J. Scott Carter Daniel E. Flath

and

i~lasahico Saito

Mathematical Notes 43

1 Introduction

2 Representations of U(sZ(2)) . . . . . . . . . . . . . . . . . . . . . . Basicdefinitions 7

. . . . . . Finite dimensional irreducible representations 7

. . . . . . . . Diagrammatics of U(sl(2)) invariant maps 12

. . . . . . . . . . . . . . . . The Temperley-Lieb algebra 15

. . . . . . Tensor products of irreducible representations 21 . . . . . . . . . . . . . . . . . . . . . . . The 6j-symbols 27

. . . . . . . . . . . . . . . . . . . . . . . . Computations 43

. . . . . . . . . . h rccu~sio~l formula for the 61-symbols 63

Itei l lh~hs . . . . .

3 Quantum sZ(2) 67

. . . . . . . . . Some finite dimensional representations 67

. . . . . . . . . . . Representations of the braid groups 70

A finite dimensional quotient of C[B(n)] . . . . . . . . . 74

The Jones-Wentzl projectors . . . . . . . . . . . . . . . 80

. . . . . . . . . . The quantum Clebsch-Gordan theory 93

Quantum network evaluation . . . . . . . . . . . . . . . 99

. . . . . . . . . The quantum 6j-symbols - generic case 106

. . . . Diagrammatics of weight vectors (quantum case) 110 . . . . . . . . . . . . . . . . . . . . . . . . Twisting rules 111

. . . . . . . . . . . . . . . . . . . . . . . . . Symmetries 123

Further identities among the quantum Gj-symbols . . . 125

4 The Quantum Trace and Color Representations 127

The quantum trace . . . . . . . . . . . . . . . . . . . . . 127

A bilinear form on tangle diagrams . . . . . . . . . . . . 130 This book discusses the representation theory of classical and

Color representations . . . . . . . , . . . . . . . . . . . 133 quantum U ( d ( 2 ) ) with an eye towards topological applications

The quantum Gj-symbol- root of unity case . . . . . . 139 of the latter. We use the Temperley-Lieb algebra and the Van- I I

151 I tum spin-networks to organize the computations. We define the 5 The Turaev-Viro Invariant

6j-symbols in the classical, quantum, and quantum-root-of-unity The definition of the Turaev-Viro invariant . . . . . . . . 151

cases, and use these computations to define the Turaev-Viro in- I Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 variants of closed 3-dimensional manifolds. Our approach is ele-

i References 160 mentary and fairly self-contained. We develop the spin-networks

from an algebraic point of view.

1 Introduction

These notes grew out of a series of seminars held at the University

of South Alabama during 1993 that were enhanced by regular e-

mail among the three of us. We became interested in quantum

diagrammatic representation theory following visits from Ruth

Lawrence and Lou Kauffman to Mobile.

We develop the Clebsch-Gordan theory and the recoupling the-

ory for representations of classical and quantum U(sl(2)).via the

spin networks of Penrose 1271 and Kauffman [16]. In these the-

ories, the finite dinlensio1la.1 irreducible representa.tions are real-

i ~ e d ill spaces oi i~oluogcr~eoil:, l,oi> i~oi~ii,il\ 111 1 \\(, \ . I : I . I I ) I ~ ~ ~ T I I

the quantum case the variables commute up to a factor of q; z.e.

yx = qxy. The tensor product of two representations is decom-

posed as a direct sum of irreducibles, and the coefficients of the

various weight vectors are computed explicitly. In the quantum

case, when the parameter is a root of unity, we only decompose

the representations modulo those that have trace 0.

We use the spin networks to develop the theory in the classi-

cal case for two reasons. First, they simplify and unify many of

the tricky combinatorial facts. The simplification of the proofs is

nowhere more apparent than in Theorem 2.7.14 where a plethora

of identities is proven via diagram manipulations. Second, the

spin networks are currently useful and quite popular in the quan-

tum case (see for example [23], [18], [28]). One of our goals here

is to explain the representation theory of quantum sl(2) in the

spin network framework. We know of no better explanation than

- --..--I-I-.. . . I .u xu,II, I U I Y I V J - 3 1 IVI,,"ba IN L ~ U U U ~ I I V I V

I

=I

1 I1 to run through the classical case (which should be more familiar), on triangulations of a 3-manifold while the orthogonality condition

I and then to imitate the classical theory in the quantum case. can be interpreted as a Matveev [25] move on the dual 2-skeleton I Here we give an overview. The set of (2 by 2) matrices of of a triangulation.

determinant 1 over the complex numbers forms a group called The Turaev-Viro invariants were based on work of Kirillov I 1

SL(2). The finite dimensional irreducible representations of SL(2) and Reshetikhin on the representation of quantum groups [19].

are well understood. In particular, it is known how to decompose This work together with Reshetikhin-Turaev [29] formed a math-

I I the tensor product of two such representations into a direct sum ematicdy rigorous framework for the invariants of Witten [34].

of irreducibles. In this decomposition one can compute explicitly Meanwhile Kauffman and Lins [18] gave a simple combinatoric

the image of weight vectors and such computations form the heart approach to the invariants based on the Kauffman bracket and

of the so-called Clebsch-Gordan theory. The finite dimensional the spin networks of Penrose [27]. Piunikhin [28] showed that the

representations of SL(2) are the same as those of U(sl(2)) which Kauffman-Lins approach and the Turaev-Viro approach coincide.

I I is an algebra generated by symbols E, F and H subject to certain Some of Kauffman's contributions to the subject can also be relations. found in the papers [14], [15], and [17]. A more traditional alge-

Furthermore, the tensor product of three representations can braic approach to quantum groups can be found in [30]; in partic-

be decomposed in two natural ways. The comparison of these ular, they discuss from the outset the Hopf-algebra structures.

two decomytosit ions is sornctimcs c,rllctl r r rn~~,, l?r iq t/)ro,~l l . c l l l f l T icl;ol.isll'q [23] d(\finit ion of tlic R t ~ s l ~ c t i k l i i ~ l - ~ ~ ~ . a o v iilval inlltb

. . -p tile rccou1)llllg t uell~c~erit:, ,ir c. I , I ~ o \ L I ~ L J ~ C G ~ ' - ~ ~ ~ l ~ ] ~ ~ l ~ , rlllese 1s of a colllbi~~ator~al nature. The I<auiinian-L~ns [IS] d e l i n ~ t ~ o ~ l 01

symbols satisfy two fundamental identities (orthogonality and the the Turaev-Viro invariants is defined similarly. Neither of these

Elliott-Biedenharn identity) that can be interpreted in terms of combinatorial approaches relied on representation theory. How-

the decomposition of the union of two tetrahedra. In the ~ l l i ~ t t - ever, the remarkable feature of quantum topology is that there are

I* Biedenharn identity the tetrahedra are glued along a single face close connections between algebra and topology that were hereto-

and recomposed as the union of three tetrahedra glued along an fore unimagined. The purpose of this paper is to explore these $1

l edge- For orthogonality the tetrahedra are glued along two faces, relations by examining the algebraic meaning of the diagrams and

It and the recomposition is not simplicial. by using diagrams to prove algebraic results.

I The symmetry of the 6j-symbols and their relationship to Here is our outline. Section 2 reviews the classical theory of

11 M ~ a h e d r a was for the most part a mystery, until & - a e v and of U(sZ(2)). There is nothing new here, but we I

I Viro 1321 constructed 3-manifold invariants based on the analo- do how the Clebsch-Gordan coefficients and the 6j-symbols I/ gous theory for quantum sZ(2). The identities satisfied by the are computed in terms of the bracket expansion (at A = 1). In

Gj-symbols are also satisfied by their quantum analogues. The Section 3 we mimic these constructions to obtain the quantum

Elliott-Biedenharn identity corresponds to an Alexander [I] move Clebsch-Gordan and 6j-symbols. In Section 4 we will define the

" r V U n L J L I L U f i U f l1 lU WUXIY 1 UM tJJ->YMB(JLS

/' i quantum trace and discuss the recoupling theory in the root of 2 Representations of U(sl (2) ) unity case. Section 5 reviews the definitions of the Turaev-Viro

I 2.1 Definition. Let C denote the complex numbers. The I ! invariants and proves that the definition is independent of the

triangulation by means of the Pachner Theorem [26]. group SL(2) is defined to be I/ i i

Acknowledgments. We all are grateful to L. Kauffman and S L ( ~ ) = {(; ;) : a ,b , c ,dEC, a d - b c = l R. Lawrence for the interesting conversations that we have had.

I \ \

Their visits to Mobile were supported by the University of south : ( / I where the law of composition is matrix multiplication. The as-

~ ~ ~ b a m a ' s Arts and Sciences Support and Development Fund. sociated Lie algebra sZ(2) consists of the set of matrices of trace

Additional financial support was obtained from Alabama E P S C ~ R 0: for funding of travel for the first named author and support of

a Conference in Knot Theory, Low Dimensional Topology, and ,,(,)={(; i ) :a,b,c ,di( : , a + d = n -

' 11 Quantum Groups in Mobile in 1994. C. Pillen, B. Kuripta, K.

Murasugi, and R. Peele provided us with valuable information. This is spanned by E =

I ( ; ) , . = ( a =

Jim Stasheff read a preliminary version of the text and provided

us with many helpful comments. Cameron Gordon's past finanical

( - y2 ) . The Lie bmdei is c~mputed via [A, Bl = AB - support of Masahico Saito was greatly appretiated. Finall!. ive all

I 1 3 - 1 . , o t i , a t [ 1 : . 1 ] = ~ ~ i . \ r ! . ~ ] = ! > - ( l i u 1-1----l7 grdtcfull) dcl\nowledgc t i l e suppo~t aild pat le~~cc L ~ ~ A L otll wlbes -

CL 1 -8 11 have shown to us over the years.

exponential function, exp : sl(2) -+ SL(2), which is defined by the

j power series: / $1 " Q3

~ X P Q = j=o

for Q E sZ(2). The function exp maps a trace 0 matrix to a matrix

I with determinant 1.

2.2- Finite dimensional irreducible representations. The

group SL(2) acts on the vector space of linear combinations of.

variables x and y by

A N D QUANTUM 6j-SYMBOLS

representation, VO = C . The index j is sometimes called the spin

( ; ) y = c x d y of the representation V3 . The associated Lie algebra d(2) acts on V 3 8s ~ ~ U O W S :

where the action is extended linearly. This is called the funda- E~~~~ = dl exp (tE)xTys = srTt1 ys-',

mental representation of SL(2). dt t=o

I i More generally, define an action of SL(2) on the space of poly- F~~ yS = rxT-I ySC1, i

and 1

= (ax

1 11 11 One way to verify that this is a group action is to consider the ' i

I t embedding

and r - s

HSTyS = - xTyS. 2

A weight vector is an eigenvector under the action of H in any

representation; its eigenvalue is called its weight. For example

xTyS E V(T+S) I~ is a weight vector of weight y. Observe that

the set of weights in ~j is {j, j- 1, j-2, . . . , - j ) , and by definition

the corresponding weight vectors form a basis for VJ.

fixed

. . , 2.2.1 well ~ l z o w n Xheorenl. (See [g] or [ 3 3 ] , hi- c ~ l m -

under the

Y and where the sum is over all

geneous ~ o l ~ n o m i a l s of degree ( r + s ) in x and y into the tensor

product of ( r + s ) copies of the fundamental represelltation space.

The tensor product V 8 W of representations V and ]/V illherits

! action

The representations of SL(2) On V 3 are irreducible.

proof. 1f W is an SL(2)-subrepresentation of V3, then W is also

invariant under the action of the algebra sl(2) that is given above.

Therefore, it is enough to show that the representation of sl(2) On

of the permutation group on the tensor factors of VB('+"), and this space is stable under the

action of SL(2).

an action via g(v @ W ) = gv 8 g w where v E V and w E W . ~h~~ if vj is irreducible.

It is customary to let ~j denote the set of homogelleous poly-

nomials of degree 2 j = r + s where j E {0,1/2,1,3/2,. . .). Note

that v1I2 is the fundamental representation, and V0 is the trivial

I' denotes the fundamental representation space, then v@(T+s) is

also arepresentation space. Furthermore, the image of the space of

~olynomials consists of the subspace of tensors that

representation is

The matrix E acts by sending a weight vector to one of higher

weight F sends such to one of lower weight. Since the image

of anv non-zero vector under powers of E and F spans V37 this

irreducible.

Remark. In the sequel, it will be more convenient to work with

the universal enveloping algebra U(sl(2)). This is an algebra gen-

erated by symbols E , F , and H that are subject only to the rela-

tions EF - F E = 2H, HE - E H = E, and H F - F H = -F.

The relations are motivated by the properties of the Lie bracket

iir 10 REPRESENTATIONS OF U(s l (2 ) )

~r of U(s l (2) ) is deter- 2. The vector v has weight j. mined by assigning to E , F, and H operators on a vector space

that are subiect to the relations above. The enveloping algebra Then j E {0,2/2,1,3/2, . . .), and there is a unique linear map

., - - - - ----e v is a vector in the 4 : V J + W such that $(x2" = v and such that .11, commutes with L -- -- - -

representation V . the action of U(sl(2)) .

In the discussion of Section 3, the representations V J of d ( 2 ) (b) Every finite dimensional irreducible representation of

and the enveloping algebra U(s l (2) ) will have quantum analogues. U(s l (2 ) ) is isomorphic to V J for some j E {0,1/2,1,3/2, . . .). There is a quantum analogue of the group S L ( 2 ) , but we will not

Proof. Let vo = v , and for r > 0 let v, = FTv. use it to describe the representations.

We assume by induction that v, has weight ( j - r ) . Since

H v = jv and [H , F] = -F , we have HV,+~ = HFv , = -Fur + should be indexed by

F H v , = ( j - ( r + l ) ) ~ , + ~ ; thus vT+l has weight ( j - ( r + 1) ) . 8 Furthermore, we inductively assume that there are constants

e3,m = 5J+m 3-m. Y y, such that Ev, = y,v,-1. By using the relation [ E , F ] = 2H,

we have that EV,+~ = EFv , = 2HvT + FEU, = ( 2 ( j - r ) + Y T ) ~ T .

The first subscript of e is the highest weight of the represen- Hence ?,+I = 2( j - r )+yT; since yo = 0 , we have y, = r (2 j - r+ 1). tatloll <111d ind~t a f t 5 t h t tl~rnel~slon of thc rcprt~.;crltdf ioll 5pr7(t\ Yo\\ , , 0 fol , o n ~ c I I,c~c,l~l\c. I f * I ? fn r te t l~mr.nsio~~,~\ and

(dl111 j l I ) = LJ $ 1) wlllle the sccond ~~ldlcates tile icelght of the vo, v l , . . . are eigenvectors for distlnct elgenvalues of H. Suppose vector. Note that j and m are both half-integers and that j + m that v, = 0 and v,-1 # 0. Then Ev, = 0 = r (2 j - r + l)v,-1. So and j - m are integers. In this notation, j = ( r - 1 ) /2 E {0,1/2,1,3/2, . . .), the subspace of W generated

by vo is spanned by the linearly independent vectors vo, . . . , v,-1, Ee3,m = ( j - m)eJ,m+l,

and this subspace is isomorphic to V J . This proves (a).

Fej,m = ( j + m)eJ,m-17 (b) Let W denote a finite dimensional irreducible represertta-

tion of U(s1(2)), and let w be a non-zero eigenvector of H . Let the and

integer r be such that E'w # 0 while E'+'w = 0. Then v = E'w Hel,m = meJ,,.

satisfies the hypotheses of (a). Hence W = V J where H v = jv.

2.2.3 Well Known Lemma. (a) Let W denote a finite 2.2.4 Theorem. Every finite dimensional representation space dimensional representation space for the algebra U(s l (2) ) . Let E for U(sZ(2)) decomposes as a direct sum of irreducible representa- W denote a non-zero vector that satisfies:

tions.

1. E v = 0. I Proof. See [33] or [8] for example.

2.3 Diagrammatics of U ( s l ( 2 ) ) invariant maps. The Pen-

rose spin networks facilitate the computation of U ( s l ( 2 ) ) invariant

maps via diagrammatic techniques. At the heart of the networks

are three elementary maps U, n, and / that are defined in Sec-

tion 2.3.1. Their relations are described in Lemma 2.3.2. The

networks or spin-nets will consist of trivalent graphs embedded 1 1 1 i

in the plane with non-negative half-integer labels on the edges. P

These labels will satisfy an admissibility condition at each vertex

that will be made explicit as we continue the discussion.

2.3.1 Definition. Consider the U ( s I ( 2 ) ) invariant maps U : I ,lit 170 + ~ 1 1 2 ,g, ~ 1 1 2 , , : v 1 1 2 fg ~ 1 1 2 + vO, : v 1 / 2 fg ~ 1 1 2 4

i I

l f 1/2fg171/2, and I : v1I2 -+ v1I2 that are defined on basis elements

I (and extended linearly) via

3. (I fg n ) 0 (U 8 1) = ( = ( n 8 1) 0 (1 fg U ) : v1I2 -+ v1I2 under the identification of C ,g, v1I2 = y1l2 fg C = v 1 I 2 .

4 . ( n 8 1) 0 ( I fg 1) = ( I @ n ) 0 (1 @ 1 ) : ( v 1 / 2 ) @ 3 v 1 / 2

where as before we identify C 8 '[fl/2 = ~ 1 1 2 8 c = 1/1/2.

/ /

X ( a f g b ) = b @ a for a , b € { x , y }

where i = G. Finally 6. f l o u : C - + C

i s multiplication by -2.

] ( a ) = a for a E { x , y ) . Proof. Items (I), (2), ( 3 ) and (6) are elementary computations.

2.3.2 Lemma (Penrose [27]). Item (4) is a general property that holds for any bilinear. form n; similarly, item (5) follows for any "co-bilinear" form U .

