aa r X i v : . [ m a t h . QA ] F e b ON CENTRALIZER ALGEBRAS FOR SPIN REPRESENTATIONS
HANS WENZL
Abstract.
We give a presentation of the centralizer algebras for tensor products of spinorrepresentations of quantum groups via generators and relations. In the even-dimensional case,this can be described in terms of non-standard q -deformations of orthogonal Lie algebras; inthe odd-dimensional case only a certain subalgebra will appear. In the classical case q = 1the relations boil down to Lie algebra relations. Classically, representations of Lie groups were studied by decomposing tensor products ofa simple generating representation. This worked very well for the vector representations ofthe general linear group and to a somewhat lesser extent also for the vector representationsof orthogonal and symplectic groups. More recently, q -versions of these centralizer algebraswere studied in connection with Drinfeld-Jimbo quantum groups which have had applicationsin a number of fields. In this paper, we study centralizer algebras for spinor representations.Some of the possible applications will be given below.We first give a description of the centralizer algebras for the classical spinor groups. If N iseven, this can be comparatively easily deduced from results by Hasegawa [8] for the action of P in ( N ). It follows that the commutant of its action on the l -th tensor power S ⊗ l of its spinorrepresentation S is given by a representation of so l . Moreover, we give a precise identificationof the action of generators which is compatible with embeddings End( S ⊗ l ) ⊂ End( S ⊗ ( l +1) ).This is important for studying the corresponding tensor categories and is not immediatelyobvious from the results in [8]. We also prove an analogous result for the odd-dimensionalcase, which is more complicated. Here the commutant is generated by a subalgebra of theuniversal enveloping algebra U so l .We then extend these results to the setting of quantum groups. In principle, it should bepossible to do this similar to the classical case, using known q -deformations of Clifford algebras,see Section 4.1. However, due to their complicated multiplicative structure we determinegenerating elements via a straightforward approach. They are q -deformations of the canonicalelement P e i ⊗ e i ⊂ Cl ( V ) ⊗ Cl ( V ) ∼ = End( S ⊗ ), where ( e i ) i is an orthonormal basis of thevector representation V and Cl ( V ) is its Clifford algebra. We obtain from this elements whichsatisfy the relations of generators of another q -deformation U ′ q so l of the universal envelopingalgebra U so l . It has appeared before in work of Gavrilik and Klimyk [7], Noumi and Sugitani[20] and Letzter [18]. Unlike the usual q -deformation of U so l , it does not have a Hopf algebrastructure. Our main result is that we again have a duality between the actions of U q so N ⋊ Z / ∗ Supported in part by NSF grants. and U ′ q so l acting as each others commutants on S ⊗ l for N is even. Again, the situation is morecomplicated in the odd-dimensional case where we have to consider a subalgebra of U ′ q so l .It is worthwhile mentioning that one of the problems for spinor representations is thatalready their second tensor power contains an increasing number of irreducible representations.This makes it difficult to characterize the centralizer algebras via braid representations, whichworked well for vector representations. A similar problem was encountered by Rowell andWang in their study of certain braid representations, which they conjecture to be relatedto spinor representations at certain roots of unity, see [23]. Our results should be useful instudying this question. This and other potential applications are discussed at the end of thispaper.Here is the paper in more detail: We first show how the commutant on the l -th tensorproduct of a spinor representation is related to so l by fairly elementary methods. While manyof the results are not new, it serves as a blueprint for the more difficult quantum group case.We then review basic material from the study of Lie algebras and Drinfeld-Jimbo quantumgroups. This is then used to prove the already sketched duality results for quantum groups,where we find generators for End U ( S ⊗ l ) in the third section, and relations in the fourthsection. Acknowledgements : Part of the work on this paper was done while the author was visitingAarhus University and the Hausdorff Institute. The author is grateful to these institutions forhospitality and support. He would also like to thank A.J. Wassermann, H. H. Andersen andG. Lehrer for references. Thanks are also going to the referee and to Eric Rowell for pointingout confusing inconsistencies in notations in an earlier version.1.
Duality for Spinor Representations
We assume throughout this paper all the algebras to be defined over the field of complexnumbers, with q not being a root of unity. For possible generalizations, see Remark 4.9.1.1. Clifford Algebras and spinor representations.
