic case and after quantizationC(S2q,q2n) is isomorphic to the finite dimensional algebra Mn(C) of matrices.

In higher dimension, there exist the so called euclidean spheres Snq introducedin [8] as quantum homogeneous spaces of SOq(n + 1); let us notice that in odddimension they coincide with the so called Vaksman-Soibelman spheres S2n+1q ([18])

introduced as quantum quotients Uq(n)/Uq(n 1), as observed in [10].This family of spheres represents a generalization of the Podles d = 1 case. In

fact they satisfy the following exact sequences (see [10, 11, 18])

0 K K C(S2nq ) C(S2n1q ) 0

0 C(S1) K C(S2n+1q ) C(S2n1q ) 0 .

This behaviour reflects exactly the Poisson level where each S2n contains an equatorS2n1 as a Poisson submanifold and the two remaining hemispheres are open sym-plectic leaves of dimension 2n. In particular all these spheres contain an S1-family

of degenerate points.

As there are no symplectic forms on higher dimensional spheres, exceptionalspheres are possible only in dimension 2.

In this paper we introduce the family of even spheres 2nq which generalize thed = 0 Podles sphere: they all share with it the same kind of symplectic foliation.The idea of the construction relies on two classical subjects: double suspension andPoisson geometry.

The suspension idea is certainly not new and, in fact, appears in many of thepapers devoted to the construction of particular deformations of the four-sphere

([5, 3, 6, 16]). When one considers double suspension, however, more interestingpossibilities appear: on one hand one could define a purely classical double suspen-sion already at an algebraic level by adding a pair of central selfadjoint generatorsand modding out suitable relations. A different kind of double suspension was con-sidered, at the Calgebra level, in [11]. There the authors consider the non reduceddouble suspension of a Calgebra A as the middle term S2A of a short exact se-quence

0 A K S2A C(S1) 0

for a suitable fixed Busby invariant. Let us remark that such a non reduced doublesuspension always has a naturally defined S1family of characters. All euclidean

2


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spheres were reconstructed in this way starting either from a twopoint space orfrom S1.

In this paper we will consider the reduced double suspension or reduced topo-logical product S2X = S2 X/S2 X where, given p S2 and q X, S2 X =

(S2 q) (p X).

It is quite natural to look at the interaction between the double suspension andPoisson geometry. One word has to be said about the fact that while suspension isan essentially topological construction, Poisson bivectors are of differential nature,so that, in principle, on the double suspension of a given manifold theres no reasonto have a manifold structure, let aside a Poisson bracket. Still whenever the manifoldstructure is there one can ask whether such a Poisson structure arises. More preciselythe double suspension of a manifold M can be seen as a topological quotient S2MofS2 M. If we are given a Poisson bivector on the 2sphere and a Poisson bivectoron M we can ask whether the quotient map S2 M S2M coinduces a Poisson

bracket on the quotient. If this is the case we then look for its quantization.

From this point of view the classical double suspension quantizes a double sus-pension with respect to the trivial Poisson structure on S2 while HongSzymanskiconstruction corresponds to the standard symplectic structure on R2 attached alongan S1 which will survive as a family of 0leaves on the suspension.

In this paper the double suspension is built by assuming the Podles d = 0 Poissonstructure on the two-sphere and considering S2n as the double suspension ofS2(n1).We will show that a Poisson double suspension of spheres exists for each n and thatit quantizes both at an algebraic level and at the Calgebra level. The spheres 2nq

that we obtain have all the same symplectic foliation of the two-sphere, and theirquantizations are topologically equivalent, i.e. they are the minimal unitization ofcompacts and satisfy

0 K C(2nq ) C 0 .

We then have a quite extreme case ofquantum degeneracy: these quantum spaces,whatever is the classical dimension, are all topologically equivalent to a zero dimen-sional compact quantum space. This is an extreme manifestation of a well knownfact, that quantum spaces associated to quantum groups have lower dimension thanthe classical one. Moreover this reminds of canonical quantization in which the Weylquantization of C0(R

2n) is K, for each n. The opposite behavior is represented by

the so called -deformation, whose behavior is almost classical (see [5, 4]).

In the case of the four-sphere 4q, the algebra that we get is that obtained in[1, 2], in the context of a quantum group analogue of the Hopf principal bundleS7 S4. It is unclear whether all even spheres 2nq can be obtained as coinvariantsubalgebras in quantum groups.

In Section 2 we introduce the Poisson double suspension that iteratively definesthe Poisson even spheres and we study their symplectic foliation. In Section 3we introduce the quantization Pol(2nq ) at the level of polynomial functions, weclassify the irreducible representations in bounded operators and then show that the

3


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universal C-algebra C(2nq ) is the minimal unitization of compacts. We introduceFredholm modules for each of these spheres. In Section 4, we give the non trivialgenerator ofK0(

2nq ) and compute its coupling with the character of the previously

introduced Fredholm modules.