1. The maps U, n, X, and ( commute with the action of U ( s l ( 2 ) ) .

2. The fundamental binor identity holds:

= I fg 1 + (U o n) : v1J2 8 v 1 I 2 -+ v 1 I 2 8 ~ 1 1 2 :

2.3.3 Remarks. Penrose and I<auffman introduced these

maps in a diagrammatic context. The domain of a map repre-

sented by such a diagram appears at the bottom of the diagram,

16 'THE CLASSICAL A N D QUANTUM ~ ~ - S Y M B O L , S I

and I The key observation about the diagram algebra is that ~ l a l l a f (1 isotopies of arcs (that are properly and disjointly embedded ill

.- i a rectangle) are generated by the topological moves that corre- hk H ( . . .

z- - I - . . I . -nL+ spend to (a) the relationship depicted in Lemma 2.3.2 (3) and (b) A - I 2 1 - L - 1

. . ,- : . , r c ~ l c l l d n ~ i l l g disf ant critical l>ointq. example, in the illustla-

tloll above the Criticdl p01111~ ~ e l ~ l e ~ e l ~ t l l l g 111 C d l l 1 1 ~ ] ) ~ l ~ i l ~ ( / (I0'' "

and those representing h3 can be pushed up. Algebraically, this

The diagra7n algebra consists of formal linear combinations of cer- interchange represents the identity h1h3 = h3hl.

tain diagrams. The diagrams are generated by the diagralns rep- The correspondence between the Temperley-Lieb algebra and

resentillg I alld hk for k = 1, . . . , 2 j - 1 that are indicated above. the algebra of diagrams shows that the dimension of T L n is the

two such diagrams can be juxtaposed vertically to represent nth Catalan number, A(:), where n = 2j . This result follows

the product of two of the elements. Having been so juxtaposed, by establishing a one-to-one correspondence between the possible

the product is rescaled vertically to fit into a standard size rect- diagrams and the collection of legitimate arrangements of n pairs

angle. Two di%pms that are isotopic via an isotopy that keeps of parentheses (See also [16]).

the top and the bottom of the diagrams pointwise fixed represent Next we let the ground ring R denote ihe complex numbers,

the same element in the diagram algebra. For example, the prod- and choose matrix representations for the symbols U and n. uct h2hlh3 is depicted below. Always the element on the left of

an ex~ressioll is a t the top of a diagram; thus the bottom most 2.4.1 Lemma. Let j be a fixed element in {0,1/2,1,3/2,. .

18 THE CLASSICAL AND QUANTUM 6 j - s y ~ ~ o ~ ~ REPRESENTATIONS OF U(sz(2)) 19

I

functions roof. The proof depends on standard facts about the represen- (~1/2)@2.1 , (~1/2)@2.1 ation theory of GL(2) and C, . We refer the reader to the excellent

The image O(I) is the identity, and for k = 1,. . . , 2 j - 1, the gen- text, [31] for details. Let n = 2j. Since dim (TLn) = Catalan(n)

erator hk is mapped to the composition Uon where these are acting we must prove that dim (B(TLn)) = (T) / (n + 1)- The binor iden-

on the (k, k + 1) factors of the tensor product as in Section 2.3.1. tity shows that B(TL,) = b(C[Cn]), where @ is the linear ex-

tension of the representation defined in the remark immediately ill I Proof. It is necessary to check that O respects the defining rela- above. We will establish that dim (@ (C[xnI)) = (',") /(n + 1). The I

tions (1-5) of the Temperley-Lieb algebra. The calculatioll follows representation 6 is decomposed as in [33]. Namely, as En repre-

from the diagrammatics of Lemma 2.3.2 part (3). I-J

As we continue to discuss U(s1(2)), we will work with the

I' (v'l2)@" = $T(m)d(T) Temperley-Lieb algebra under this representation witllout explic-

llill I itly mentioning the map 8. Our justification for this notational

abuse is given in Theorem 2.4.3. (1) the index T ranges over all 2-row Young frames with n

(i.e. T = (r,s), n = r + s, and 0 L r I s); 2.4.2 Lemma. Let 6 = -2. f i r a n y j E {0,1/2 ,1 ,3/2 , . , .I,

(2) the s u l 1 3 $ a ~ d \ITrr is t h e irrc~I,,cil,lc rel)resentatioll of l l l c l t i 5 11 ~ J ( ~ ~ I ~ o I I J O ~ ~ ) / I I ~ ~ ~ ~ ( 1 1 , , , , l / c , / , o , j q J , J ,,,, y2! ,,,, 2 ,

corresponds to the toung lldllle 1 ; letters zlzto TL23 that 2s yzven by p(ak) = I + hk where ak zs the

transposition that interchanges and k + 1. (3) the exponent d(T) is a positive integer that, incidentally, is

equal to the di~nension of the representation of GL(2) correspond-

Proof. Clearly, the images (under p) of distant transpositions ing to the Young frame T. , commute. Furthermore, jyow c [ x n ] is a semi-simple algebra because the group En is

l1 finite. The Wedderburn theory of semi-simple algebras [g], applied p(ukokklok) = I+ h i + h x t ~ + hihi51 + hkil hk = p(okklokok*l 1.

/I 1 to C[Cn], asserts that as algebras

Finally, ( I t hk) 0 (It hk) = 1 $ 2hk + 6hk = 1.0 - @(C[Cn]) % $ ~ M a t ( n ~ X n ~ ) Remark. The binor identity shows that the 1~omomorphism p is -

-a factor of the representation : CZ3 -+ A U ~ ( ( V ~ / ~ ) @ ~ ~ ) where the where nT = dim WT and Mat(- x -) denotes the algebra square

permutation group acts on ( V ~ / ~ ) @ ~ J by permuting tensor factors. matrices. It follows that

2.4.3 Theorem. In case 6 = -2, the representation, B of TLz3

on ( V ~ / ~ ) @ ~ J , is faithful for evert, i E -!0.1/2.1.3/? 1 I dim ( j (C[C, ] ) ) = n;.

For T = (r, n - r ) with 0 5 r _< Ln/2] (where 1.1 denotes the

greatest integer function), we have

This is the number of ways of filling in the n boxes in the Young

frame T with the integers 1 ,2 , . . . , n in such a way that numbers

increase across both rows and increase down all columns. The Young frame has n - r boxes on the top row and r boxes on the

bottom row.

The proof will follow from the following interesting combina-

torial identity for Catalan numbers:

Let G(r, n - r ) = n(,,,-,) denote the number of legitimate

filling.; of the Young frame nxilh I , I ~ n ~ c > i nri f 1 1 ~ 1101 l o r ? ] J.(IU. ;I n d

n - 1 . boxes on the top. For a two ron. rectangular array,

so in fact G(r, r ) is the r th Catalan number. We wish to show

that

Each term in the sum on the right is the square of the number

of ways of W g g in a smaller Young frame. For s [n/2j, we consider the Young frame (n, n) to be decomposed as the union

of a frame (s, n - s ) and its mirror image. For example,

A filling of the frame (s, n - s ) with the integers 1,. . . , n to-

ther with a filling of its mirror image with the integers n $

. . . ,2n yields a filling of the rectangular frame. Therefore, the

m on the right is no larger than G(n, n).

On the other hand, let a filling of the rectangular frame be

iven. Then consider the subset of the rectangular array that

ontains the numbers 1,. . . ,n. This subset is convex and forms

a smaller frame of type (s ,n - s). Thus we have a filling of it

and a filling of its mirror image. Therefore, G(n, n) is no larger

than the sum on the right. This proves the combinatorial identity.

Consequently, the representation is faithful.

2.5 Tensor products of irreducible representations. Re-

call that if V and W are spaces on which the group SL(2) acts,

I then there is an action given on the tensor product by g(v @ t u ) =

911 ~ I L ) . .\1i element .Y in the assoriatctl 1,ic alge111.a. d ( 2 ) . ac ts

on tensor protluc~s via the Leibniz rule, . Y ( u LI ru) = St?) ,I iu $

. v @ X(w) since the action is determined by differentiation. Notice

that if v and w are weight vectors, then so is v @ w, and its weight

is the sum of the weights of v and w. Recall that V3 is isomor-

phic to a sub-representation of the 2j-fold tensor product of the

, fundamental representation via the map

1 (P3 : X I . ' ' ' ' x2j ++ - C p xu(1) @ . ' @ Xu(2j)

2.5.1 The projectors. The projection of (V1/')@'j onto the

image (Pj(Vj) can be written in terms of the Temperley-Lieb ele-

ments as the map

Observe that +2j 0 + 2 j = 4 2 , So that this map is indeed a projec- 2.5.3 Lemma.

proof. The proof follows by induction.

2.5.4 Definition. Suppose that a , b E { 0 , 1 / 2 , 1 , 3 / 2 , . Let

j E { a + b,a + b - 1 , . . . , l a - bl + 1,la - bl). Such a triple of

half-integers ( a , b, j ) is said to be admissible. Notice that admis-

sibility is a symmetric condition in a , b, and j . Define an U ( s l ( 2 ) ) In 3-5, the quantum analogues of these projectors are de- invariant map

: ( ~ ' / 2 ) @ 2 3 - ( ~ 1 / 2 ) @ 2 a ( ~ 1 / 2 ) @ ~ ~

I 2.5.2 Definition. Let u= u : c ~ 1 1 2 a ~ 1 1 2 . H~~~~~

defined C -+ ( v " ~ ) @ ~ ( " - ~ ) , define 5 to be the composition

where 1, is the identity map on the m-fold tensor power of v 1 l 2 - ab

The map ( p a @ p a ) o Y 0q5~ -where de is the isomorphism of 3

ve with the symmetric tensors while pe(xl@. . - @ ~ 2 e ) = 5 1 - . . ' 'x2e

for 1 = j , a , or b - is called the Clebsch-Gordan map: V 3 -+

The map fi is defined dudy .

Let C ( 1 , . . . , n}. Define 2.5.5 Theorem. There is a direct sum decomposition

24 T H E CLASSICAL A N D QUANTUM 6j -SYMBOLS

where the sum is taken over all j such that ( a , b, j ) is admissible.

Furthermore, i f ( a , b, j ) is an admissible triple, then any U(s l (2) )

invariant map V j + V a @ V b is a scalar multiple of pa @ pb o v 0 q5j. Finally,

Proof. The map y :" is U(s l (2 ) ) invariant because it is the

Example. Consider v ~ / ~ @ v ~ / ~ . According to Theorem 2.5.5,

this tensor product decomposes as the direct sum of V0 and V1. 112,112

The map coincides with U while 0

and finally

112,112 y I ( ~ x Y ) = ~ I ~ ( x @ Y + Y B x ) .

2.5.6 The Clebsch-Gordan coefficients. Let ej,t denote the

weight vector xj+tyj-t in ~j of weight t . We have maps pa @ pb o ab

( V o $; : vj -+ V a 4 vb provided that ( a , b, j ) are admissible. composition of l T ( s / ( 2 ) ) invariant maps and G. TIle f o l l n u ~ ~ l

for (Pa @ 1.. ( ,' (4. ( x ' J ) ) ) / follows by cornputatiol~ using in the sum ab

Lemma 2.5.3; thus y # 0 for ( a , 6, j ) admissible. The tensor 8 pb ( y 3 (A (e3,t)l) = u+v=~

c:;t:ea,U 8 eblv.

3

product V a €3 vb has dimension (20 + 1)(2b + 1) while the image 2.5 -7 Lemma, The Clebsch-Gordan coeficients satisfy the fol-

of V3 has dimension ( 2 j + 1). Since lowing recursion relation

x ( 2 j + 1) = (2a + 1)(2b + 1) 3

( j + t + l)ct:% = ( a + U + l)~:;b:,v,t+l t (b + v + l)'::b:l,t+~.

( j t t ) ! ( j - t ) ! (a+ b - j ) !

P ( ( 3 ) is the only s p a c e of v 8 v is ( a + u + ~ ) ! ( b + v + w)! isomorphic to v'. Consequently every U(s l (2 ) ) invariant

z!w!(a - u - z ) ! (b - v - w)! ' V 3 --+ V a 8 V b must be a multiple of this map. Z,w: ~+w=3-t

26 THE CLASSICAL AND QUANTUM 6j-SYMBOLS REPRESENTATIONS OF U ( ~ 1 ( 2 ) ) 27 1 11

Proof. The recursion relation is found by applying F to both

sides of the equation that defines the Clebsch-Gordan coefficient.

The closed form is determined by solving the recursion using the

2.5.8 Diagrammat ics for weight vectors. By definition of I 4, the weight vector $t(ej , t ) is the image under -kzj of

m n

where m = j + t , n = j - t .

We represent x (resp. y) by a white (resp. black) vertex with

a string coming out from the top: 6 (resp. ). Then the neight

1

a + u a - u a ~ 6 1 - v b - 3

Y=----T I These conventions have been known to physicists (see 1161). E I It is convenient to introduce similar diagrams for dual vectors.

Consider the dual vector space (v1l2)*. We represent the dual

11. basis vectors x* and y* of tkis dual space diagrammatically by

7 a i d 7 , respectively. For parallel strings representing tensor

products of the fundamental representation, putting one of these

dots on the top of the strings algebraically means that we take

the values of the pairing among vectors and dual vectors. In par-

ticular, X means the pairing between x and x* which is equal to

one. We have, then, that the dual to

j + t j - t

From the closed formula given in Lemma 2.5.7 for the Clebsch-

Gordan coefficients, one cannot easily see the symmetry properties "P<

of the coefficients under replacing ej,m with ej,-,. However, this

symmetry and the probabilistic nature of the coefficients is more

apparent in the network evaluation.

2.6 T h e 6j-symbols. Here we consider the space of U(sl(2))

invariant maps vk -+ Va @I V b @I VC. We will construct such maps

THE CLASSICAL A N D QUANTUM SYMBOL^ REPRESENTATIONS OF U(SZ(~))

in two different ways. First, consider the composition (a, b, n), and (n, e , k) form admissible triples, form bases for the

\ \ N vector space of U(sZ(2)) invariant linear maps vk -+ va @vb @VC. fiere pabc = pa @ pb @ pc is the tensor product of the multiplication

( I2a €3 ',' -- maps and d3 : ~j -+ (v1I2)B2j sends a a homogeneous polynomial f in x and y to a symmetric tensor.

k I1 li i I

Proof. The triple tensor product Va @ vb @ VC decom-

' I (v1I2)@2k ( v 112 ~ 2 a @ ( 112 ) 8 2 6 @ ( ~ . l / ~ ) @ 2 c poses as ($,Vn) €3 VC = $,(Vn €3 VC) where the direct sum

/ 11 /I is taken over all n such that (a, b, n) is admissible, by Theo-

I I for various values of j. Second, consider the composition rem 2.5.5. For each such n, (Vn @ Vc) contains at most one copy

II~I 1 1 If of vk. Thus h ~ m ~ ( , ~ ( ~ ) ) ( v ~ , V a @ v b @VC) decomposes as a direct

i t /I y.' a b sum of the 1-dimensional spaces h ~ m ~ ( , ~ ( ~ ) ) ( v ~ , vn €3 VC). Sim- nc

= ( y 812C)o y ilarly, it decomposes as a direct sum of the 1-dimensional spaces k

I . h ~ m u ( ~ i ( z ) ) ( V ~ , Va €3 V3 ).

I I & 2.6.2 Definition. Define the Gj-symbol to be the coefficient

for various values of n.

The values of j and n are restricted so that (b, c, j ) , (a, j , k),

(a, b, n), and (n, c, k) all form admissible ' '

,------ "&.-" I A

is admissible, then so are the triples obtained by permuting a , b,

2.6.1 Lemma. The sets

a V b A ) a\ b \ / c )

as the indices j and n range in such a way that (6, c, j ) , (a , j, k),

{ ] in the following equation.

Lrlples. Alternatively, if one of these triples is not admissible, then we may declare the

corresponding map V to be the zero man. (Recall t h a t if f n h ;\ I Pa @ Pb @ PC'

a b n By convention, { } = 0 if any of the triples (0,c,j),

( a , j , k), (a, 0, n ) , j n , c, k ) is not admissible.

// 30

I/ 1 In the 'paces h o m ~ ( ~ l ( 2 ) ) (Vk7 Va @ Vb @ Vc), we have the two

bases that are defined by these trees, and the ~ i - ~ ~ ~ b ~ l is the

change Of basis matrix. For example, consider the case when = = =

= l /2- One can compute direct]y from the definitions that the possible values for j and n are 0 and 1, and that:

{ :;: :;; ] = -lP, Recall that a spin-net is an embedding in the plane of a graph

with edges labeled by non-negative half-integers in which each

vertex has valence 3. The three edges coincident at a vertex must

{ :;: ;;: } = 1, form an admissible triple. The half-integer labels represent the

spin carried by an edge. When we need to emphasize the number

{ ti; :;: P } = 3/47 of strings represented by an edge (and hence the number of tensor

factors of the fundamental representation carried by an edge), we

will label the edges with natural numbers that are twice the half-

{ ;;; ;:; } = 1/2. integers. The suffers from this minor inconsistency, but

we have found that the meaning of the labels is clear within the J i ' S('rfioil 2 fi u r ~ yivc A I Y I I I 5i1 o :,,(,t I,(,(!

( c,,, , coll~c'xt 111 ~t il:t h !i I - \' I 1 l 1 ~ ' l l -

and

addition, we assume that the embedding is in general Po- we need t~ define an u ( sz (~ ) ) invariant map sition with respect t o a fixed height function: Thus each vertex

(V112)@2a @ (v1'2)@2b -+ (v112)B2J, for admissible triples (a , b, j ) appears at a distinct level, the critical points on each edge are as follows: non-degenerate, and these critical points are at distinct levels from

j a+b--j the vertices. Furthermore, some edges may be marked with 'ym-

nl I = +2J ( /a+j-b 8 n 8 /j+b-a ) 0 (42,

+2b) . a b , I metrizers: +. More precisely, we include valence 2 vertices in 3 which the two incoming edges have the same hbel. The compositi~n irj 0

0 (+a 8 mb) : va 8 vb , vj is also ab

U(sz(2)) invariant. ~h~ principal results of the current section are the orthogo-

2.6.3 computations. ~h~ elegance of the nality and the EEott-Biedenharn identities that are satisfied

spin-net notation that we have developed so far is it facili- the ~ j - ~ ~ ~ b ~ l ~ . We will give proofs of these relationships (and tates Otherwise tedious calculations. There is slight disadvan- others) that are simply manipulations of diagrams. To this end tage in that the calculations are performed in the tensor power, we state a diagrammatic lemma.