Let { e , ... e N } be an orthonormalbasis of the finite dimensional inner product space V . Then the Clifford algebra Cl = Cl ( N )corresponding to V can be defined via generators, also denoted by e i , and relations e i e j + e j e i = 2 δ ij , for 1 ≤ i, j ≤ N. It is well-known that Cl ( N ) has dimension 2 N . It is isomorphic to M N/ for N even, andto M ( N − / ⊕ M ( N − / for N odd; here M d denotes the d × d matrices. The action of anelement g in the orthogonal group O ( N ) on V induces an automorphism α g on Cl ( N ), foreach g ∈ O ( N ). As any automorphism of the d × d matrices is inner, we obtain a projectiverepresentation of O ( N ) on a 2 N/ dimensional module S in the even-dimensional case . Byrestriction, the module S becomes a projective O ( N − S .It decomposes into the direct sum of two simple projective O ( N − S + ⊕ ˜ S − . Thesetwo modules are isomorphic as projective SO ( N − SO ( N ) module for N odd goes asfollows: We replace the full Clifford algebra, which is not simple in the odd-dimensional case, ENTRALIZER ALGEBRAS FOR SPIN REPRESENTATIONS 3 by Cl ev ( N ), the span of all products of e i ’s with an even number of factors. Observe thatfor N odd, the element f N = e e ... e N is in the center of Cl ( N ), and ( γf N ) = 1 for asuitable γ ∈ { , √− } . Hence the map induced by e i γe i f N defines a homomorphismfrom Cl ( N ) into Cl ev ( N ). If one restricts this homomorphism to the simple subalgebra M ( N − / ∼ = Cl ( N − ⊂ Cl ( N ), it is obviously not the zero map. Hence it becomes anisomorphism, by dimension count and simplicity of Cl ( N − α g inducedby an element g ∈ O ( N ) restricts to the subalgebra Cl ev ( N ) of Cl ( N ). Hence, again, we obtaina projective representation of O ( N ) on an 2 ( N − / dimensional module S ; it is isomorphic asa projective SO ( N ) module to the modules ˜ S ± of the last paragraph.The projective representations just mentioned extend to honest representations of suitablecovering groups. It will be convenient to consider these modules over P in ( N ) for N even,and over Spin ( N ) for N odd. Here P in ( N ) and Spin ( N ) are the two-fold covering groups of O ( N ) and SO ( N ) respectively.1.2. Hasegawa’s results.
Let now V = C Nl = C N ⊗ C l . Then we have obviously commutingactions of O ( N ) and O ( l ) on V , acting on the corresponding tensor factors. Theorem 1.1. (Hasegawa [8] ) The algebras generated by the actions of O ( N ) and O ( l ) on Cl ( N l ) are each others commutant. It is easy to check that Cl ( N l ) ∼ = Cl ( N ) ⊗ l ∼ = Cl ( l ) ⊗ N as vector spaces. This stronglysuggests a relationship between the commutant of the action of Spin ( N ) on the l -fold tensorproduct of its spinor representation, and the group Spin ( l ). Observations to this extent havebeen made at the combinatorial level in several papers before, e.g. [1], [2]. However, theprecise result we need is a little bit more subtle and does not immediately follow from theresults above. In particular, in our context there are nontrivial distinctions between the oddand even-dimensional cases which do not occur in [8].1.3. Some elementary lemmas.
Let f m = e e ... e m ∈ Cl ( m ) for m ∈ N . Moreover,consider the map(1.1) Φ : 1 ⊗ ... ⊗ e i ⊗ ... ⊗ ∈ Cl ( N ) ⊗ l ( f ( j − N e ( j − N + i if j is odd, f jN e ( j − N + i if j is even,where e i is in the j -th factor of Cl ( N ) ⊗ l . Then we have the following easy lemma: Lemma 1.2. (a) f m e i = ( − m e i f m for i > m , and f m e i = − ( − m e i f m for i ≤ m .(b) If N is even, the map Φ above extends to an algebra and O ( N ) -module isomorphismbetween Cl ( N ) ⊗ l and Cl ( N l ) , and Φ( f N ⊗ f N ) = ( − N ( N − / f N .(c) For N odd, the map Φ defines an embedding of Cl ev ( N ) ⊗ l into Cl ( N l ) . It maps theelement e r e s , in the j -th factor of Cl ev ( N ) ⊗ l , to e ( j − N + r e ( j − N + s . P roof.