2 The standard Poisson structure.

Let us define a point on S2. . .S2 by giving to it coordinates ((1, 1), . . . (n, n)),where |i|

2 = i(1 i). Let M be the matrix whose entries are Mii = 0, Mij = 1and Mji = 1/2 if i < j. Let

ai = ik Mikk t = i i . (1)

Since i

|ai|2 =

i

i(1 i)k

2Mikk = 1(1 1)22 . . .

2n+

2(1 2)123 . . .

2n + . . . + n(1 n)1 . . . n1

= t(1 t) ,

then relation (1) defines a projection into S2n. Let us call : S2 . . . S2 S2n

such projection. One can verify that this map is equivalent to the iterated reduceddouble suspension of a two-sphere, with preferred point the North Pole = = 0.In fact the map from the cartesian product S2X to the reduced topological productS2X is the unique continuous map which is a homeomorphism everywhere but onthe counter image of a point over which its fiber is S2 X. Starting from the two-sphere and iterating this procedure one defines a map from S2. . . S2 to S2n that is ahomeomorphism everywhere but on the North Pole, where its fiber is the topologicaljoin of n copies ofS2 . . . S2

n1

. This map is the projection .

Let us equip S2 with the standard Poisson structure, i.e. the limit structure ofthe Podles standard two sphere ([13, 15]), and S2 . . .S2 with the product Poisson

structure. The brackets among polynomial functions are defined by giving

{i, j} = 2ijii , {i, j} = 2ij(

2i i

i ) . (2)

We prove the following result.

Proposition 1 The map is a Poisson map. The coinduced brackets on S2n read:

{ak, a} = aka (k < ) , {ak, a} = 3aka

(k = ) ,

{ai, t } = 2ait , {ak, ak} = 2t

2 + 2


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Proof. The result is obtained by explicitly computing the brackets on S2 . . . S2.The only relation that deserves some attention is the last one. By direct computationwe obtain

{ak, ak} = 2

2k i

2Mkii 2akak .

Let us show by induction on k that

2ki

2Mkii = t2 +


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=

d2n . . . d 1 e

12

2(n1)ij S

(n)ij ije

2(n1)i=1 iJieS

(n)2n1,2n2n12n ,

where Ji = S(n)i,2n12n1 + S

(n)i,2n2n. By using standard rules for fermionic integration

(see for instance [14]) and taking into account that JiJj = 0 we get

P f(S(n)) = P f(S(n1))

d2nd2n1(1 + S

(n)2n1,2n2n12n)

= P f(S(n1))S(n)2n1,2n = P f(S

(n1)){zn, zn}

= 2P f(S(n1))(1 +n

|z|2) .

Since P f(S(1)) = 2(1 + |z|2) we conclude that P f(S(n)) = 0 and that R2n is sym-plectic.

The Poisson structure on the other chart R2n = S2n \ {t = 1} is symplectic every-where but the origin. In [19] Zakrzewski introduced a family of SU(n)-covariantPoisson structures on R2n with this foliation. It is an interesting problem to under-stand the relations between them.

3 Quantization of the standard Poisson structure.

The algebra Pol(S2q,0) of the Podles standard sphere is generated by {,

, } ,where is real, with the following relations:

= q2 , q2 = (1 ), = q2 + (1 q2)2 .

Let 0 < q < 1. There are two irreducible representations of Pol(S2q,0) with boundedoperators, the first is one dimensional () = () = 0, the second : Pol(S2q,0) B(2(N)) is defined by

()|n = qn1(1 q2n)1/2|n 1 ,()|n = q2n|n . (5)

Let us define n : Pol(S2q,0)n B(2(N)n) as the nthtensor product of the

representation , we denote by {i, i , i} the generators of the ith Pol(S2q,0).

Proposition 3 Let us define Pol(2nq ) the algebra generated by {ai, ai , t} with rela-

tions

ai t = q2t ai , ai aj = q

1aj ai , ai aj = q

3aj ai (i < j) ,

ai ai = q

2ai ai + q2(1 q2)


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The mapping n : Pol(2nq ) B(

2(N)n) given by

n(ai) = n(i

k

Mikk ) , n(t) = n(

i

i) , (6)

is a representation of Pol(2nq ).