32 THE CLASSICAL AND QUANTUM 6j-SYMBOLS

2.6.4 Lemma. The following relationships hold among the

U(sZ(2)) invariant maps )\ , y , 6, Inl and fi. (Here we iden-

tify C 8 V and V for any U(s l (2) ) space V . )

1. ( I n 8 ; ) 0 ( 6 8 l , ) = ~ n = ( f i ~ l n ) o ( l n 8 6 ) .

Proof. Part 1 follows from Lemma 2.3.2 part 3, and induction

since the cancellation of a U with a n can occur regardless of the

tensor factors on which those maps are acting. Part 2 follows by

induction and part 5 of Lemma 2.3.2. Part 3 follows similarly

using part 4 of Lernma 2.3.2. Part 4 and part 5 are proved using

2.6.5 Remark. Consider the collection of proper embeddings

of two and three valent graphs in a rectangle whose edges are

parallel to the coordinate axes in the plane. The free end points

of the edges of the graph are embedded in the top and bottom

edges of the rectangle. If two such embedded graphs are isotopic

via an isotopy that keeps the boundary fixed, then there is an

isotopy between them that can be decomposed as a sequence of ~h~~ we have the following calculation.

moves that are the diagrammatic descriptions of items 1 through

5 in Lemma 2.6.4. The valence two vertices are represented in

the Lemma by the projectors, and the valence three vertices are

represented by the Clebsh-Gordan maps.

To find such a nice isotopy, one replaces a given isotopy by

one that is in general position with respect to the height fuliction

defined on the rectangle. The existence of the generic isotopy is

guaranteed by a transversality argument, and a similar transver-

, , i ~ , d ~ C I I I ~ I ( \ I I ~ ( ~ ( ~ ( ~ ~ ~ ~ t : ~ ~ ~ ~ ~ ~ . : I I , > l , O ~ o l j \ I I I I ~ ~ ~ I I ! I I ~ ~ - 1 , ,

of pieces each of which is of the diagrammatic form specified.

2.6.6 Theorem (Orthogonality). Suppose that (a, b, n),

(c, k,n), (a, b, m), and (c, k, m) are admissible triples. Then then

6j-symbols satisfy the following relation:

Proof. Define

Therefore,

THE CLASSICAL A N D QUANTUM SYMBOLS REPRESENTATIONS OF U(sz (2 ) ) 39

. This complex is depicted in Figure 1. Each face

of the Zdimensional complex arises as the cartesian product of an

edge of the tree and a unit interval; the interval factor is thought of

as a time parameter in the deformation between the two trees. The

vertex of the complex occurs as the edge labeled b passes through

the lower junction of three edges. The 2-dimensional complex has

I vertex, 4 edges, and 6 faces; thus it is the dual complex to a

tetrahedron as indicated in the figure.

Let us associate to this complex a 6j-symbol. Then consider

the orthogonality conditions and the Elliott-Biedenharn identities.

Either side of each equation can be similarly thought of as a 2-

tllllle~l~lo~ldl cell co~iil)lcx. 1'01 C ~ ~ I I I ~ ~ C , 111 I ' I ~ L I I C ' 2 ulle \icil: O [ t11e

orthogonality relation is depicted both as a movie description and

as a 2-dimensional complex; the Zcomplex is the "time elapsed"

version of the accompanying movie. The sucessive stills of the

movie differ by a 6j-symbol and these symbols are associated to

the vertices of the 2-dimensional complex on the right of the fig-

ure. The trees in the stills are the trees that represent the maps in

2.6.8 Associating t h e 6j-symbol t o t h e dua l skeleton of the orthogonality relation. Figure 3 depicts the other side of the

orthogonality relation, and the 2-dimensional complex here differs

that occurs when the branch labeled by b'is moved from the from the previous one by one of the Matveev moves [25, 16, 321.

Simaarly, Figure 4 depicts in a movie fashion the three trees that

cessive stills in the movie differ by a 6j-symbol. In Figure 5 the

stills in the movie are the trees on the other side of the Elliott- right side of the tree to the left side of the tree Biedenharn identity. The 2-complex in either figure represents the

THE CLASSICAL AND QUANTUM 6j-SYMBOLS

Figure 1: A movie of a 6j-symbol and the Matveev complex

Figure 2: A movie of Gj-symbols and ort.hogonality (left hancl side)

Figure 3: A movie of 6j-symbols and orthogonality (right hand

side)

dual to the union of tetrahedra - two tetrahedra glued along a

face for Figure 4 and three tetrahedra glued along an edge for

Figure 5. These two complexes are related by a deformation that

is also one of the Matveev moves [25, 16, 321.

Thus the identities expressed among the 6j-symbols are dia-

grammatically expressed as deformations between 2-dimensional

cell complexes. So the use of 6j-symbols in the construction of

the Turaev-Viro invariant appears quite natural in the context of

the representations and their diagrammatic realizations.

However, there are two obstacles to overcome before the defini-

tion of the Turaev-Viro invariant can be made. First, the invariant

Figure 4: A movie of 6j-symbols and the Elliott-Biedenharn iden- is a sum over representations, and since we have irreducible rep-

tity (left hand side) resentations for all non-negative half-integers j, such a sum would

be infinite, so we cannot use the classical theory of U(sl(2)) to

obtain an invariant. Second, the Gj-symbols that we have defined

( I J J 1 1 0 1 r1111 I ( ~ ~ ~ ~ I I ~ ( ~ C I ~ ~ I I ,) l l l l l l ~ ~ t I I I ( I 1 IIII, t ill,\ ( , 1 1 1 ~ 1 0 ~ I ) ( >

associated to tetrahedra in any meaningful way. We will overcome

the first obstacle in Section 4, by passing to the representations of

U,(s1(2)) for q a root of unity. Professor Biedenharn informs us

that this is the physicist's notion of renormalization since we are

converting an infinite sum to a finite sum. We will overcome the

second obstacle by normalizing the 6j-symbols and by showing

that the normalized versions possess the desired symmetr'y while

still satisfying orthogonality and Elliott-Biedenharn relations.

2.7 Computations. We now express the 6+symbols in terms

of evaluations of certain spin-nets. In particular, we determine

Figure 5: A movie of 6j-symbols and the Elliott-Biedenharn iden- some of the symmetry properties of the 6j-symbols in the current

tity (right hand side) normalization, and we find a normalization that has full tetrahe-

dral symmetry.

44 THE CLASSICAL A N D Q U A N T U M 6 3 - S Y M B O L S KEYKESLN'l.A'l.lVNS O F U (S1 (L)) 45

2.7.1 Topological invariance. Let a spin-net be given. This

network is a graph with its edges labeled by non-negative half-

integers that has only 2-valent and 3-valent vertices. Two edges

incident at a 2-valent vertex must have the same label, and if

edges with labels a , b, and j are incident at a vertex, then the

triple (a, b, j) is admissible. Choose an embedding of the spin-net

into a rectangle such that the endpoints of free edges (if any) are

on the top and bottom of the rectangle. Suppose that the labels

of the edges that appear on the top are a l , . . . , a,, and those that

appear on the bottom are bl, . . . , b, where these labels are read

from left to right once the rectangle has been embedded in the .

plane.

The embedding of the spin-net allows us to define a map

Proof. This follows from the remarks 2.6.5.

2.7.3 Lemma. The value of f i : ( v ~ / ~ ) @ " @ ( v ~ / ~ ) @ ~ -i C is

given by

if ~ j = f j forsome j = 1, ... n, in(- l)#{"'"k'~) if {xj, f i} = {x, y} for all j = 1, . . . , n

1 where x j , q E {x, y} for all j = 1,. . . , n.

Proof. This follows by induction, let us exemplify the formula. 2

In case n = 2, the non-zero values of n are as follows:

jectors are associated to the 2-valent vertices and maps

and )\ are associated to the 3-valent vertices. At the bottom of

the rectangle the v b s are mapped via q5 into the tensor powers of 2.7.4 Lemma. For u + v = j , consider the map,pj o

the fundamental representation, and at the top the tensor powers -

(Pa @ q5b : Va @ vb -i Vj. We have are projected back onto the Vas. Some care has to be taken with

regard to the indices along the edges as we indicated in the defi-

nition given in Section 2.6.3. Finally, the embedding of the graph

' is to be in general position with respect to the height function on

2.7.2 Lemma. The map does not depend on the isotopy Proof. Recall that e,,, = xa+" y , that a similar formula for

class (rel. boundary) of the embedding of the given graph in the eb,, holds, and that the image under 4 of a weight vector, ej,t, is

a symmetrized version of it.

THE CLASSICAL A N D QUANTUM 6j-syMBOLS REPRESEN l A l l U l V 3 U p u \J1\"J/

proof. Since the given composition maps an irreducible repre-

sentation V J into vk, it is O when k # j. And when k = j it is a

constant multiple of the identity. In Theorem 2.5.5 we computed

4a(xa+uya-u) 4 4 b ( ~ ~ + ~ €3 yb-')))

= x 2 ~ ia+b-~ia+b-~ ( - I ) ~ - u u+u=1

(c, + 1, - ? ) ' ( ( L t 7 1 ) ' ( h t !))I ( - ] ) ' - i ' -- .-

(2a)!(Lb)I Proof. Part ( 1 ) is a direct calculation that can be achieved by

evaluating either side on the highest weight vector, 4J ( 2 2 ~ ) . part

( 2 ) follows from ( 1 ) and Lemma 2.7.2 by rotating the vertex of the

right side of the diagram for ( 2 ) 180'. Parts (3) and (4) follow by ( ( a f b - j)!)2(a+b+j+l)!(a+j-b)!(b+j-a)!

on the tensor products of appropriate weight vectors. ( 2 ~ + l ) ! ( a t b - j ) !

1 b + j - a)!(b + j + a + I ) ! ( a + b - j ) ! (a + j - b).( (2a)!(2b)!(2j t I ) !

2.7.6 Theorem. ~h~ next to the last equality is a combinatoric identity. The

proof that follows was indicated to us by Rhodes Peele. Consider

where is a Kronecker 6 function, A, = ( - 1 ) 2 ~ ( 2 j + 11, a + b + j + 1 ) such that the value f ( a + j - b + 1 ) is

@ ( a , b, k ) =

every element of f ( { a + j - b + 2 , . - ., 2j + 1 ) ) - We count the

elements of the set B in two ways-

50 THE CLASSICAL A N D QUANTUM 6j-SYMBOLS

There are ('+,'tl) possible choices for the image f ({1, . . . , 2 j+

I)), and each such choice can be arranged in (a + j - b)!(b + j -a)!

distinct ways while f ({2j + 2,. . . , a + b + j + 1)) can be arranged

in (a + b - j)! distinct ways. Thus

Alternatively, B = UeBe where Be = { f E B : f (a+ j - b f 1) = a+b-j' f + I), so # B = xe #Be. There are (e-(a+j-b) ) possibilities for

the set f -'({l,. . . ,!)) since the inverse image must exclude the

integers in the closed interval [a + j - b + 1 , 2 j + 11 and include the

integers in the closed interval [I, a + j - b]. Each such set can be

arranged in l! distinct ways. Furthermore, f -'({l+ 2,. . . , a + b + j + 1)) is determined by f -l({l,. . . , !)) and this can be arranged

in ( u + b + j - e)! distinct ways. So that

strings (= number of tensor factors of v'f2 involved). The loop

closure of 'Czj gives the value Azj = (-1)~j(2j + 1) which is the

denominator of the right hand side. The value AZj gives the spe-

cial case when a = 0 of Theorem 2.7.6. The closed network in the

numerator on the right has the value @(a, b, j ) - this is why the

function is named theta.

2.7.7 Corollary. Let (a, b,j) denote an admissible triple of half-

integers. Then we have equality between the following spin-nets.

Now let C = a + u, and let j = u + v, we have

2.7.8 Lemma. The 6 j symbols possess the following symmetry

(a t j - b)!(b + j - a)!(a + b - j)!.

The required identity follows by rearranging the factors in the Proof. Embed the spin-net in a rectangle with the edges S

above equation. This completes the proof. labeled m and p attached to the top edge while the edges labeled

The spin-net version of Theorem 2.7.6 is indicated in Corol- s and t are attached to the bottom. By Lemma 2.7.2, the eval-

lary 2.7.7. Here we have labeled the spin-nets with the number of uation of the spin-net remains the same when the cross bar of

52 THE CLASSICAL AND QUANTUM 6j-SYMBOLS

the mH is rotated clockwise 180' while the end points of the s t

boundary remain fixed.

Having performed such a rotation, recouple; the 6j-symbol I

that appears is , where a sum is being taken over

u. Then rotate the (now vertkal) cross bar 180' counterclockwise.

We obtain,

Thus the coefficients are equal since pm @ p p o u 04, @ I & forms ,If a ba.sis for the set of U(sZ(2)) invariant maps V S @ V t i V m @I VP.

2.7.10 Lemma. Let TET(a, b, c, d,e, f ) denote the value of the

spin-net depicted below:

1 Then,

Proof. The proof follows by recoupling and then applying Corol-

1. <IIJ . . 2.7 I . , - and L C I I ~ L I ; ~ 2.7.9 ( ~ f . [!(i]. ;). I IS). .\ il<i:ti.l~ of tlic

Sagrammatic proof is shown below.

n

Proof. Any of these constants is the coefficient Z in the ecluation f

That there is such a constant Z follows since the space of U(sZ(2))

invariant maps vk -+ V m @ VP is 1-dimensional and is spanned mP -

by pm @ pp o Y o 4k. We leave it to the reader to draw the -

k e d c

corresponding diagrams. a

54 THE CLASSICAL AND QUANTUM ~ ~ - S Y M B O T 9

"-- - 1; 2.7.11 Remark. In [18] Kauffman and Lins give a closed for- i gives that isoto] I mula for the value TET(a,b,c,d,e, f ) in both the case at hand

[ ! and in the quantum case. Furthermore, there are methods to corn- affect the spin-net evaluation. Lemma 2.7.5 indicates how the spin Pute the values of the 6j-symbols based on the Efiott-Biedenharn net evaluation changes when a crossing is introduced. Changing I

11 identity [2] or the recursive properties of the projectors [24]. rn orientations can be achieved by twisting the trivalent vertices, and Section 2.8, we define four fundamental 6j-symbols, and use their

by strings through as indicated in the diagrams below. 11 values and the ELliott-Biedenharn identity to compute the values Here we have indicated crossing information So that the reader

of the 6j-symbols in general. ie ,

can see how to pull the diporam nn t h e r i ~ h t out to reverse the

I' i* 2.7.12 Lemma. The symbol,

orientation of the projectec . . .

I ---

= T E T ( ~ , 6, c, d, e, /I/ J I W , 6, I)W, e, m a , c, ~ ) o ( B , c, e ) ~

8 is invariant under all permutations of its columns and under

the exchange of any pair of elements in the top row with the

corresponding pair in the bottom row. Equivalently, the symbol

$1 [ 1 : ] kinvariant under the permutations of the set I (b+f-a)+(a-c-d)+(d+e-f ) + ( c - ~ - ~ )

(I

({a, b, f >, {a, c, d>, {b, c, e l , {d, e, f}}.

(1 Proof. This set -is the set of vertices of a tetrahedron with l i

edges hbeled by half-integers a, b, c, d, e, f , such that any element /I in the set forms an admissible triple. We choose an embedding of

B~ Lemma 2.7.10, the normalized 6 j coefficient

the 1-skeleton of this tetrahedron into a rectangle (see for example [: r ]

is the value of the embedded tetrahedral network divided by

I the diagram below). This embedded labeled graph is a spin-net the factor J(@(a, b, f)O(d, e, f )@(a, c, d)@(b,c, e)l which possesses

I and as such determines a U(sl(2)) invariant map c + c (such tetrahedral symmetry by definition. This completes the Poof.

56 T H E CLASSICAL A N D QUANTUM 6j -SYMBOLS

2.7.13 Theorem. The orthogonality relation and the Elliott-

Biedenharn relation hold for the normalized 6j-coefficients in the

following form.

Orthogonality,

Elliott-Biedenharn:

Proof. These r rs~r l ts follorv from Tbrorerns 2.6.6 r~, , , r l 2.6.7

subst i tu t , io~~, allti ck~,~iceliilg Os OIL eitlier side oi. tile equations.

2.7.14 Theorem.

.{'J p } d k n

a n p = x ( - 1 ) 2 n ( 2 n + 1 ) [ b f n ] [ n d j c d k j ]

~ L A S S I C A L AND QUANTUM 6j-syMBOLS REPRESENTATIONS OF U(d(2))

Therefore, comparing the coefficients for fixed m and p,

P r o o f of p a r t (6).

a

Next we observe that part (2) follows from (I), part (4) follows

recoupling using the 6 j-coefficient

2.8 A recursion formula for t h e 6j-symbols. In

Biedenharn-Louck [2], a recursive method to compute the 6 j -

symbols is presented. We summarize this method here.