Part (a) is straightforward. For part (b), one first checks that the map Φ indeeddefines a nonzero algebra homomorphism. This is straightforward. As both source and targetalgebras are simple and have the same dimension, Φ is an isomorphism. As Φ is an O ( N ) HANS WENZL module morphism on the linear span of the generators, and O ( N ) acts via algebra automor-phisms, Φ is also an O ( N ) morphism. The second part of (b) is checked easily using (a) andthe definition of Φ, 1.1. Part (c) again is straightforward.Recall that the Lie algebra so l is isomorphic to the subset of l × l matrices spanned by L rs = E rs − E sr , 1 ≤ r < s ≤ l , where the E rs are matrix units. It can also be definedvia generators L , L , ... L l − and relations [ L i , [ L i , L i ± ]] = − L ± and [ L i , L j ] = 0 for | i − j | >
1. Indeed, it is easy to check that these relations are satisfied for L i = L i,i +1 . Alsoobserve that one can replace − L i ± by L i ± on the right hand side of the first relation aftersubstituting L i by √− L i . Let now N = 2 k be even . We define the elements C rs ∈ Cl ( N l )by C rs = P Ni e ( r − N + i e ( s − N + i and C ′ rs = P N − i e ( r − N + i e ( s − N + i for 1 ≤ r < s ≤ l .Then we have Lemma 1.3.
The elements C rs and C ′ r,s satisfy the commutation relations of the generatorsof the Lie algebra so l . P roof.
If for indices p, q, r, s the set { r, s }∩{ p, q } is empty or has two elements [ L rs , L pq ] = 0.Otherwise, if, say s = p , we get [ L rs , L sq ] = L rq . But then we also have [ C rs , C sq ] == 14 X i = j e ( r − N + i e ( s − N + i e ( s − N + j e ( q − N + j − e ( s − N + j e ( q − N + j e ( r − N + i e ( s − N + i + 14 X i e ( r − N + i e ( q − N + j . One checks that the first sum is equal to 0, and the second one is equal to C rq , which is therequired relation. The proof for the C ′ rs goes the same way.We shall need the precise preimages of C and C under the isomorphism Φ. It followsfrom Lemma 1.2 that they are given byΦ − ( C ) = ( − N ( N − / N X i =1 e i f N ⊗ e i f N ⊗ − ( C ) = N X i =1 ⊗ e i ⊗ e i . Similarly, the elements Φ − ( C ′ ) and Φ − ( C ′ ) are given by the same sums, now only goinguntil N − Corollary 1.4.
The elements Φ − ( C ) and Φ − ( C ) are in End
P in ( N ) ( S ⊗ ) , and the ele-ments Φ − ( C ′ ) and Φ − ( C ′ ) are in End
P in ( N − ( ˜ S ⊗ ) , where ˜ S is S viewed as a P in ( N − -module. P roof. As g ∈ O ( N ) fixes P e i ⊗ e i , viewed as an element in V ⊗ , one deduces thatconjugation of Φ − ( C ) by a lift of g in P in ( N ) leaves it invariant. The other statementsfollow similarly. ENTRALIZER ALGEBRAS FOR SPIN REPRESENTATIONS 5
Eigenvalues of C . It remains to determine the structure of the representation of so l in Cl ( N l ). For this, we define the elements C , C ′ and ˜ C m in Cl ( N ) ⊗ by(1.2) C = 12 X e i ⊗ e i , (1.3) ˜ C m = m ! X i
1. Observe that ˜ C = 2 C .Moreover, observe that the elements e i ⊗ e i and e j ⊗ e j commute also for i = j . We will alsoneed the polynomials P m ( N, x ) defined inductively by P ( N, x ) = 1, P ( N, x ) = x and(1.4) P m +1 ( N, x ) = xP m ( N, x ) + m ( N + 1 − m ) P m − ( N, x ) . Then we have
Proposition 1.5.
The element C ∈ Cl ( N ) ⊗ has the eigenvalues iN/ , i ( N/ − , ..., i (1 − N/ , − iN/ . The same statement holds for the element C ′ , with N replaced by N − . P roof.