Proof. From the relation n(kMikk ) = q

Mikn(Mikk k) where Mij is the matrixwith Mii = 0, Mij = 1 and Mji = 1/2 for i < j, it is straightforward to verify thefirst line of relations. In order to prove the relations in the second line we need thefollowing equality (we will omit the application of n):

2ik

2Mikk = t2 + q2


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Let Sp((t)) \ {0, 1} and let {s} be a set of approximate unit eigenvectors,i.e. unit vectors such that lims ||(t)s s|| = 0. By writing t t2 = (1 ) + (t )(1 t) we get

||(t t2

)s|| |(1 )| | |(t )(1 t)s|| |(1 )| | |(t )s|| ||(1 t)|| C|(1 )| ,

for s bigger than some so and for some C > 0. Moreover we have

||n

k=1

(akak)s|| n

k=1

||(ak)|| ||(ak)s|| Cn||(ak(s))s|| ,

where C and k(s) are such that ||(ak)|| C and ||(ak)s|| ||(ak(s))s|| for

all k. We conclude that for each s > so there exists 1 k(s) n such that||(ak(s))s|| C

|(1 )|. We can define s = (ak(s))s/||(ak(s))s|| and verify

that they are approximating unit eigenvectors for q2; in fact

||((t) q2)s|| =q2

||(ak(s))s||||(ak(s)(t ))s||

q2(1 )

C||(ak(s))|| ||(t )s|| .

We then showed that if Sp((t)) \ {0, 1} then q2 Sp((t)). In order to keepSp((t)) bounded it is necessary that for each there exists k such that q2k = 1,i.e. Sp((t)) \ {0} = {q2k, k N}. Since each q2k is isolated, we conclude that it isan eigenvalue.

Let the eigenvector corresponding to = 1: since it is the biggest eigenvaluewe get that (ai) = 0. By direct computation one recovers:

(ai)

j

(aj)mj = q3

jimjq2(mi1)(1 q2mi)

(

ji

a mjj ) . (7)

Let us define, with m = (m1, . . . , mn),

m = Cm (

ia mii ) , C

m = q(

imi)2+

imi(mi+1)/2i

(q2; q2)1/2mi ,

where (; q)s =s

j=1(1 qj1 ) and (; q)0 = 1. Formula (7) implies

||(ai )m||2 = q2

jimjq4

j>imj(1 q2(mi+1))||m||2 ,

from which we conclude that m = 0 for each m. The space generated by {m}is invariant and it coincides with H. Finally it can be verified that the mappingT : 2(N)n H defined by |m1, , mn

m intertwines the representations

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n and . Since one can verify that the ms are orthonormal, we conclude that T

is unitary.

The universal C-algebra generated by Pol(2nq ) is then the norm closure of

n(Pol(

2n

q )). Since n(ai) and n(t) are trace-class operators, then n(Pol(

2n

q ) \C) K. By using Proposition 15.16 of [7], which states that a norm-closed -subalgebra A of K, such that the representation A K is irreducible, coincideswith K, we prove the following result.

Corollary 6 The Calgebra generated by Pol(2nq ) is isomorphic to K, the mini-mal unitization of compacts.

Remark 7 The Calgebra of quantum even spheres is independent of the classicaldimension. This is not as strange as it may appear; the Calgebra level usually

reflects the topology of the space of leaves on the underlying Poisson bracket whichis the same in all cases.

Remark 8 As already hinted in the introduction, starting from any Calgebra Awith at least one character x and from a quantum two-sphere B with a character0 one could define a topological double suspension:

S2qA := {f A B (x id)(f) = (id 0)(f) C} . (8)

It is then a trivial remark that such construction applied to standard Podles sphereis stable. What is less trivial is the fact that such algebras quantize a whole family

of polynomial quantum even spheres.

Thanks to Corollary (6) we can conclude that K0(C(2nq )) = Z2 for each n, see

[12]; each polynomial sphere Pol(2nq ) will provide different representatives of thesame class in K-homology. Let us describe them explicitly, along the same lines of[12].

The first one [] is the pullback by : C(2nq ) C of the generator of K0(C).

By analogy with the classical case we say that its character computes the rank ofthe vector bundle.

Let us describe the second and more interesting generator. Let the Hilbertspace be H = 2(N)2 2(N)2 and =

n 0

0

, F =

0 11 0

. We have

that (H , , F ) is a 1summable Fredholm module whose character is the zero cycliccocycle trn = tr (n ). Let us denote with [trn ] its class; we say that trncomputes the charge. In particular, from (6) we get

trn(1) = 0 , trn(t) =1

(1 q2)n. (9)

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4 Algebraic projectors and the ChernConnes

pairing

Let us now come to the construction of non trivial quantum vector bundles onspheres. Since C(2nq ) = K we have that K0(C(2nq )) = Z

2. In this section wewill introduce two algebraic generators, i.e. two projectors with entries in Pol(2nq )whose classes generate K0. The first one is the trivial one [1]; in order to get thenon trivial generator let us go back to the classical case.