First, we define the following four 6j-symbols to be the fun- On the other hand,

damental 6j-symbols. Their values can be computed by means of

THE CLASSICAL A N D QUANTUM 6 j - s y M B O L S REPRESENTATIONS O F U ( s l ( 2 ) ) 65

the recursion relation for + given in Section 3.5 a t A = 1. see [241

for details. An alternative computation in the classical case (the

case a t hand) is given in [ la] .

sibility conditions force j = f f 112 and k = g f 1/27 yields

2s 112 112 x + 1 / 2 g + 1 / 2 e - 1 1 2 f + 1 / 2

2 - 112 112 z + 1 / 2 g + 112 e - 112 f - 112

2 + 1 / 2 112 x - 1 / 2 g - 112 e - 112 f + 112

2 - 112 112 x - 112 g - 112 e - 112 f - 112

( e + c - j ) ( y + h - ~ + l )

This pushes everything back to the case ( e - c + f ) ( e + c + f + l ) ( f + d - g)(g + h - + 1 )

( 2 e ) ( 2 f )(% + 1 ) ( 2 f + 1 )

( e + c - f ) ( f - d - g )

( e - c + f ) ( e + c + f + l ) ( f - d + g ) ( f + d + g + l )

( 2 e ) ( 2 f ) ( 2 f + + 1 )

The coefficients A, B, C , and D are values (with suitable

choices for j and k ) of the product

which can be using the four fundamental 6j-symbols-

2.9 Remarks. In the above discussion, we have reproduced

proofs of the important identities among the Clebsch-Gordan co-

/ 66 THE CLASSICAL AND QUANTUM SYMBOL^ I ' iR

I) 3 Quantum efficients and the 6j-symbols by means of the spin network anal-

! !/: ysis of Penrose [27] and Kauffman (16, 181. The diagrammatic or 3.1 Some finite dimensional representations. In Set-

F $1 graphical method seems to be well known [3, 21 but only with the tion 2 we showed how the tensor products of finite dimensional work of [19] and its sequels has this been brought to the fore-front

of sZ(2) can be decomposed. of the theory. The identities 6.16-6.19 given in [19] for the repre-

/ I / In this section we mimic this classical theory in the so-called sentations of the quantum group are given above (for the classical

quantum case where the representation spaces are spaces of hO-

i i case) as Theorems 2.6.7, 2.6.6, and parts of Theorem 2.7.14 (but mogelleous polynomials in two variables that only c ~ m m u t e UP

not respectively). / il to a parameter. ~t is these representations that give invariants of The fact that these proofs are expressible in terms of simple

h '$dimensional manifolds, and physical applications are found in diagrammatic manipulations points to various levels of generaliza-

statistical mechanics where they provide solutions to the Yang- tion. In the next section we explore one such level: knot theoretic

Baxter equation. quantization, in which the braid group replaces the permutation

Recall we have ~ o u l ~ u I I I C U U L I I J ~ ~ ; r = y l = - r r u w v A u r r v -- - - \ - I - -

group, and a parameter is inserted into the binor identity. This V J = {llomogeneous polynomials of degree 2.7 in x and Y). Namely,

notion of quantization coincides with the notion of a quantuln l f (1 + I ) = 27

Z ~ O I I ~ I < \ \ ( I ( > \ c I < ~ j ) ( l ( \ I > \ . J I I I I I ) O I ~ I , 1 0 ,-,, < , I t < / ( , , [, , , + I !>-I

L ; [ L " ~ J " ) = / ) I tJ for example [ll] for further references.)

a-1 b+l Some puzzles remain about these techniques. Combinato- F ( x ~ Y ~ ) = ax y

rial identities arise - for example, recursion relations among the ( a - b ) n(xayb) = -xayb.

Clebsch-Gordan coefficients. Are there simple discrete probabilis- 2

tic reasons for these formulas, and if SO, how are the more elemen- n.. ... :+L +Lo nnt-ltinn h o t t ~ ~ adanted to the theorv of weights, tary combinatorial formulas expressible diagrammatically?

.- z3+m 3-m In the next section we turn to the finite dimensional repre- el,m -- Y

sentations of the quantum group Uq(s1(2)). We will see the direct

analogy-with the classical case as the theory develops, and in Set- Eel,m = ( j - m)ej,m+l

tion 4, we examine in detail the special case when the quantum - Fe3,,,, = ( j + m)e3,m-1

parameter is a root of unity.

Hel,m = me,,,

from which it follows that

IE, F]ej,m = 2me3,m

6 7

Ill THE CLASSICAL AND QUANTUM ~ ~ - S Y M B O ~ ~

/ I

We want to "quantize," which, in our context, means introduce second, the action of E , F and K is given by

a parameter. Roughly speaking, we want to replace integers n 2 0 EeJ,m = [ j - m]e~,m+l

by

n-1 + qn-3 + . . . + q-(n-l) = qn - q-n Fe3,m = [ j + mIe3,m-1 [nlq = 9 q - q-l '

Kel,m = ~ ~ ~ e 3 , m If q # 0 is fixed, we will write [n] for [n],. By definition [0] = 0 and

[I] = 1. The theory of the quantum group U,(s1(2)) is particularly from which it follows that interesting in case the quantum parameter q is a root of unity

[E, F]e3,m = [2mIe~,m- because [n] = 0 when q2n = 1. This case will be covered in detail

in Section 4. ~h~ representation V i is irreducible provided /I4' # 1 for 5 5

3.1.1 Definition. Let q # 0,1, -1 E C . A representation of 2 j . i

Uq(s1(2)) is a vector space W with operators E, F , I< ( l i " = " o H ) Prnnf Comnute d such that

I{2 - I{-2 q2H - q-2H [E, Fl = 11 - ,, -

4 - 4-I Q - 4-I

.. A ---.. irectly the actions of E F , FE, K E , E l i , K F ,

1 and FIi. The identities for the l i e brackets follow by manipdat-

ing the rational functions of q that result. To prove irreducibility, C . . ^ " _ , _ _ .

Ti I? = q E r , lilese[orc, tile illla,ge ally null-Lc.!.o \ c C l i ) l ~ l i ; i ~ ( ' i I ) O \ ~ ( ' l ~ 1:

l i F = y - l ~ ~ <

and F spans vi. where [E, FI = EF - FE. The quantum group Uq(s1(2)) is the al- one of our present goals (Theorem 3.4.1) is to give concrete gebra over C that is generated by E, F, I<, and K-1; the elements realizations of the abstractly given representations V i . The real-

i : are subject to the relations specified above. For the tirne being, izations will be !generated from the fundamental representations we will not deal with this algebra directly, but instead we v 1 l 2 by tensor products. We can take tensor products of quantized work with it via its representations. ~ h u s we will only consider representations because the algebra Uq(s1(2)) has a comultiplica- the entire algebra (and co-algebra) ea: post facto. tion. Rather than specifying the comultiplication, we wig specify

the action of uq(s l (2) ) on tensor products of representations; from 3.1.2 Theorem. . The representations vj -of s1(2) can be quan- these formulas the comultiplication can be derived.

tized i n the following sense: Let q # -0,1, - 1 E C . Let j {0,1/2,1,3/2, . . .), and let A E C , where A2 = q. ~h~~~ is 3.1.3 Theorem. Let U and V be representations of U q ( ~ l ( ~ ) ) -

a ( 2 j + l)-dimensional representation ~i of ~ , ( ~ 1 ( 2 ) ) given ab- ( A ~ always, # o , l , - l ) . Then there is a representation of

stractly as follows: First, a basis for V; is ~ , ( ~ 1 ( 2 ) ) on U 8 V given by the following formulas-

'1 THE CLASSICAL A N D QUANTUM SYMBOL^

Proof. Three relations must be verified. 0

3.1.4 Remarks. The representations for the quantum case wiU .d 1 be made explicitly analogous to the those of the classical case. In ,u

71

lassically, the obvious representation of the symmetric group

2j (permuting factors) on ( ~ ' / ~ ) @ ~ j commutes with the action of

sz(2) on the same space. The subrepresentation on which EZj acts

trivially, namely the space of symmetric tensors, is isomorphic to

the representation VJ of sl(2). Since representations of the braid

situation, rather than

particular, we will find the quantum analogues of the spaces of

homogeneous polynomials as irreducible representations and the quotient representations in the quantized

identification of these with subrepresentations in the tensor prod- subrepresentations.

uct of copies of the fundamental representation for generic values d j i ~ i of the Parameter 9. These subrepresentations are the images of 3.2.1 The Artin braid group B(n) is given by gen-

projectors that are deformations of the projectors in the classical 1 ' erators and relations as follows: case. 1 ; (sl ,s z,... s : sxs, =s , s t if l k - j l > l ;

Via such identifications, we will be able to perforlll the dia-

1"' ~ , r a n i n i a l i c . ( ( ~ ~ i ~ p 1 1 1 ~ 1 i o r i ~ 1 1 1 ~ 1 1 I I I C r l ~ l c ~ l ~ s ~ l l , 10 f j l O , f , 1 j i C ,

l i r sical case. Specifically, we will show in the quantum case how Sksk+lSk = ~ ~ + ~ s k s k + l if k = l " A - - - , n - 2,

111 to decompose tensor products of representations (Clebsch-Gordan Remark ~h~ braid group is depicted graphically as indicated

Theory), and develop the 6j-symbols and verify their important 'I!

below. ~ ~ l t i ~ l i ~ ~ t i ~ n is achieved by vertical juxtaposition braid

I'

the so-called trace 0 representations that arise.

3.2 Representations of the braid groups. To construct the concrete realizations of the quantum representation spaces,

we will find a natural quotient representation of (v;/')@~J that

is U,(sl(2))-isomorphic to V i . To describe the quotient, we in-

troduce an action of the braid group B(2j) on ( ~ ; ' ~ ) @ ~ 3 that

commutes with the action of U,(s1(2)). Furthermore, for generic

values of q or for q an r th root of unity (where r > 2j), we will The braid diagram for s l s z s l s ~ l

72 THE CLASSICAL A N D QUANTUM SYMBOL^

3.2.2 Definitions. Let A # 0 E C be fixed, and let

Here we redefine the matrix representation of the maps U and n to apply to the quantum case.

Define nn = n : vii2 @ viI2 + C = V i via

?/ @ x I--+ - iAW1,

and

Y @ Y + + O .

Define U A = U : V j = C - v;" @ vlI2 by the formula

b

QUANTUM sl(2)

;

Proof. This follows by computation.

3.2.4 Definition. We define a Uq(s1(2)) invariant map (called

a positive crossing)

as follows:

-1

Observe that ( A ) = A-1 [ ] 4- A [ 1 @ 1 ] . Let the negative crossing be defined as follows:

k

3.2.3 Lemma. Let A2 = q. i \ L A /J .

The UA and n~ are Uq(s1(2)) invariant. in section 2, the symbol I denotes the identity map, but here

2. ~ A O U A = O A : C - + c the domain is v:I2.

is multiplication by - ~ 2 - A-2 = -[21.

3.2.5 Theorem. There is a representation r~ of B ( n ) on 3. U A o nA = u : vl/2 V;/2

n -+ vy2 ,g, viI2 ( v : /~ )@~ defined by is given by

- . U . I . . . I Y n ( x @ x ) = n l-J ( Y @ Y ) = ~ , rA(sk) = -/

k-1 n-k-1

4 * + fork = 1,2 ,..., n - 1.

and Proof. It is clear that the images of distant braid generators U n ( Y @ x ) = X @ y - q - l y @ x . commute. That the relations sksk+lsk = sk+lsksk+l hold in the

representation for k = 1, . . . , n - 2 is a direct computation wl

11 t 74 THE CLASSICAL A N D QUANTUM 6 j - s ~ ~ ~ ~ ~ Q U A N T U M d ( 2 ) 75

li i I is usually performed diagrammatically and depends on the far proof. The situation is analogous to Lemma 2.4.2 and

t ' that rem 3.2.5. So details are omitted. 8

iJ il (n @ I O ( I @ OX(* )) = ( I @ n) 0 ( X ( r f ) €4 ( )

3.3.2 Remark. The relation where X(&) denotes the positive or negative crossing depicted

above. (See [16] for the diagrammatic version, or compare with h i = ( -A2 - ~ - ~ ) h k Lemma 2.3.2.)

I in T L , ( - [ ~ ] ) cuts the braid algebra down to a finite dimensiona1 3.2.6 Theorem. Let A E C , q = A2 # O,1, -1, The acttons -

of Uq(si(2)) and of B ( n ) (via T A ) on (v:/~)@" are commuting algebra. 1" 1 (J I actions.

3-3.3 Remark. The formula for p ~ ( s k ) is referred to as bilii ,I; i Proof. This f~llows because T A is defined in terms of the ~ ~ ( ~ l ( 2 ) ) bracket identity. This identity encapsulates Kauffman's simplifica- .r

ji .: maps and 1 4 1 . tion of the Jones polynomial 1161. Recall that the bracket identity ';

I l 1 3.3 A finite dimensional quotient of C [ B ( n ) ] . The rep- is given diagrammatically as:

l'p%'llf~t~ions Y..l 1~'Pl.p tliscovcrrt] via f h c . i l l tc l . r ! lc~c] i ; i l . i . c. , , , . : ; , i i ,

finite dimensiona.1 algebras that are quotiellts of tile (inlil,ite di- 'y =jL( X ) + 1 I ) . mensiona.1) group algebra of B ( n ) .

/

Given a knot diagram, K, of a knot, K , the bracket identity Next we consider the Temperley-Lieb algebra T L , ( ~ ) where

I@ is used to compute a polynomial (K) in A*' by removing each :& ,j;) I 2 and where 6 = - (A2 f A-') for some A # 0 E C . Recall crossing via the bracket identity and associating the loop value ig that T L n is generated by elements I and hk for k = 1,. . . , n - 1

- ~ - 2 - ~2 to each of the simple closed curves that results in that are subject to the relations given in Section 2.4.

any of the daughter diagrams. The bracket is an invariant of the

3.3.1 Theorem. For n 2 2 and A # 0 E C there is a surjective regular isotopy class of the diagram, but not of the knot type A'- algebra homomorphism TO obtain an invariant of K, define

- -

PA : C [ B ( n ) ] -, TL,(-[2]) L ( K ) ( A ) = ( - A ) - ~ ~ ( ~ ) ( K )

that is defined on generators as f o ~ l o ~ ~ : . .

76 THE CLA'SSICAL A N D Q U A N T U M 6j-SYMBOLS

It is clear that the representation r~ of the braid group defined

in Theorem 3.2.5 factors through PA. The algebra TLn(-[2]) is

represented on (v;I2)@'" via the map

as in the classical case but with = . We have the follow-

ing result: A

3.3.4 Theorem. The representation

QUANTUM d(2)

3.4 A model for the representations Vi. Let

be the tensor algebra of v;I2, and let

LA c en(^;/^)

be the two-sided ideal generated by qx €3 y - y @ x (where q = A2,

as always), and let

112 @2j . ~ 2 A j = LA n (VA )

Then there is an isomorphism

is faithful. ~ e n ( v : / ~ ) / ~ a C[x, Y]/(YX - qxy)

The case when A = 1 is covered in Theorem 2.4.3. The case

i = -1 i t l l l c > ,<IlllC*<l, -4 = 1 l ~ ~ ~ < < l L l ~ < ~ I)OLIL 1 f J , L ( - [ 2 ] ) <l l l< l I t \

representation OA depend on A only through q = A2. The proof

of the generic case and of the case that A is a 4rth root of unity

and r > n is explained in Section 4.2. The remaining case is

proven in [7].

3.3.5 Notation. Let IA(n) denote the two-sided ideal in

TLn(-121) that is generated by the set {hl,. . . , hn-l}.

3.3.6 Lemma. The ideal IA(n) is a proper ideal, of C-vector

space codimension 1 in TL,'(-[2]).

Proof. The space TLn(-121) = TL, is spanned by 1 and the

monomials in the hk, so the codimension of IA(n) can only be 0 or

1. We need only prove that 1 6 IA(n). This is easily seen from the

representation of TLn(-[2]) on ( ~ 1 ' ~ ) ~ " . For l (x 8 x 8 . - -8 x) =

x @ x @ . . . @ x , but X @ X @ - - . @ x @ I A ( ~ ) ( x @ x @ . . . @ x ) .

1 between the quotient algebra and the algebra of polynomials in

tile I L ~ I I C ~ I ~ I I ~ I , , ~ I I I ; \ . , I 1.11)1(~\ 1 r ~ i l ( l t/ \\ I I O I C 11 1 = (1 I I / (TI\(>

parameter q commutes with x and y as it IS a member of the

ground field.) We will identify these two algebras in the sequel.

Moreover, we have

T ~ ~ ( v ; / ~ ) / L ~ = @ (v1l2)@2j/ L; j = O , l , ...

where (~1l")ej / L"Aj

% can be identified with the space of homogeneous polynomials of

degree 2 j in x and y, where yx = qxy.

A direct computation shows that

112 82,. ~2 = I A ( ~ ~ ) ( V A )

112 @23 Because IA (2 j ) is an ideal in T L2, (- [2]) whose action on (VA )

commutes with the action of Uq(s1(2)), it is clear that L? is a

78 T H E CLASSICAL A N D Q U A N T U M 6j -SYMBOLS

subrepresentation of ( ~ l / ~ ) @ ~ j for the actions of all three of T L Z j ,

B ( 2 j ) , and Uq(s1(2)).

3.4.1 Theorem. The actions of Uq(s1(2)), B ( 2 j ) , and T L z j on

(v1I2 A 1 @ 2 J / ~ y can be described as follows:

1. For T L 2 j :

IW = W , hiw = O for all w E ( v A ~ ~ ) @ ~ ~ / L Z

2. For B ( 2 j ) :

S ~ W = A-lw for all w E ( V , / ~ ) @ ~ ~ / L : :

3. For Uq(s1(2)): There is an isomorphism

3.4.2 Lemma.

1. q X a y b ) = [ a ] ~ a - b - l x a - l Y b+1

3. l i ( x a y b ) = AaVbxayb

Proof. Notice that v:/' consists of the set of linear combinations

of x and y. Thus Ex = Fy = 0 , whi1e.F~ = y, E y = x , K x = A x ,

and K y = A-lx . These computations form the initial steps of

inductions that will follow.