Let us first prove the following recursion relation:(1.5) ˜ C ˜ C m = ˜ C m +1 + m ( N + 1 − m ) ˜ C m − . Observe that if we define y j = e j ⊗ e j , then y j y i = y i y j and y i = 1. Moreover, ˜ C m = m ! P i
This subsection serves to calculate the eigenvalues of the poly-nomials P N +1 ( N, x ). Moreover, we do some additional calculations which are useful for anexplicit description of End ( S ) pin ( N ) ( S ⊗ l ) also in the quantum case. All of this is obtained ina fairly straightforward way from the representation theory of sl , which is well-known (seee.g. [10]). Presumably, most of the results in this section are known to experts.Let H, E, F be the usual generators of the Lie algebra sl , and let V N be its ( N + 1)-dimensional simple representation. It can be defined via a basis { e o , e , ... e N } of eigenvectorsof H which satisfies(1.6) E.w r = ( N − r + 1) e r − , H.e r = ( N − r ) e r , F.e r = ( r + 1) e r +1 . HANS WENZL
The following lemma is well-known and easy to check (e.g. (a) follows from the fact that E − F is conjugate to iH ). Lemma 1.6. (a) The element E − F has eigenvalues ( N − r ) i , ≤ r ≤ N in the represen-tation V N .(b) The elements iH/ and ( E ± F ) / satisfy the relations of the generators L rs of the Liealgebra so for ≤ r < s ≤ . Lemma 1.7. (a) The polynomial P N +1 ( N, x ) has the roots ( N − r ) i , ≤ r ≤ N .(b) Let x = ( x ( λ ) r ) and y = ( y ( λ ) r ) be the right and left eigenvectors of E − F for theeigenvalue λ , with respect to the basis ( e r ) and normalized by x = 1 = y . Then x ( λ ) r = ( N − r )! P r ( λ ) N ! and y ( λ ) r = P r ( λ ) r ! .P roof. Writing E − F as a matrix given by 4.8, we obtain from ( E − F ) x ( λ ) = λ x ( λ ) therecursion relation x o = 1, x = λ/N and x r +1 = 1 N − r ( λx r + rx r − ) . Similarly, one obtains from y t ( E − F ) = λ y t the recursion relation y o = 1, y = λ and y r +1 = ( − λy + ( k + 1 − r ) y r − ) / ( r + 1). Comparing this with the recursion relation 1.4, onecan easily check claim (b). Moreover, we obtain from the last coordinate in the equation( E − F ) x ( λ ) = λ x ( λ ) that λx N ( λ ) + N x N − ( λ ) = 0. Hence0 = ( λP N ( N, λ ) + N · P N − ( N, λ )) /N ! = P N +1 ( N, λ ) /N !for any eigenvalue λ of E − F . This together with Lemma 1.6 implies statement (a). Proposition 1.8. (a) Let N be even. Then we obtain a representation of so l in End
P in ( N ) S ⊗ l by mapping the element L rs to the inverse image of C rs under the isomorphism Φ . For l = 3 ,it contains an irreducible ( N + 1) -dimensional representation of so .(b) If N is odd, we obtain a representation of the subalgebra of the universal envelopingalgebra U so l generated by the elements L rs , ≤ r < s ≤ l , by mapping these generators to theinverse images of the elements ( C ′ rs ) in End
Spin ( N ) S ⊗ l . For l = 3 , it contains an irreducible ( N + 1) / -dimensional representation of this subalgebra. P roof. If N is even, we have End( S ⊗ l ) ∼ = Cl ( N ) ⊗ l , and End P in ( N ) ( S ⊗ l ) ∼ = Cl ( N ) ⊗ l , thecomponent of Cl ( N ) ⊗ l ∼ = Cl ( N l ) on which the multiplicative action of O ( N ) is trivial. Byconstruction, the elements C rs are fixed by the action of O ( N ), and they define a representa-tion of so l by Lemma 1.3. By Prop. 1.5 and Lemma 4.8, the largest eigenvalue of the imageof H is N , which shows the existence of an irreducible ( N + 1)-dimensional representation of so ∼ = sl in Cl ( N ) ⊗ .For N odd, we obtain a representation of so l in End P in ( N ) ( ˜ S ⊗ l ) by Lemma 1.3 and its corol-lary. Up to a common sign, the elements Φ − (( C ′ rs ) ) coincide with the elements Φ − ( C rs ) ∈ Cl ev ( N ) ⊗ l ∼ = End( S ⊗ l ). Moreover, it is easy to see that ( E − F ) and H leave the spans ofthe even and of the odd basis vectors invariant, and that these are irreducible submodules. ENTRALIZER ALGEBRAS FOR SPIN REPRESENTATIONS 7
Definition 1.9. (a) We define the algebra U ( l, k ) via generators D , D , ... D l − and relations[ D i , [ D i , D i ± ]] = D i ± , [ D i , D j ] = 0 if | i − j | > Q kj = − k ( D i − j ) = 0.(b) We define the algebra U o ( l, k ) as a subalgebra, generated by D , D , ... D l − , of analgebra with generators D , D , ..., D l − , where the D i satisfy the relations [ D i , [ D i , D i ± ]] = D i ± , [ D i , D j ] = 0 if | i − j | > Q kj = − k +1 ( D i − j − /
2) = 0.Observe that the algebra U ( l, k ) is a quotient of the universal enveloping algebra U so l ofthe Lie algebra so l , while U o ( l, k ) is a quotient of a subalgebra of U so l . Corollary 1.10. (a) If N = 2 k is even, the image of U so l in End
P in (2 k ) ( S ⊗ l ) is a quotientof U ( l, k ) .(b) If N = 2 k − , the image of U so l in End
Spin (2 k − ( S ⊗ l ) is a quotient of U o ( l, k ) . Lie algebras and quantum groups
Quantum groups.