The non trivial generator of K-theory for the classical even sphere can be ex-plicitly written in the following way. Let G2n2k M2k(S

2n) be defined iterativelyby

G2n2(k+1) =

G2n2k a

k+1

ak+1 1 G2n2k

G2n0 = 1 t .

It is easy to verify that G2n G2n2n is an idempotent for n 0 defining the vector bun-

dle E2n of rank 2n1 and charge 1. This construction has the following geometrical

interpretation. Let i : S2n S2(n+1) defined by i(t, a1, . . . , an) = (t, a1, . . . , an, 0) bean embedding ofS2n in S2(n+1) and let i(E2(n+1)) the pullback vector bundle on S

2n.It is clear that i(E2(n+1)) = E2n E

2n, where E

2n is the conjugated vector bundle

on S2n of charge 1.

In the quantum case not every step of this procedure can be obviously extended.In fact there is no embedding of 2nq in

2(n+1)q , i.e. there are no algebra projections

from Pol(2(n+1)q ) to Pol(2nq ). It is possible to adapt the procedure in order to pro-

duce a rank 2n1 idempotent for 2nq ; but it is convenient to lift the construction to

R2n+1q , a deformation of the odd plane where the even sphere lives. Let us introducePol(R2n+1q ) as the algebra generated by {xi, x

i , y = y

}ni=1 with the relations

xi y = q2y xi , xi xj = q

1xj xi , xi xj = q

3xj xi , i < j ,

xi xi = q

2xi xi + q2(1 q2)


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We need the following Lemma.

Lemma 9 For each such that k < n we have that

M2n2k, e

2n2kC

2n2k x

C2n2k x

(e2n2k) = 0 . (12)

Proof. We prove the result by induction on k. For k = 0 it is equivalent toyx = q

2xy. Let us suppose (12) true for k and let us show it for k + 1. Bydirect computation we get [M2n2(k+1),]11 = M

2n2k,, [M

2n2(k+1),]22 = q

1n(M2n2k,)

and [M2n2(k+1),]12 = q(C2n2k )

2(xk+1x qx

xk+1). Using the inductive hypothesis and

relations in Pol(R2n+1q ) we conclude that M2n2(k+1), = 0.

We now prove the main result of this section.

Proposition 10 For each k n we have

(e2n2k)2 e2n2k = [q

2ki=1

xi xi y(1 y)]q2(C2n2k )

2 . (13)

Proof. We show the result by induction on k. For k = 0 it is easy to see that itis true. Let us suppose it true for k. By direct computation, using the inductivehypothesis and equation (12) we get

[(e2n2(k+1))2 e2n2(k+1)]11 = [q

2k+1

i=1xi xi y(1 y)]q

2(C2n2k )2

[(e2n2(k+1))2 e2n2(k+1)]22 = (C

2n2k )

2xk+1xk+1 + n(e

2n2k)

2 n(e2n2k)

= (C2n2k )2[xk+1x

k+1 +

ki=1

q4xi xi y(1 q2y)]

= (C2n2k )2[q2

k+1i=1

xi xi y(1 y)]

[(e2n2(k+1))2 (e2n2(k+1)]12 = e

2n2kC

2n2k x

k+1 C

2n2k x

k+1(e

2n2k) = 0 .

Recalling the iterative definition of C2n2(k+1) we finally get the result

(e2n2(k+1))2 e2n2(k+1) = [q

2k+1i=1

xi xi y(1 y)]q2(C2n2k+1)

2 .

It is then clear that for each n > 0, G2n = p(e2n2n) M2n(Pol(

2nq )) is a projector;

let us denote with [G2n] its class in K-theory (both algebraic and topological). Byusing the recursive definition of e2n2k it is easy to compute the matrix trace of G2n.In fact the equation

Tr(e2n2(k+1)) = 2k + Tr(e2n2k n(e

2n2k)) , Tr(e

2n0 ) = 1 y ,

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is solved by Tr(e2n2k) = 2k1 (1 q2)ky (for k 1), so that we have

Tr(G2n) = 2n1 (1 q2)nt . (14)

By recalling the definition of the Fredholm modules [] and [trn

], we compute theirChernConnes pairing with G2n:

[], [G2n] = 2n1 , [trn], [G2n] = 1 .

Corollary 11 The projector G2n defines a non trivial class both in K0(Pol(2nq ))

and K0(C(2nq )).

Aknowledments. F.B. wants to thank L.Dabrowski, G.Landi and E.Hawkins foruseful discussions on the subject.

References

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[15] Sheu A.J.L. (with an appendix by Lu J.H. and Weinstein A.): Quantization ofthe Poisson SU(2) and its Poisson homogeneous space the 2-sphere, Commun.Math. Phys. 135, 217-232 (1991).

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