We compute, I i (xBa @ yBb) = (Kx )Ba @ ( ~ y ) @ ~ = Aax@" @

A-by@b = Aa-bxBa 8 yQb where, for example, xga is the tensor

product of a factors of x. Thus the third identity holds.

By Item ( I ) , we mean that F(xga @ yBb) [ a ] ~ ~ - ~ - ~ x @ ~ - ' @

- ,. , : 1 .;: - j 1 . I / ' )'???.i, / 2 i . I 1 .

4 : Y ~ ~ ~ + ~ ( m o d L,t).

First consider the case whe~e = 0. Illcll lllduct 011 (1. I.0'

F(xBa) = F(x@~-' B x >

Henceforth, we will identzfy V i with ( V : " ) @ ~ J / L ~ via the iso- = ~ - 1 ~ @ ~ - l 8 F X + F(x@"-') 8 K x

[ a ] ~ a - l ~ @ a - ' @ y (mod L A )

Proof. Item ( 1 ) follows because ~ , ( ( V : ~ ~ ) @ ~ J ) c L?. Item ( 2 ) because A1-a + [a - l]Aa+l = [a]Aa-l.

follows from (1) and the bracket identity. Item (3) follows from For general b 2 1, the next Lemma which indicates that in the case at hand E, F ,

and I i act as quantum differential operators; so these actions are F(x@" @ yBb) = F ( x @ ~ @ yBb-' @ Y )

analogous to those given in Section 2.2. - 8 Y @ ~ - ' ) @ Fy + F(xBa €3 ymb-l) @ ICY -

- - - [ a ] ~ a - b x @ a - l @ yb @ A-ly (mod LA.)

The of ( 2 ) follows along the same lines. This completes

the proof of the Lemma 0.

80 1 THE CLASSICAL AND QUANTUM SYMBOLS

d 3.4.3 Remark. Let XA be the one dimensional character o ~ B ( ~ )

Y such that xA(si) = A-'. (If A = 1 or -1 then XA factors through

I 4

the ~ermuta t ion group En, giving the trivial character (A = 1) or

111' the sign (A = -1). Thus XA is a sort of "generalized sign".) Part 1: (2) of Theorem 3.4.1 asserts that the representation of B(2j) on 4 1

'ijir the space of homogeneous polynomials of degree 2 j in x and y is

through the character XA. 3 -%

In the next section, we will define for certain values of A, a : Uq(sl(2)) invariant projector $5 : (V;/~)@U -+ ( v ~ / ~ ) @ ~ J whose 4

I image is a subrepresentation that is isomorphic to Vi. The pro-

jection is analogous to the projection in the classical case, and "!

Observe that +,A is not defined when [n]! = 0; thus we assume

further that A4T # 1 for 15 r < n.

3.5.2 Theorem [13]. cf. [18, 23, 161. The element 4-1 t TL,

satisfies the conditions:

1. h, o(+" O/)O h,o(+! B I) = ( - l ) w h n o ( + k €3 1).

2. +; 04; = 4;.

3. + a o = . . 4; = o for all u E I A ( ~ ) .

Moreover, any non-zem element in TL, safisfying these condi-

tions must be equal to 4;. it we will be able to construct the quantum Clebsch-~ordan

coefficients and the quantum 6 j-symbols. proof. The recursion relation above defines the projector, 4:.

We will mimic the algebraic proof presented in [16], but here we

3-5 The Jones-Wentzl projectors . 111 analoRv wit], t hc clas- use diagrams. The diagral1lma,tic version of tlle recilr5ion relation

sica.1 ca.se, we will define idempotents, called the Jortes- 13Ve1ztzl , , I * . [ 1 1 , - 1 1 , 1 , i - , , , , L.4 (.,., I i

1 projectors,

!I in the algebra of Uq(s1(2)) invariant transformations on (v ; /~)@~.

3.5.1 Definition. Let A # 0,1, -1 denote a complex number, -3 9

and let A2 = q. There is a canonical embedding of TL,-l into 5 * TL, obtained by juxtaposing a straight string on the right of

every generator in TL,-l. We will consider all of the elements in . 'j rn

1 - TL, to also be in TL,+k for the rest of this section. Define the

7M

is depicted as follows:

We will assunie, by induction, that +;-l+al = + a l , and

that +i-lhi = hk+!-, = 0, for 1 < k < n - 2 the case of +f '-8 Jones- Wentzl Projector, 4: E TL, via the recursion relation:

being trivial. The

+a =+A, €3 I + ln - 1 1 / [ ~ ] ( + 5 ~ €3 I ) h n - ~ (+fl I )

n we show that

where

4: = I = id : vii2 , y;l2. by applying h, to the recursion for 4; to obtain hn 0 (+A O 1) =

83

ost horizontal line in the second icon from the left was absorbed

e single vertical strings in the second and fourth icon into the

ring to the left. In this way, the labels on the strings change

om n - 2 to n - 1. (The labels on the vertical strings on the last

on are omitted for type-setting reasons) Furthermore, we have

composition of idempotents. So the above sum reduces to the

- (-PI + En - 11/Inl).

+ [n-l] /[n] n - 2

'I'HE CLASSICAL A N D QUANTUM 6 j -SY

w e obtain hn 0 (4: @ 1 ) 0 h n ( e 8 1 ) = -[n + l]/[n]hn(+f 8 11 and the projectors involving n - 1 strings are absorbed since these by the identity are idempotents by induction. So far we have shown that +; are

dempotents. Furthermore, 4; # 0 because the coefficient of 1 , among quantum numbers that is easy to verify. Next we use this n the sum is 1, as can be seen by induction.

to show that +f+; = +A n . By induction and by this recursion relation, h k e = 0 = e h k

if k = 1, . . ., n - 2. Next we compute hn-l+i =

The last sum is obtained again by the fact that (hn-1e-,)2 =

- [n] /[n - 1] hn-l+i-l and the sum of these two terms is clearly . .

0. By the top/bottom symmetry of the diagrams, it follows that

' I ' , A ~ , - ~ = 0. Thus we have an inductive construction for the

Next we show that these elements are unique. Any given ele-

ment in the Temperley-Lieb algebra can be represented as al, +U

where a is in the ground ring, 1, denotes the identity, and U E

IA(n) so that U is a linear combination of products of the hj .

Suppose that g: = g, is a non-zero element of T L , such that

gnIA(n) = (0). Then g , = a ( , +M = a21, + 2 o l ~ + M 2 . In particu- In the fourth term, the icon that represents (hn-l o 8 lar, a2 - a = 0 since I n 6 IA(n). If a = 0, then 0 = gnu = g i = g,. is replaced by the icon representing -[n]/[n - l ]h , (+ ,~_~ 8 1 2 ) , Hence a = 1. Thus any non-zero idempotent fn that kills IA(n)

86 THE CLASSICAL A N D Q U A N T U M 6j-SYMBOLS I

I!

can be written in the form fn = In+U1. Finally, gn = gn(ln+,gl) = 4 ( I , + U) fn = f,. This proves uniqueness 0.

.=

QUANTUM sZ(2) 8 7

First we recall the canonical inductive construction of the set

of permutations. Let (k, . . . , n) denote the cyclic permutation of

the elements k through n in C,; this cycle (k, . . . , n) can be written

as a product of adjacent transpositions: (k, . . . , n) = (k, k + 1)(k + To a bijection t : (1,. . . , n) -+ (1, . . . , n), we

), where t(n) = k, t = (k, k+ 1,. . . , n)tk, and

e regarded as a permutation of (1,. . . , n - 1). . -

!r of adjacent transpositions that

equal to me minimal number that it takes to

lding path that starts from the lower left and

3.5.3 Notation. For every permutation a E C,, define a

braid 13 E B(n) as follows. Write a in any way as the product of 1, k+2)-. . -(n-1, n). a minimal number of adjacent transpositions (i.e., transpositions

t associate a pair (k, tk P4 of the form a k = (k, k + I)), and let T(a) denote this minimal

tk(n) = n so tk can bl t I

number. Then lift the product to B(n) by replacing each of the

I* In this way, the minimal numbf transpositions a k in the product by the corresponding braid gen-

it takes to write t is 4 ' ' "

14 erator s k . The minimality of the product for a insures that the lift t r write tk plus n - k. I b will depend only on the permutation a and not on the particular

Consider an ascel 14 I*' product representation chosen.

travels upward through the triangle depicted below. Diagrammatically, the transposition ok is represented by n

'I arcs running down the page in which the kth and (k + 1)st arcs (0 , l ) (1,2) ( n - 2 , n - 1) ( n - 1,n)

cross. The corresponding braid generator s k can be represented (0 , l ) (1,2) ... ( n - 2 , n - 1) . . .

13v t he same arcs, where the kt11 arc crosses over t he k + 1st.

Next rep~esent the b ~ a l d g ~ o u p I I I ~ O the dlgcbld 01 L I ~ I I S ~ O I - ( 0 , l ) ( 1 , 2 )

mations on (v:/~)@~ via the bracket identity. Let [ X I denote the (071)

quantum integer that corresponds to the integer x. Finally, let A permutation can be represented by such a path as product

[n]! = [n][n - 11 - - [I]. of the switches to t

the product of the E 3.5.4 Proposi t ion [18]. The Jones-Wentzl projector is also

given by the formula:

P r o o f of Proposition. We must show that the formula above (0, I) , (1,2), . . . , (n - 1,n). In fact, such descriptions of permuta- .

defines an element that kills each of the hk, 1 < k < n - 1, and tions by paths always use the minimal number of adjacent trans- that the coefficient of 1, is 1 when the sum is expanded in terms positions necessary to write the permutation. of the standard basis for the Temperley-Lieb algebra. The proof We compute the coefficient of 1, in the sum ~uEz , , (~ -3 )T(u )3 . we present follows [18]. Rarh n~rmnta.t inn in t h e sum contributes a term of the form

88 THE CLASSICAL A N D QUANTUM 6j-syMBOtS U A N T U M ~ 1 ( 2 ) 89

?.

A - ~ ~ ( ~ ) to the coefficient of 1, by the bracket identity. we need 3.5.5 Remark. The formula given above for the projector in-

the computation icates precisely how 4: is analogous to the classical symmetrize !.' ,

ing projection 4,. Moreover, the classica~ projection satisfies the = A*~T(o). same recursion relation with quantum integers replaced by inte-

i < a€Cn gers. such a replacement is an evaluation of the quantum Pro-

To prove this, we use long multiplication to multiply out jector at A = 1, and for that value of A, positive braiding is

I A ~ ( ~ - ~ ) [n~! = indistinguishable from negative braiding. When the Jones-Wentzl projector +,A is defined ( e - g . whell A

is transcendental), it is a nonzero central idempotent such that - (A4(,-') + A4tn-') + . . . + A ~ ( ~ - ~ ) + . . . + 1) . . . ( ~ 4 + 1) 42 . IA(n) = (0). So the Temperley-Lieb algebra has a direct sum

The terms in each of the factors are arranged along the horizon- decomposit~on as scalar multiples of the Jones-Went21 projector

tals in the triangular array that is depicted above. A term in this and the ideal I A ( ~ ) :

expansion corresponds to an ascending path. Moreover, the co- TL,(-[2]) = c+! (33 Ia(n).

efficient of when like terms are combined, is the number of

paths that have k: points to their right, and tllis is the number of lve have the direct sum decomposition

r ) ( : ~ , t ~ l ~ ~ t a t intis 111a t 111itti111i111,~ I ISO 1; ; i (~, j i ,c(t l l t I , . ~ , ~ , 5 , ~ ~ ~ s j ~ j ~ l , , 5 ~ .~.i,,~,> 1 1 2 ,.:.,, - +;A(\,;/~),<JIL d, 1,; (\,i ) - coefficient of I,, in our expressioll for 4: is 1.

Now for any given k, the minimal product representations of for representations of B(n) and Up(sl(2)) where

the Permutations can be chosen in such a way that the set of 112 8" L; = IA(~) (VA ) all permutations is partitioned into a set of words W that do

end wit11 a k = (k, f I), and the set Wok. (For example, is the kernel of the projector 4:. This leads us to the following:

// the triangular scheme above does this for k = 1.) Clearly, these ii 3.5.6 Definition. A Uq(s1(2))-equivariant map 11 two sets have the same number of elements, and the number of i 112 8 2 j

transpositions that it takes to write an element wak is one more $j = df ;": V: - (vA )

I than the number of transpositions that.is takes to write for is defined in terms of 'the projector as follows: w E W.

4 , ( e . j = A(j+m)(j-m)+A.(x~~+m 8 y~j- 'n) . The computation that 0 = (~ , , zn (~ -3 )T tu )d ) hk f o ~ o W S be- 3 3,m 23

cause 6khk = --A3hk, and SO the contribution of word w E w is This formula only makes sense when [2j]! # 0, and thus the

canceled by the contribution of wak. A similar argument value of the quantum parameter is important. When it is de-

that 0 = hi, ( Z , E ~ ~ ( A - ~ ) T ( ~ ) ~ ) . This completes the proof. fined, the map $j is a lift of the map w j (which was defined in

90 THE CLASSICAL A N D QUANTUM SYMBOLS

Theorem 3.4.1); in other words, pjq5j = wj where pj : (v1I2)@2j _, ( V ~ / ~ ) @ ~ ~ / L T is the projection.

The proposition that follows indicates that the image of $ j is

the quantum analogue of the symmetric polynomials in x and 51.

3.5.7 Proposition. For a + b = n,

where

t , (S) is the minimal nunzber of adjacent tran,spositions that it

) Q U A N T U M d ( 2 )

I 3.5.9 h m m a .

-

where

for R =

~ ( I L ' T . ~ 112 ~ ~ ~ o i ~ c - ." ! m ! / ? r . ~ i ! / , s r / f n -L 1 . . . ..!! -!.- 1): (, j. 1 , ; . ! Proof of Proposition 3.5.7. .4ssrilning Lemma. 3.5.9 nrc. indrlct.

1 on b. In case 6 = O the result Sollows by usi~lg inductioll on ,n a.liti

. #,- the recursion relation for the projector 4:. X I /2 8 2 3

$ For general values of b we use the maps 4j : V i -4 (VA )

defined above, where j = n/2. On the one hand,

k q5j (Fej ,m> = F43 (e3,m) i$ = A ( J + " ) ( ~ - ~ ) ~ ( + A ( ~ @ J + ~ n 8 y@3-m > 1 5

where +i is evaluated by means of the inductive hypothesis. On

the other hand,

dr

$J(Fe,,m) = h ( Z j + m]e , ,m-~)

= [ j + m]~(j+m-l)(~-m+l)+~(x@j+m-l n q y@~-m+l 1.

The proof will follow by comparison of the two sides of the

equation once we establish the following:

Proof. We have for example,

3.6.2 Lemma.

C q-2tn(R) = 9 -k(n-k)

R C {1, ..., n}

q P p l y Lemma 3.6.1 to complete the proof. 1 1

1 3.6.3 Definition. n-1 n-1 Let U=UA= U* : C 4 n @ n vii2. Hav- ing defined U = U A : C 4 (v:/~)@~("-'), define U = U A to be the

1 ' composition n-1 112 5 f i . ~ 3 v,1I2a ... @VA

where tn(R) is the minimal number of transpositions that it takes

to move the subset R to the subset {n + 1 - k, . . . , n).

Proof. The proof will follow by induction. We indicate the proof

of the first formula; the second follows similarly.

n n The map n=nA is defined dually, and is also in analogy with

the classical case.

3.6.4 Lemma.

&dl)= ( ~ A ) " - ~ I S I ~ ~ . . @ Z.s 32 . . . @ f

where

and

Also we have,

( ( x 1 8 . . - @ x n ) @ ( ~ n @ . . . @ Z 1 ) ) . .

= { O i f ~ ~ = ~ ~ f o r s o m e I c = l , . . . , n (iA)n-2S if {xk, Zk) = {x, y) for k = 1,. . . , n

where xk, Z k E {x, y} for all k = 1,2,. . . , n and s = #{k : xk =

Y).

96 THE CLASSICAL A N D QUANTUM SYMBOLS

Proof. As in the classical case, the proof follows by induction

(See Lemmas 2.7.3 and 2.5.3). In case n = 2 the non-zero values

are as follows:

Q U A N T U M sl(2) 9 7

I 3.6.6 Theorem. (1) Let A E C. Let (a, b,j) denote an admis-

k sible triple such that if A is a primitive 4rth root of unity, then

1 max {2a, 2b, 2j) < r. Then the Uq(s1(2)) invariant map y :":

(v:'~)"~' -+ (v ; '~ )@~~ 18 (v:'~)@'~~ is defined and

f i ( y @ y @ x @ x ) = ( i ~ ) - ~ . I j - c i(b-~)-(a-~)A(b-u)(btutl)-(a-u)(a+~+l) -

u+v=j I (2) Let A E C . Choose a,b E {0,1/2,1,3/2 , . . .I so that if A is

3'6'5 Definition. Let j E {0,1/2,1,3/2,. . .). Define a map a primitive l r t h mot of unity, then 2a t 2b < r. There is a direct p j : ( ~ j ' ~ ) @ ~ j + vj via

sum decomposition

/c j (X, ' @ ~2 @ . . . @ 2 : ~ ~ ) = X I . X 2 . . . . ' ~ 2 j

I!,? @ lfj = @ /LC, CJ /lf,

I where the lnultiplication on the right occurs in the ring of polyno-

mials in non-commuting varia.hlps r ?, F . ~ ; J ~ ~ + ~ ~ ~ ,, . ;" , rr / ,J/Q\\ where the sum is taken over all j such that ('7 b , j ) is

Furthermore, if (a, b, j ) is admissible, then any Uq(s1(2)) invariant

Next we mimic the classical case to define, for (a, b, j ) an ad- map V; -+ V; @ V; is a scalar multiple of Pa @ Pb

Q ------- - 7 3 - -"-"LL"L J 7 Pj "q\OO\'='))

map and its kernel is L?.

missible triple of half-integers, a Uq(s1(2)) invariant map

via the formula

Proof. Recall that ej,t denotes the weight vector in Vi of weight

t, and that ~ ~ ( e i , ~ ) = A(J-" ) (~+~) xjtm b-'. The restrictions on a b

and j insure that 'I' i is defined.

v ab (4, (ejj))) follows by computa . ' i / Ima 3.6.2, and Proposition 3.5.7.

The formula for pa @

,tion using Lemma-3.6.4,

J 1s

where Im is the identity map on m tensor factors of Vi/2 (Cf. defined and non-zero for all 3 such that (a, b, j ) is admissible and Section 2.5.4).

that the representations Vi are irreducible since j 5 a -k b < r /2* -

ii The argument that V j @ VA splits as a direct sum of the images

I r of the V j follows the same lines as in the proof of Theorem 2.5.5. i; I i

3.6.7 Definition. There is a Uq(s1(2)) invariant map h l b :

(v; '~)"~~ @ -+ ( ~ ; ' ~ ) " ~ j , defined for admissible triples (a, b, j ) as follows:

3 I The composition pJ o ,/, o (4, 8 &,) : V; 8 V; + V j is also l

a b Uq(s1(2)) invariant, and it corresponds to the projection of V$@\<

II

i onto the direct summand that is isomorphic to Vj. I

3.6.8 Lemma. For u t v = j, (a, b, j ) admissible, and max 120,

2b, 2 j ) < r if A is a primitive 4rth root of unity, 7ue have

/ 1 \

3.6.10 Lemma. The quantum Clebsch-Gordan coefficients sat-

isfy the following recursion relation

Hence,

(- l)z~(z-w)(i+t+l) [ a + u + z]![b + v + w]!

z,w: z+w=~-t [z]![w]![a - u - z]![b - v - w]!

The sum is understood to be over all integers z, w such that z+w =

j - t and all the factorials are of non-negative integers. Further-

more, u, v, and t are weights of V$, v;, and V;, respectively, and

u + v = t .

Proof. The computation is analogous to the proof of Lemma 2.7.4,

but relies on Lemma 3.6.2 and Proposition 3.5.7 to take care of

the powers of A.

3.6.9 The quantum Clebsch-Gordan coefficients. a b

We have maps pa 8 pa 0. Y 0 4i : V; -+ V j @ V; when the i

triple (a, b, j') is admissible. Define the quantum Clebsch-Gordan

coeficient CZ;;$ to be the coefficient in the sum

by applying F to the equation that defines the Clebsch-Gor dan -i

coefficient. The closed form is determined by solving the recursion

using the value

a ,b . j - i(h-u)-(a-u)A(b-~)(b+v+l)-(a-u)(a+u+l) [a + - jl! Cu,v,~ - [a - u]![b - v]! '

3.7 Quantum network evaluation. Recall that in the clas-

sical . case . the key computations were the ev,aluations of the closed

"theta-nets" and the closed "tetrahedral" networks. In [18], these

network evaluations are given in case A = ei"l(2~) and their di-

agrammatic computations can be applied to the case of generic

values of A as well. We use the machinery developed above to

give alternative computations.

100 THE CLASSICAL A N D QUANTUM 6 j - s ~ ~ ~ ~ ~ ~

3.7.1 Theorem. Let A E C . Choose a, b, j, k E {0,1/2,1,3/2,. . such that

1. (a, b, k) and (a, b, j) are admissible triples;

2. If A is a primitive 4rth root of unity, then assume that

i a, b, j, k < r/2. a + b - j t

( a - u 1 $a(ea,u) 8 $b(eb,v)) )

where 6; is a Kronecker S function, AJ = (-1)2~ [2j + 11, and ([a + b - j]!)" q(b- ' J J (b tv t l J la 4 = e3,3(-l)aSb-3

[2a] ! [2 b] ! U+V=J q(a-~)(a+~+l) r- u]![b - zlj!

@(a, b, k) = - (-l)"+b-" [a + b - j]![a + j - b]![b + j - a]![b + j t a t I]! - e3,3

1 ,, ,T, [(I + 1) - 1:lt[rt - b t XIt[ - (1 t b -1 X]~ [O + b + 1, 4- 1 ' 1 [2a]![2b]!\21 + I]!

k, , \ - 1 ) [2a]![2b]![2l;]! 'l'llo last ecluahty 1s a ( ~ U ~ I I ~ U I ~ I C O ~ ~ I ~ I ~ I ~ ~ U I 1C ~ d e ~ l t i f \ 111~11

analogous to the identity used in the proof of Theorem 2.7.6. Thus 3.7.2 Remark. In case k = j = 0 and a = b, the formula

to complete the proof we need use the following: reauces to A, = (-l)""[Za + 11 and this 1s the value of a closed

3.7.3 Lemma.

L J I . \ I . . . . \ _ - ul![b + vl!

8% -i- q value O(a, b, k ) is the value of a closed network as in the ~ a r a ~ r a ~ h 3 L " -

C L q(a-u)(a+u+l) [a - u]![b - v]! % u+v=j ::

Proof. By the assumption on A, the representations V; and V: [a + j - b]![b$ j - a]!

are irreducible. Ther

and when k = j it is a multiple of the identity. Proof. The proof is a yantization of the argument that we gave

during the proof of Theorem 2.7.6. We rely on the quantum combi- In Theorem 3.6.6

natorial identities stated in Lemma 3.6.2. Furthermore, we recall

weight vector, and in from the proof of Proposition 3.5.4 that

these results and manipulate the expression for the exponent of A c qk2T(u) = A*"("-1) [.I! while recalling that A2 = q and obtain the following: aECn

102 THE CLASSICAL A N D QUANTUM 6j-SYMB 103

where T(a) is the minimal number of transpositions of the for an every element of f ((1,. . . , a + j - b)) and less than every 2T(u)

(k, k + 1) that it takes to write a. merit of f({a + j - b + 2,. . . , 2 j + 1)). We evaluate LEB q

Consider a sequence 0 = a0 < al < . . . < a, = n, and I two ways to obtain the desired identity-

f : {I, . . . , n) -+ (1, . . . , n) denote a bijection for which the , on the one hand, when S C {I , . . - 7 a + + j + is a subset

strictions f + , . . - , aP+l) are increasing for 0 5 p 5 -

Thus when T = 2, the permutation f acts like a shuffle to a deck B~ ={a E B : a({2j+2, . . . , a + b + j + l ) ) = S } .

cards- An arbitrary permutation up of the set {a, + 1,. . . , ap+ll, is extended to a permutation of 11,. . . , n) by the identity. ~h~~ For such an S, let fs be the unique permutation of

- + b +

compute the number of adjacent transpositions that it takes to + 1) such that fs E Bs and the restrictions f \{I,. . . ,23 + 1) write f 0 (goal . - to obtain are both increasing.

a n d f I { 2 j + 2 , . . . , a + b + j + l } T-1 The set B can be written as the disjoint union of the sets Bs

T(f (0001 . . . a ,-I)) = T(f) + x T ( a p ) . as s ranges over subsets of size a + 6 - j- Furthermore,

p=o

(The preceding formula follows rather easily from the depiction of B~ = {fs o (alanaa) : 01 E x{l ,...,a+j-b),

permutations as strings crossinq in the planc.)

111 a si111ilar f a s l ~ i o ~ ~ if' J F 1 I i ~ p + 1 , . . . , ~ p + l } is increasing for

O < p < r - 1 , t h e n

If f : {I,. . .,n) + {I,. . . ,n) is a bijection such that the

restrictions of f to (1,. . . , m) and to {m + 1.. . , n) are both in-

creasing, then

because t,( f ({m + 1, . . . , n))) is the minimal number of trans-

positions that is needed to put f({m + 1, . . . , n)) back into its

standard position.

Consider the set, B , of bijections f : {1,2,. . . , a + b + j + 1) -+

{1,2, ..., a+b+ j+ l ) such tha t t heva lue f (a+j -b+l ) i sgrea ter

From this formulation we have,

where the exponent is given by

a + b-3) (a + b-j-1 N = (a+j-b)( .+j-&l>r(b+j-a)(b+j-aa1M -

Next we sum over all the subsets S to obtain

1 104 THE CLASSICAL A N D QUANTUM SYMBOL^ i

On the other hand, we compute cEB ~ ~ ~ ( " 1 by examining the

images of the various a. Specifically, for a + j - b < C < 2a, let

Then B is the disjoint union of the Bes.

F o r a + j - b I C I 2 a , a n d f o r S ~ - { 2 j t 2 , . . . , a + b + j + l )

of size IS1 = C - (a t j - b), let

Let ge,s be the unique bijection on {1,2, . . . , a + b + j + 1) such

that ge,s E Be,s and the restrictions of gii to each of (1,. . . , e l

and {C + 2 . . . . . n + b + j + 1) are both increasing. Then Br is the

( l~ , io~ i~ i l i l l $ 1 *IS 1 I I P /I, ,. < \ I ~ ( Y

E the number required to move S into the final position, i.e. t(S).

' Recapitulating,

F i T(ge,s) = (C - a - j + b)(a + b + j + 1 - C) - t(S).

We recall from Lemma 3.6.2 that

Putting everything together, we get

As before, we examine the number of adjacent transpositions re- = AP C q2e(.i+l) [[]![a + b + j - C]! quired to write various permutations, and obtain

e

q 2 ~ ( u ) = (q2~(ge , s ) ) (= q2~iul ) ) (. q 2 ~ ( u 2 ) ) where the exponent E is given as

oEBe,s

= ~ ~ q ~ ~ ( g e , s ) [ C ] ! [ a + b + j - C]! and the related exponent P is the following

- . where theesponent M = C ( C - l ) + ( a + b + j - C ) ( a + b + j - C - I ) . P = ( a + b + j ) ( a + b + j - 1 ) - 4 ( a + j - b)(b + j_+ 1).

observe that T(ge ,~) is the number of transpositions of the

form (k, k + 1) that are needed to move {a + j - b + 1, . . . , C} Set e = a+u and set j = u+v, and compare the two expressions

into the set S. This is the number of transpositions required to for CuGB q2T(a). We get

move {a + j - b + 1, . . . ,C) into the final position in the interval q2("+")(j+l) I" + ~l ! [b J- "'"

{I , . . . , a + b + j + I}, which is IS[(a + b + j + 1 - e), minus u+v=j [a - u]![b

106 THE CLASSICAL AND QUANTUM SYMBOL^

Lemma 3.7.3 follows by manipulating the exponents. This corn-.

pletes the proof of Theorem 3.7.1.

3.7.4 Remark. In case A is a primitive 4rth root of unity,

@(a, b, j ) # 0 precisely when a + b + j 5 ( r - 2). We will give the

representation theoretic meaning in Section 4.

3.8 The q u a n t u m 6j-symbols - generic case. In this

section we define the quantum 6j-symbols in case A is not a root

of unity, and verify that the Elliott-Biedenharn identity and the

orthogonality identity hold here. In Lemma 3.10.9 we establish

the diagrammatic rules that are necessary to prove the analogues

of the identities give11 in Theore111 2.7.14. We esar-tlinc the svm-

for various values of j. Second, consider the composition

for various values of n.

The values of j and n are restricted so that (b,c, j ) , (a, j , k) ,

(a, b, n), and (n, c, k ) all form admissible triples. Alternatively, if

one of these triples is not admissible, then we declare the corre- \ I

I 108 THE CLASSICAL AND Q U A N T U M SYMBOL^

I

! 3.8.3 Definition. Define the quantum 6j-symbol to be the !

coefficient { } in the following equation.

4

and

Observe that these calculations are analogous to the classical case.

Moreover, the recursive method given in Section 2.8 that is used

to compute the values of the 6j-symbols in general also applies to

the quantum case.

3.8.4 Identities among diagrams. The key to simplifying the

computations of the classical case was t o derive identities among

the diagrams that represented various maps. To this end we ob-

serve that Theorem 3.7.1 gives the evaluation of the closed theta

network for certain values of (a, b, j).

The identities that are expressed in Lemma 2.6.4 and their

proofs go over to the quantum case as theg are stated. The essence

(a , j , k), (a, b, n), (n, C, k) is not admissible. The matrix on the left represents the maps U and n, and this

For example, consider the case when a = b = c = = 112. element squared represents the composition of the operators fl €4

One can compute directly from the definitions that

3.8.5 Theorem (Orthogonality). Suppose that qr # 1. The

quantum 6j-symbols satisfy the following relation: .

Proof. Identical to the proof of Theorem 2.6.6.

3.8.6 T h e o r e m (Elliott-Biedenharn Identity). The follow-

ing relation holds among the 6j-symbols when qT # 1.

THE CLASSICAL AND QUANTUM 6j-SYMBOLS UANTUM ~ l (

Figure 6: A tangle diagram

3.10.1 Definition. A tangle diagram is a diagram of a collection

of knotted and linked proper arcs and circles in a rectangular box

in which the arcs in the diagram terminate on the top and bottom

Two tangle diagrams are regularly isotopic if one can be ob-

tained from the other by a sequence of the type I1 and type I11

Reidemeister moves. More precisely, when a height function is

chosen on the plane of projection, then the 3-dimensional ana-

logues of the identities expressed in Lemma 2.3.2 together with

the diagrammatic representations of the identities s,syl = 1, - sjsj+l sj = sj+lsjsj+l, and their obvious variants obtained by

changing appropriate crossings are the moves that generate regu-

lar isotopy of tangle diagrams. These last two moves are depicted

below. Thus a regular isotopy is a sequence of these diagrammatic

moves, and each such move corresponds to an identity between

f maps represented by the respective diagrams.

rectangular faces of the box. The box is projected to the plane of I l l the paper, and the diagralll is a.ssiuucd to b(! 111 ~ c ~ l c r a l posibion

meaning that a t most two arcs cross at a given point, and such

a crossing is transverse. The diagram depicts over and under

crossing information in the standard way: a t a crossing point, the

arc that is farthest from the plane of projection is broken. An

example of a tangle d.iagram is given in Figure 6.

A tangle diagram for which n arcs come out of the bottom

of the diagram and m arcs come out of the top of the diagram

represents a map We have established the following:

112 8 n , (Vl/2)8m T:(V, 1 3.10.2 Lemma. (cf. Lemma 2.7.2). Regularly isotopic tangle

via the association of maps n and U to generic maximal and min- diagrams with n strings at the bottom and rn strings at the top

imal points and via the association of the bracket identity to each represent the same map

crossing. In the sequel, we will identify the diagram and the rep-

resented map. T : (v:/~)@~ -+ (v:/~)@~.

114 THE CLASSICAL AND QUANTUM 6j-SYMBOLS 115

3.10.3 Definition. Let a, b E (0,1/2,1,. . .). Define maps Suppose that 2b strings come out of the top left edge of the

I;:(+) "nd I;:(-) that are quantizations of the map that switches tangle S , suppose that 2a strings come out of the top right of S,

factors defined in Lemma 2.7.5. Thus, and 2 j strings come out of the bottom of S. The top-right, top-

left, and bottom of the tangle are called the a, ,f3 and y regions of

the boundary, respectively. Thus S represents a map ( V ; I ~ ) @ ~ ~ -+

The (unquantized) switching map Xt: can be written as a prod- (v;/~)@~' @ (v:'~)@~~. The strings that come out of the coupon

uct of adjacent transpositions since it represents a permutation; are represented by the three ribbons in the picture below.

choose a minimal such product. Each transposition uk =' (k, k t 1) We will compare the composition Kt:(+) o S to the composi-

in the product is lifted to a braid generator ci,, = s k to define tion (Tz,(-) @ T2b(-)) 0 2 0 Tzj(+) where each of these tangles

&$(+), and it is lifted to &-I = s i l to define Kt:(-). Hence, in is associated canonically with a map among tensor powers of the

the diagrammatic representation of Kt:(+) the 2a strings on the fundamental representation.

left cross over the 2b strings on the right as one reads from top to

bottom; in Xt:(-) the strings on the top right cross over those on 3.10.5 Observation. The diagrams depicted below are isotopic

the top left. but not necessarily regularly isotopic diagrams. The isotopies fin:

. 1 / 2 . . > , 11.2 . . , L , , ])cf i lLe i~kaps I ; , , ( & ) : J - " - 1 tl~i1.1, i i ~ . ~ ' I .O~II 'C~- / / I ( ( ( I < / , .q (I-f !h( O I I / ~ O ; I I ~ / ~ < / i ~ t ? , % ,

sented by half-twists in the cable of n-strings. Specifically, the

half-twists can be spelled out in terms of braid generators as the

products:

Tn(-) = (S;~)(S;~S;I) - . . (S,~~S,:~ - .ST')

while Z

Tn(+) = (sl . . .sn-2~,-1)(sls2 - - .sn-3sn-2) . . - ( s I s ~ ) ~ I .

To obtain the desired maps, It:(&) and Tn( f ) apply the bracket

identity to each braid generator in the product. -

3.10.4 Notation. Suppose that S is a planar tangle with no

closed loops. Thus S consists of properly embedded arcs in the

rectangular region containing S. We depict S as a rectangular Demonstration. Cut out a piece of paper in the shape of a

coupon as in the illustration on the left below. The tangle is a thick Y with fairly long edges. Arrange the edges with a crossing

180' rotation of S through a vertical axis. as in the left hand diagram and tape the ends to a table. Write

THE CLASSICAL A N D QUANTUM 63-SYMBOLS QUANTUM 4 2 )

the letter S on the neighborhood of the vertex of the Y. Then f and

rotate the vertex about the vertical axis. This gives the isotopy. 6 OX.(*) = (-aA3). o(Tn(+) 8 Tn(+))

The maps that are represented by the diagram on the left and where X,(f) denotes n strings crossing from top left to lower right

the diagram on the right differ by a factor of ( - A ~ ) ' ~ , for such a over/under n strings running from lower left to upper right, re-

factor measures the difference between isotopy and regular isotopy. spectively.

We next quantify that difference. Proof. Bracket aficionados will recognize that In the planar tangle-S (with ,8 region on the upper left, a region

on the upper right, and y region on the bottom edge) suppose that X(f)u = (-A)*~u x strings start and end in the p region, r strings start and end

I in the a region, p strings start and end in the y region, w strings and

run between the ,8 and the cr region, u strings run between the p nX(k) = ( - ~ ) * ~ n ,

and y regions, and t strings run between the cr and the y region. with signs read res "- -

Otherwise the diagram for S can be quite arbitrary, and there is

~ i o , I ~ ~ ~ I I I I I ) ~ ion ~1)o11t 1 11c 1 1 c ~ \ 1 I I I P , 01 i ~ l f ~ l . l i ~ ( i11g of ~ ~ P S F ~ t r i l l q \ Tlle LI.ILI\ 1s to l egu Ia l Iy 1 5 o ~ o p c

beyond the condition that no strings cross. The tangle Z denotes ;d A, L , ,a - the map represented by a rotation of S by 180' about the vertical

axis.

i 3.10.6 Lemma. In the notation above

Proof. We establish some special cases then we will proceed to

1 the general result. I t

i Corresponding to the case x = r = p = u = t = 0 and

w = 2a = 2b = n in the notation above, we have the first equation I

in the following:

3.10.7 S u b l e m m a (Case 1).

pectively. (Non-ahcionados are encouraged to

111 follows I,y ind~~ct.io~t.

01 Lne alagram. u

The diagrams representing Xn(f) ?J and f i In(&) will be called

For the case in which x = r = w = a = b = 0 and p = n

we have the first equation below. The second equation below

corresponds to the case when all strings start and end in either

the a or the /3 region; i.e. j = 0 and either a = 0 or b = 0.

3.10.8 Sublernma ( c a s e 2).

?I= ( - A ~ ) ~ ~ I? oT2,(f)

6= ( - A ~ ) * ~ T ~ , ( + ) o I?

where the signs are read respectively.

118 THE CLASSICAL A N D QUANTUM 6j-SYMBOLS

Proof. We induct on n. The case n = 1 is the same as Case

1 above when n = 1. For larger values of n, one small kink can

be canceled from the diagram representing the map on the right

(of either equation) at the expense of multiplying that factor by

( - A ~ ) * ~ . Because T represents a complete half-twist, the arc from

which the kink has been canceled can be isotoped away from the

rest of the diagram.

We turn now to the proof of the general result. Let S' denote

the tangle obtained by twisting the w curlicues out of )$:(+) o S. Specifically, in each string that runs between the CY and P regions,

the small loop is removed. These loops are removed successively.

Thus we replace Xw(+) o Uw by (Tw(-) @ Tw(-)) o U,. Then

as in Case 1.

Similarly, let Z' denote the result of removing the curlicues

from the T , X , and p strings that run back and forth to the a,

p , and y regions, respectively. The nesting among arcs that run

back and forth to a given region is not relevant, as can be seen by

examining the figure below. In this way we obtain, Furthermore, Z' and-S' are regularly isotopic, and thus rep-

resent the same map from ( ~ i / ~ ) ' @ ~ j to ( ~ i / ~ ) @ ~ ~ @ (v1I2 8% A ) .

(T2,(-) @ T2b(-)) o Z o T2j(+) = ( - A ~ ) ~ - ~ - ' Z ' )$:(+) 0 S = ( - A ~ ) ~ s '

= ( - A 3 ) ~ ~ ' = ( - A ~ ) ~ + " + T - P (T2a(-) 8 T2b(-)) 0 0 T2j(+)-

This completes the proof.

1

by Case 2.

120 T H E C L A S S I C A L A N D QUANTUM ~ ~ - S Y M B O ~ ~

3.10.9 Lemma. The following identities hold among the Uq(s1(2))

invariant maps.

Proof. \,\'(, 1 1 5 t L!tl~iil~a. 3.10.6 t.o provc it.cn.1 I as follo\~~s. I\!(.

restrict to the case of Xi:(+), the negative crossing case is similar. ba

The map Y can be written as a linear combination of maps

fohvs . Expand each of the projectors %a, 4tb, and +fj in terms

of the bracket identity. Replace any closed loop that might appear

in the expansion by the factor -A2 - A-2. Thus we have

where f s (A ) is a polynomial in Af and the sum is over all planar

tangles S with 2b + 2a outputs on the top and 23' outputs on the

bottom. Next we consider the exponent on the factor -A3 on the

right hand side of 3.10.6. We have

x + r + w - p = a + b - j

122 T H E CLASSICAL A N D QUANTUM 6 j -SYMBOLS

regardless of the planar tangle S. This gives

because the projector +,A is invariant under 180' rotation for n =

2b, 2b, 2j.

The second step of the proof is now relatively easy. Observe

that

Tn(-) 0 +; = A"("-')/~+A n -

This follows by-applying the bracket relation to each of the neg-

ative crossings in the half-twist. In the bracket relation either

stra.ig11t strings or lnasilnum and minimum points result. The op-

Item 2 follows from item 1 by techniques of regular isotopy.

Items 3 and 4 follow trivially from the invariance of the map

represented by regularly isotopic diagrams.

3.11 Symmetries. Here the twisting rules will be used to

adjust the 6j-symbol to one that has full tetrahedral symmetry.

3.1 1.1 Lemma. The quantum 6j-symbols possess the followiny

symmetry

Proof. Identical to Lemma 2.7.8. U

3.11.2 Lemma. \

annihilated by the Jones-Wentzl projectors. Consequently, all of

these terms with a coefficient of A-' vanish in the bracket expan-

sion. Similarly we compute

where the functions @(-, -, -) and A are defined as in Theo-

Combining results we have

3.11.3 Lemma. The symbol, -

a+b-j~3(a+b-j)~a(Pa-I)+b(2b-l)-j(2j-l)

= (-l)a+b-i~2(a(a+l)+b(b+l)-j(j+l))

This completes the proof of item 1.

THE CLASSICAL A N D QUANTUM 6j -SYMBOLS

where a choice of each square root is made once and for all, is

invariant under all permutations of its columns and under the ex-

change of any pair of elements in the top row with the correspond-

ing pair i n the bottom row. Equivalently, the symbol [: t ] 9 is invariant under the permutations of the set

{ {a , b, f 1, { a , c, d l , {b,c, e l , {d, e, f }1.

Proof. The proof of Lemma 2.7.12 needs to be modified as fol-

lows. When the closed tetrahedral network depicted in that proof

is rotated through space to reverse its orientation, four twists at

the vertices are added. Each of these twists contributes a factor

of An where n , by Lemma 3.10.9, depends on the labels at the

vertices. A careful count indicates that these powers of A cancel. n

where Ak = ( - 1 )~~ [2k + 11.

Proof. This is a direct computation. EI

3.11.5 Remark. These identities are slightly different than

they appear in the classical case. In that case, our choices of the r -

b c j symbol were motivated by the desire to make con-

L - tact with the existing literature, specifically [2]. In the quantum

case, these identities coincide exactly with those in [32].

3.12 Theorem.

U

In [24], Masbaum and Vogel use the recursion relation for the A., Jones-Wentzl projectors to give a formula for the quantum 6j-

svmbols. See also Section 2.8.

3.11.4 Theorem. The orthogonality relation and the

Elliott-Biedenharn relation hold for the normalized quantum 6j74

coefficients in the following form.

b Orthogonality,

Elliott-Biedenharn:

THE CLASSICAL A N D QUANTUM 6j-SYMBOLS

4 The Quantum Trace and Color Repre-

l)k+c+n+m A2(k(li+l)+c(c+l)+n(n+l)+m(m+ 1)). sent at ions = E(-

n

In this section we will define the quantum trace and use it in the

{ ; } , { . } , { ; 1 , . case where A is a root of unity to distinguish representations that

are not used in the topological applications that we have in mind.

The representation theory of Uq(s1(2)) in the root of unity case

3. 12a12b = z3 @(a, b, j ) is worked out in [20] under the assumption that E and F are

nilpotent and I(; is of finite order where the orders are determined

4. by the value of q. We will not need to make that assumption as it

(-1) a+b-kA2(k(k++l)-a(a+l)-b(b+l))~, b, k)

holds on all the representations that we consider, nor do we need

- - 'y(-1)a+b--3 to classify all of the representations of Uq(s1(2)).

3

4.1 The quantum trace. We will define the trace of a linear ,12(a(b+l)+b(b+l)-1(~+1))@((~, b 1 3 ) {U Z } map f : W - 147 in case Mr 15 a lT,(s1(2)) ~epresentation. This

2'

notion of trace is quite versa.tile. In particular, such a, tra.ce a.1-

Proof. The proof follows along the same lines as in the proof lowed Jones [12] to define an invariant of knots via braid group of Theorem 2.7.14, but crossings are lifted to over and under cross- representations. Lickerish's [23] combinatorial definition of the ings in a consistent manner so that the diagrams can be isotoped in Reshetikhin-Turaev invariants [29] is given via the trace. Lick- 3-dimensional space. The remaining details are left to the reader. orish's techniques are employed in Section 4.2 in order to prove

1 0 that the matrix representation of the Temperley-Lieb algebra is I I 3.12.1 Remark. Similar identities hold for the normalized 6j-

faithful for generic values of q.

Finally, we will use the trace to characterize certain represen-

I symbols as well. g tations in the case when A = eZ"/(2T) (or any other primitive 4rth

root of unity). The tensor power (vif2)@In decomposes as a direct

sum of a trace 0 summand and a sum of representations Vi where

j is taken from the finite set (0,112, . . . , ( r - 2)/2). By excluding

the representations that have trace 0, we will be able to define the

Turaev-Viro [32] invariant as a state summation.

128 T H E CLASSICAL A N D QUANTUM 6j-SYMBOLS

4.1.1 Definition. Let W be a representation space for Uq(s1(2)).

Then the quantum trace of a linear map f : W + W is the ordi-

nary trace of the map K 2 o f .

In particular, if

112 @n L : (vy2)@n -+ (VA )

denotes a linear map, then

tr,(L) = t r ( l i 2 o L ) .

Consider the bases { f s : S C {1,2, . . ., n ) ) and {fs : S c {1,2, . . . , n ) ) for ( v : / ~ ) @ ~ where f, = xf 8 - - . 8 x z and f, =

and

T H E QUANTUM TRACE A N D COLOR REPRESENTATIONS 129

"..--'

Figure 7: The braid closure of a tangle

z,S 8 . . .8 ~f with Proof. The second equality follows because Ir' fs = A ~ ( ' / ~ - " / ~ ) fs =

x if j @ S ~ ~ - ~ 1 ~ l fS where T = # { k : x z = x ) and s = # { k : x: = Y). For X j S =

y if j E S the first formula, wc compute using Lemma 3.6.4

- ( z if ~ F S i4 o ( L @ In)o ilr ( 1 ) = ij o(L 9 1,)

1 and the sum is taken over all subsets R c ( 1 , . . ., n).

1 4.1.2 Lemma.

I S <

where the sum is taken over the set of all subsets S C {I,. - . . "1.

= (-1)n- C ~ 2 ( n - 2 1 s l ) ~ ~ ~

S C { ~ , ..., n}

This completes the proof.

4.1.3 Remark. The point of the Lemma is, for maps L defined

on ( v ; / ~ ) @ ~ in terms of Temperley-Lieb elements, the trace of the

130 T H E CLASSICAL AND QUANTUM 63-SYMBOLS

operator can be computed by the bracket expansion of the braid

closure of the operator; the braid closure of a tangle diagram is

depicted in Figure 7. In particular, we apply the Lemma to the

Jones-Wentzl projectors to obtain the following:

4.1.4 Proposition. Suppose that A4T # 1, or if A is a primitive

4rth root of unity, then suppose that n < r - 1.

In particular, i f A is a primitive 4rth root of unity and i f n = r - 1,

then

trq(C-") = 0.

Proof. The proof follows from comments made in Remark 3.7.2.

T H E QUANTUM TRACE AND COLOR REPRESENTATIONS 131

the association of the bracket identity to each crossing. Regu-

larly isotopic diagrams where the end points are fixed during the

isotopy determine the same maps.

Let I, denote the set linear combinations of such tangle dia-

grams. The bilinear form

is defined as the h e a r extension of the map defined on tangles S

and M by:

(S, M) =; o(S @ M ) o ir

where W is a rotation of the tangle M through 180" about an axis

perpendicular to the plane of the dia.gra.m. The second cqua.lity

4.1.5 Definition. Now suppose A = e"il(27'), and as usual,, let follows because the diagrams representing these tangles a.re regu-

larly isotopic. The image of (., -) is C by evaluating the diagrams

trace 0 if trq( f ) = t r ( K 2 o f ) = 0 for every Uq(s1(2)) invariant map via the bracket identity; the parameter A is assumed to be a non-

zero complex number.

The Temperley-Lieb algebra, TL,, has dimension (:)/(n+ 1). 4.2 A bilinear form o n tangle diagrams. Recall that a A vector space basis is a certain set of monomials in the generators

tangle diagram is a diagram of arcs embedded in a thin rectan- (1, hl, - -, h,-I}. Because of the relations among the products of gular prism obtained by projecting in the short direction onto a these generators, any monomial can be put into a normal form. rectangular face and depicting crossing information by breaking We will not list the normal form monomials here, but we will the arc at an intersection point that is farthest away from the assume that these form a basis. plane of projection. Now suppose that such a diagram h s the We can mod out the free module generated by regular isotopy same number of incoming and outgoing strands, say n. Such a classes of tangle diagrams by the ideal generated by the bracket tangle diagram (in general position wi.th respect to a height func- relation and the relation that the loop value is 6. In this way, every tion) represents a map (v:'~)@" -+ (v:l2)gn via the association n-string tangle diagram can be written as a linear combination of of n, and U to the maximal and minimal points respectively, and elements in the basis of TL,. A matrix representation of the inner

132 THE CLASSICAL A N D QUANTUM 6j-SYMBOLS

product (., 0 ) is given as the collection

where w j and wk range over the set of normal form monomials.

For example, when n = 3, the matrix is the matrix

- S 3 S2 S2 6 6 - .

S2 6 63 S2 S2

S2 S3 6 .S2 6'

6 d2 S2 6 S3

6 S2 S2 S3 S3 -

where S is the value associated to any simple closed curve, and

the ordered basis of the algebra TL3 is (13, hl, h2, hlh2, h2hl).

because this is the loop value associated to the identification

U I+ [0, iA, - i ~ - ' , 0It. [0, iA, - i ~ - l , 01. n

More generally, 6 is the loop value in C of the Temperley-Lieb

algebra regardless of the representation.

4.2.1 Proof of Theorem 3.3.4. This proof was indicated to

us by Paul Melvin. Let T, denote the matrix of the inner product

(., .) defined on n-string tangles with respect to the basis of normal

form monomials in the generators 1, hl , . . . , h,-l. Observe that

det T, is a polynomial function of thk loop value 6. Under the

given representation 6 = -A2 - A-2.

The representation,

of the Temperley-Lieb algebra into a matrix algebra is faithful

when the matrix Tn is non-singular.

THE QUANTUM TRACE AND COLOR REPRESENTATIONS 133

In [21] KO and Smolinski show that all the roots of the equation

det T, = 0 are of the form 6 = 2 cos kn/(m + 1) where 1 < m < n

and 1 5 k < m. So T, is non-singular unless A is a 4rth root of

unity and n > r - 1. This completes the proof.

4.2.2 Remark. The singularity of the pairing (., .) when n > r - 1 is a key ingredient in Lickerish's [23] construction of the

Reshetikhin-Turaev invariants. KO and Smolinsky give an ele-

mentary combinatorial proof of this singularity via a recursive

construction of the Temperley-Lieb basis. The proof of singular-

ity is originally due to Jones [13].

Our proof of faithfulness breaks down in case the matrix Tn is

singular, but according to [7] the representation BA is faithful in

a.11 ca.ses except possibly when A = -1. Their proof follows along

the sa.nic li~ics as ours ill t , l~c cla.ssical ca.scl n.llic.11 ;~iso c.o\.cr:, ;.!lo

case of A = -1.

4.3 Color representations. We keep to the case that A is a

primitive 4rth root of unity with the integer r 2 3. In this case

[T] = 0. For such values of A define the set of colors to be the

set {0,1/2,1,3/2, . . . , ( r - 2)/2). We will consider representations

vi where j is a color, for these representations are irreducible and

by computing modulo the trace zero representations, we will still

have a Clebsch-Gordan theory.

4.3.1 Lemma. The representation V i is not of trace 0 if a is

a color. I t is of trace 0 if a = ( r - 1)/2.

Proof. A computation with weight vectors shows that the rep-

resentation V i is irreducible when a < ( r - 1)/2. By Schur's

Lemma, any Uq(s1(2)) map is a multiple of the identity. It suf-

fices to compute trq(+&) = [2a + 11 (Lemma 4.1.4); this trace is

non-zero unless a = (T - 1)/2.

134 THE CLASSICAL A N D QUANTUM 6j-SYMBOLS THE QUANTUM TRACE AND COLOR REPRESENTATIONS 135

4.3.2 Definition. A q-admissible triple is a triple (a, b, j ) of col- Proof. Let 4 : M 8 v;/' + M 8 v;/' be a Uq(s1(2)) invariant

ors such that map. Define an invariant map 4 : M -t M by the formula

1. a + b + j is an integer; 4 = ( 1 ~ 8 n) ( 4 8 I) 0 (IM 8 u)

2. a + b - j , b + j - a, and j $ a - b are all > 0; mapping

The role of the last condition is as follows. When A is a prim- An algebraic computation shows that trq(4) = -trq($). Since M itive 4rth root of unity and when (a, b, j) is q-admissible, then the

- has trace zero, trq(+) = 0, and so trq(+) = 0, which completes the

a b

map y is non-zero (Theorem 3.6.6). Consider the computa- proof. j

tion of the closed 0-network, @(a, b, j), given in Theorem 3.7.1. 4.3.5 Lemma. For colors a and 6 , there is a subrepresentation

In the root of unity case, @(a, b, j ) = 0 when w + b + j > r. - 2. , . I , , . ~ b Ua,b of 63 V: such that

The inter~retation is tha.t t.he corllpositioll 1 o \/ va.uislles " ab I j

a b

because y maps into a summand of ( v ' / ~ ) @ ~ ( ~ + ~ ) that is of j

trace zero.

4.3.3 Lemma. For colors b,

if b = 0; v; €4 v;I2 2 2 - 2 i f 1 1 2 5 b 5 ( r -2)/2.

- .

Observe that vF1l2 has trace zero if b = ( r - 2)/2.

Proof. The relevant maps y Y2 are defined and non-zero

j = b f 112; so the result follows by a dimension count.

4.3.4 Lemma. If M is a finite dimensional Uq(s1(2)) repres

tation of trace zero, then M 8 ~ 1 ' ~ also has trace zero.

1. Ua,b has trace zero, and

2. V j 8 ~1 I UaC $ (ej vj) where the sum is over all colors

j such that (a, b, j) is a q-admissible triple.

Proof. The proof is by induction on b. The case b = 0 is trivial.

The case b = 112 is Lemma 4.3.3.

Now suppose that 1 5 b 5 ( r - 2)/2. On the one hand, the

inductive .hypothesis applied to b - 1 shows that

b-112 vj 8 v,

2! vj 8 (v; @ vj - l )

c2 v; 8 v; $

j:(a,b-1 j)g-admissible

where Ua,b-l has trace zero.

138 THE CLASSICAL A N D QUANTUM 6j-SYMBOLS

where the sum is over all colors j such that (a, b, j ) is a q-admissible

triple, and the summand U' is a subrepresentation of trace 0.

Proof. First we observe that in the case of colors k the Jones-

Wentzl projectors +& are defined. Thus is defined, for

112 8 2 j is (a, b, j ) a q-admissible triple, and the map $j : Vi + (VA )

defined.

Let V j @ V; = W' $ U' be a direct sum dec~mposition where

W' is a direct sum of color representations and U' is a subrep-

resentation of trace 0. By Lemma 4.3.5, we just have to prove ab

that under the stated conditions (pa @ pa)( y (4j(Vi))) is not j

contained in U'.

Let U1' = da@db(U1) c ( l ~ i ' ~ ) @ ~ ~ + ~ ~ . Since + , @ d b ( ~ i @ ~ l ) is

THE QUANTUM TRACE A N D COLOR REPRESENTATIONS 139

is a Uq(sE(2)) endomorphism of (v;'~)@~("'~). Its quantum trace

is the value @(a, b, j ) # 0 (See Remark 3.7.4). Let W $ U denote

a direct sum decomposition of (v:'~)@~("+~) where W is a subrep-

resentation complementary to U, and let pw and pu denote the

projections onto the indicated summands. Now we compute

The first term is 0 because the image of f is assumed to be in U ;

the next two terms are 0 because in a block matrix representation

of these maps, the two diagonal blocks are 0. The last term is

0 because it equals the quantum tra.ce of the l1 , (s1(2)) inva.ria.nt

for any color t, the multiplication map peld,(v;) is an isomorphism

is an isomorphism that inverse to 4e. So (pa @ ~b)l~,@+,(v;gv,) 4.4 The quantum 6j-symbol - root of unity case. In

is the inverse of 4, @ 4 b . Now the image of order lo define the 6j-symbol in the case that A is a primitive

4rth root of unity, we will need to decompose the tensor product

of three color representations into a direct sum of colors and a

summand that is of trace 0. We will mod out by the trace 0 piece

and construct bases for invariant maps into the quotient. The 6j-symbol, then, will be defined as a change of basis matrix in a

family of maps from a color representation to the quotient of the

tensor product by a maximal trace 0 summand. Let us turn to the construction.

4.4.1 Proposition. Let A denote a primitive 4rth root of pose it were. Consider the composition f =

unity, and let a, b, c E (0,112,. . ., ( r - 2)/2) - the set of colors.

140 THE CLASSICAL A N D QUANTUM 6j-SYMBOLS

Let pabc = Pa 8 pb 8 pc. Then

v,. 8 v; 8 v,. =

where Ul and Uz are trace 0 subrepresentations of V i @ V: @ V j .

Proof. \,\:e lla.\:c i s v ~ ~ ~ v r p l l i s ~ ~ ~ s (*)

THE QUANTUM TRACE A N D COLOR REPRESENTATIONS 141

ab - -

n:(a,b,n) is q-ad.

= ( y :8 1) o ( + n ~ + c ) ( v ~ 8 v j ) @ ( u 8 v A " ) n:(a,b,n) is q-ad.

- - €3 n,k:(a,b,n)&(n,c,k) are q-ad.

(v; 8 v;) 8 vi = @ v,"@Vj@(Ua,b@V;) By computation (*) and the Remak-Krull-Schmidt Theorem, {n:(a,b,n) is q-ad.}

the subrepresentation Ul must be of trace 0. This completes the

) proof of the first equality.

N - ( v j a u n , c @ ( u a , b @ v i ) {n:(a,b,n) is q-ad.} {k:(n,c,k) is q-ad.) Now we turn to establish the second equality. We claim that

- @ v,k up to isomorphism,

-

{n,k:(a,b,n)&(n,c,k) are q-admis.}

) -

v,. 8 v; 63 vz;l 2 ( @ v - @ ut

@ ( @ u n , c @ ( u a , b @ v i ) {~,k:(bc,~)&(a,~,k)are q admis.) {n:(a,b,n) is q-ad.}

) where U' has trace 0. By the computation (*) and the Remak-

where the term @{n:(a,b,n) is q-admis.) Un,c @ (Ua,b 8 V;) On the Krull-Schmidt Theorem, it suffices to show (*)

right of the last equality is of trace 0. Furthermore,

03 k v* 2 @ VA" (v," 8 v;) 8 vi {~,k:(b,c,~)&(a,~,k)are Q admis.) {n,k:(a,b,n)&(n,c,k)are q admis.)

THE CLASSICAL A N D QUANTUM 6 j -SYMBOLS

be a U,(s1(2)) invariant map. Then there are unioue comvlex num.-

T H E QUANTUM TRACE A N D COLOR REPRESENTATIONS 145

maw. Thus S = S', and by the linear independence of the maps bers d, and map S : VJ -+ V j 8 V; 8 V, such that

{ n : ( a , b, n )&(n , c, k) are q-admis.)

4.4.5 Definition. In light of the preceding Lemma, we can

define in the case that A is a primitive 4rth root of unity the 6 j - I \

symbol to be the coefficient 1 : } in the following epua-

and s ( v ~ ) is contained in a trace zero summand of V j 8 V; 8 V,L. q tion:

Proof. The existence of the numbers d, and map S is immediate

from Proposition 4.4.1. Suppose that dk and S' were another

solution. so that

By the preceding lemma, the image of the right side S' - S lies

within any maximal trace zero summand of V j 8 V; @ V,L, and

by Proposition 4.4.1 the image of the left side must intersect

such a summand trivially. Therefore both sides are the zero

where S maps into a summand of trace 0, pat,, = pa €3 pi, 8 p,,

and the sum is taken over the set of colors n such that the triples

( a , b, n ) and ( n , c, k) are q-admissible.

4.4.6 Theorem. ~ e t A denote a primitive 4rth root of unity.

Then the quantum 6j-symbols satisfy the Elliott-Biedenharn and

orthogonality identities as stated in Theorems 3.8.6 and 3.8.5 where

the sums are over q-admissible indices. Moreover for such val-

ues of A, the normalized 6j-symbols are defined and the identities

stated in Theorem 3.1 1.4 hold.

c k j 9

T H E CLASSICAL A N D QUANTUM 6j-SYMBOLS

We form a closed network by reflecting

the horizontal axis to obtain the diagram a b c

This is juxtaposed (with the upper index n changed to s ) to

(.he top os I 1; t,o C ~ I . ~ I L t . l~c! I I C ~ I V C , I . I < :

T H E QUANTUM TRACE AND COLOR REPRESENTATIONS 149

from which it follows that

Now we establish the Elliott-Biedenharn identity using the

same technique. As in the proof of Theorem 2.6.7, we have the

identity -

where T is the tree depicted on the left below - that holds in the

N = N(a, b, c, n, j, k) = a sense of insertion into closed networks.

Then the network, N , is closed by joining the arcs that are labeled

by k. Let N denote this closed network, or its value in C . We

Let T* = T*(a, b, c, d, s, t, g) denote the mirror image of T through

a horizontal axis as shown on the right of the diagram above.

Consider the closed network Closure(T*T).

150 T H E CLASSICAL A N D QUANTUM 6j-SYMBOLS

We compute

Closure(T*T) = Sh,tO(t, d, c)Ss,kO(t, b, s)O(a, s, g)/(AtAs).

.@(t, d, c)@(t, b, s)@(a, s, 9)/(AtAs).

It follows that for any choice of s and t so that (t,d, c), (t, b,s),

and (a, s, g) are q-admissible:

This completes the proof. CI

5 The Turaev-Viro Invariant

In this last section we explain how to use the normalized 6j-symbol

(in the case that A is a primitive 4rth root of unity) to give the

definition of the Turaev-Viro invariants of 3-dimensional mani-

folds. Computations of this invariant can be found in the paper

[32] and the book [18]. Specialization to the root of unity case is

necessary so that the sum in the definition of the invariant is a

finite sum (physicist's renormalization).

While the invariants have not distinguished 3-manifolds that

cannot be distinguished in other ways, new applications of these

invariants are expected. Furthermore, the framework of a topolog-

ical quantum field theory - into which the Turaev-Viro invariants

fit - is quite general, and is currently being explored in its own

right. In particular, there is hope that interesting Cdimensional

g c ~ ~ ~ l a l i ~ a t i o n s call Ijt. foulld. n ~ r d t ha t thc Do~la ldson irir.;lr.ia~~t<.

for example, can be defined as state summations [4]. We men-

tion the following interesting problem: Suppose two manifolds of

dimension 3 have the same Turaev-Viro invariants, in what ways

are they similar? In other words, what qualitative features do the

Turaev-Viro invariants distinguish?

5.1 Definition. Fix an integer r 2 3, let A = eTil(2T), and

let q = A2. Let M denote a triangulated 3-dimensional closed

manifold. Let t denote the number of vertices, let {El,. . . , E,)

denote the set of edges, and let {TI,. . .,T,) denote the set of

tetrahedra of the triangulation. Let C = (0,112,. . . , ( r - 2)/2)

denote the set of colors associated to the integer r. A coloring of

M is a mapping f : {El,. . . , E,) + C. An admissible coloring is

a coloring such that for each triangle with edges Ee, Em, and En

the triple (f (El), f(E,), f (E,)) is a q-admissible triple of colors.

152 THE CLASSICAL AND QUANTUM 6j-SYMBOLS

Suppose that a coloring is admissible, and consider a tetrahe-

dron T with colors a , b, c, j, k, n associated to its edges so that the

triples (a, b, n), ( n , ~ , k), (a, j, k), and (b, c, j ) are admissible, and

these are the labels on the bounding triangles of the tetrahedron. r -I

Associate the symbol T~ = I : : ; 1 to this tetrahedron.

0

The value associated to the coloring 1 of the 3-manifold M is

the quantity. 7L ?,I

where

A = AT' 3 C AkAl.

{k,e:(j,k,l)is q-admis.)

It is a consequence of the orthogonality identity that the quantity

on the left can be manipulated to the figure on the right and vice

versa.

(I;,~:(~,rl-,l)is q-admis.} '4 \y

is independent of j (see [32, 181 for a proof).

The T~raev-Viro invariant of the 3-manifold M is the state In the first figure two tetrahedra with edges (a, b,e,g,h, k)

sum and (,-, d, e, f , g, h) are glued along their common triangular face

IMI = I M I f (g, h, e). Then an edge labelled j is inserted, so the polyhedron

f is the union of 3 tetrahedra: (a, b, e, e , f , j), (b, e, di h, j, li), and

where the sum ranges over all admissible colorings f of the given (,, d , f , g, j, k). The Elliott-Biedenharn identity

triangulation.

5.1.1 Theorem (Turaev-Viro [32]). The value IMI E C is independent of the triangulation chosen; as such it is an invariant

[; : :I., . [ R "1, of the manifold M. b e j l [ j d k 1 d h ]

Proof. The Pachner Theorem for triangulations of $manifolds,

states that any two triangulations of a given 3-manifold are related

by a sequence of the two moves depicted below where the figure

156 THE CLASSICAL AND QUANTUM 6j-SYMBOLS

5.1.2 Definition (Kauffman-Lins 1181). In Kauffman-Lins f the following alternative definition of the Turaev-Viro invariant

is presented. First a triangulation of a 3-manifold is chosen and

dualized. The vertices of the dual correspond to the tetrahedra of

the triangulation, the edges in the dual correspond to the faces of

the triangulation, and the faces in the dual correspond to edges

in the triangulation. Colors are associated to the faces of the dual

in such a way that three colors coincident to an edge form a q-

admissible triple. To a vertex, at which six faces meet, the value

of a tetrahedral spin network is associated. The spins on the edges

of the tetrahedral network are the colors associated to the faces

of the dual, and the edges of the dual correspond to the vertices

of the tetrahedral network. To an edge in the dual with colors a ,

b and 7 coincident the value @(a, b , ~ ) is associated. Since these

for111 a q-admissible triple, the value of 0 is defined and 1s non-

zero. To each colored face in the dual we associate the value A,

where j is the color on the face.

Thus a state, S , of the triangulation is an assignment of colors

. . . to the "faces7' of the dual such that the colors coincident at an edge

. , . form a q-admissible triple. (The term "face" is in quotes above

because as the dual complex to the triangulation is deformed,

some of the 2-dimensional pieces may not be 2-.cells. For example,

the Matveev bubble move introduces an annular face.) To such

a state, we have TET(v1S) the value of the tetrahedral spin net

associated to each a vertex v dual to the given cell. To an edge, E ,

with colors (a, b, j ) coincident, we associate the value O(EIS) =

@(a, b, j). And to a face, f , with color j ( f ) = j ( f IS), associate

the value Aj(f). Let ~ ( f ) denote the Euler characteristic of the

2-dimensional face f where the term face is interpreted as above.

If a given edge, e, forms a simple closed curve with no vertices

from the dual complex, then let ~ ( e ) = 0; if the edge has a vertex,

then let ~ ( e ) = 1. With these conventions, a state sum is defined

by the formula

The proof that this does not depend on the choice of triangu-

lations is given in [18]; the proof depends on expressing the tetra-

hedral networks in terms of the 6j-symbol. In Kauffman-Lins a

nice glimpse of the shadow world is presented, as well.

5.1.3 Theorein (Piunikhin [ZB]). ' The Iicr~<ffnza~n-Liizs (1e.b-

~zi t io~z u i ~ t the Y L I . U ~ U - lfiro definitioizs coi~lcide.

Proof. This is a computation dependent on the definition of the r -!

5.2 Epilogue. In the current work we have presented the

diagrammatics of the classical and quantum representation theory

with topological applications in mind. One major focus has been

the Clebsch-Gordan theory and the explicit construction of the

Gj-symbol in all three cases - classical, generic quantum, and

quantum root of unity. In the process of developing this theory,

we have touched on some other topological aspects that deserve

mentioning.

The Jones polynomial [12], which is an invariant of knotted

and linked curves in 3-dimensional space, is the starting point

of the quantum topology invariants. The bracket identity leads

158 THE CLASSICAL AND QUANTUM 63-SYMBOLS

directly to a definition, and this construction can be found in [16],

for example.

The quotient of the quantum group Uq(s1(2)), when A is a

primitive 4rth root of unity, by relations Er = Fr = 0 and K4' =

1 has the structure of a modular ribbon Hopf algebra as defined

by Reshetikhin-Turaev [29]. Rather than explicitly describing this

structure we have worked with the algebra via its representations.

However, in doing so, we have verified that the ribbon structure is

present. The modular ribbon Hopf algebra gives the Reshetikhin-

Turaev invariant, and Lickorish [23] has presented the definition of

the invariant in a diagrammatic form. From the present point of

view, there is not much more that needs to be done to get to that

formulation. The details of that construction can also be found in

[161.

THE TURAEV-VIRO INVARIANTS 159

known algebraic structures. We have seen this happen already in

the passage from classical sl(2) to quantum sZ(2). For computa-

tions of the Turaev-Viro invariants see [18].

Finally, there are deep connections to theoretical physics that

require much further study from the mathematical, theoretical,

and experimental sides. The mathematical aspect that is the most

problematic is the definition of functional integration - which

might be thought of as a continuous version of the state sum

method. A rigorous definition of the functional integral will lead

to analytic interpretations of the algebra and of the topology, and

such intepretations will certainly shed light on all of the aspects

of the theory.

The Turaev-Viro illvariant is an esanlple ol a topologica.1 quan-

tum field theory (TQFT), namely a functor from the category of

smooth manifold cobordisms to the category of Hilbert spaces. A

great deal of effort is currently being exerted towards finding new

extended topological field theories, and towards finding higher di-

mensional analogues. One formulation of the higher dimensional

theories is found in Lawrence [22]; from that point of view the

structure that is associated to a 3-manifold is a "3-algebra," and

the quantum 6j-symbol gives an explicit construction of such an

algebra. In dimension 4, an example of a 4-algebra would give rise

to a state sum invariant of the type constructed here. There are

very good reasons for searching for such higher dimensional invari-

ants. Evidence for their existence is given by the solutions to the

tetrahedral equation [35] which is an analogue of the Yang-Baxter

equation. Furthermore, non-trivial examples of these higher alge-

braic structures will give new and in-depth meaning to the well-

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