We list some basic information about quantum groups (see e.g. [12],[19]). Let g be a symmetrizable Kac-Moody algebra given by a Coxeter graph X with k ver-tices, with generators e i and f i , 1 ≤ i ≤ k ; eventually, we will only be interested in orthogonalLie algebras. We denote the simple roots by α i , i = 1 , ..., k . Fix an invariant bilinear form h , i on h ∗ , and define ˇ α i = h α i ,α i i α i . If all the roots have the same length, we assume h , i tobe normalized such that ˇ α i = α . If h , i is nondegenerate, we define the fundamental weightsΛ j by h ˇ α i , Λ j i = δ ij . If α ∈ h ∗ , the reflection s α on h ∗ is defined by s α ( λ ) = λ − h λ, ˇ α i α . Wedenote by U = U q g the Drinfeld-Jimbo quantum group corresponding to the semmisimple Liealgebra g . It is well-known that for q not a root of unity, the representation theory of U isessentially the same as the one of g , i.e. same labeling set of simple representations, characterformulas etc. So we will sometimes state results only for g when its generalization to U isobvious.2.2. Gradation via Lie subalgebra.
Let g be a Lie subalgebra of g corresponding to thegraph obtained from X by removing the vertex labeled by 1. If λ = P Ni =1 a i Λ i is a weight of g , we denote by ˆ λ = P Ni =2 a i Λ i the corresponding weight of g (after obvious identificationsof the fundamental weights of g with a subset of fundamental weights of g ). Let V be afinite dimensional module of g such that h Λ , Λ i = c , a constant for all highest weights Λ of V ; here Λ is the fundamental weight corresponding to 1. We denote by V [0] the g -modulegenerated by the highest weight vectors of V . More generally, we define the level i subspaces V ⊗ n [ i ] for tensor powers of V and for i = 0 , , , ... by V ⊗ n [ i ] = span { V ⊗ n [ µ ] , h n Λ − µ, Λ i = i } . It is easy to see that V ⊗ n [0] = ( V [0]) ⊗ n for all n ∈ N . Conversely, we say that a weight µ has level i in V ⊗ n if V ⊗ n [ µ ] ⊂ V ⊗ n [ i ]; in this case we denote the level of µ by lev n ( µ ) or just lev ( µ ) if no confusion arises. Lemma 2.1.
Let V be a g -module as just described. Then(a) V ⊗ n [0] ∼ = V ⊗ n ˆΛ as a g -module. HANS WENZL (b) Let W be the g -module generated by V ⊗ n [0] . Then End g ( W ) ∼ = End g ( V ⊗ n ˆΛ [0]) . Inparticular, mult V µ ( V ⊗ n ) = mult V ˆ µ ( V ⊗ n ˆΛ )[0] for any weight µ with lev n ( µ ) = 0 . P roof.
Part (a) was already shown in [30], Lemma 1.1 if V is irreducible. The proof carriesover easily to our slightly more general setting. Part (b) is an easy consequence of part (a).2.3. Traces and contractions.
The following material can be found in e.g. [15], [27] and[21], Section 1.4. Let W be a U -module, and let a ∈ End U ( W ). Then the categorical trace or q -trace T r q ( a ) is given by T r q ( a ) = T r ( q ρ a ); here T r is the ordinary trace on End( W ), and q ρ acts on the weight vector w ∈ W with weight µ by the scalar q h µ,ρ i . Let W = V λ be anirreducible module with highest weight λ . Using the notation [ n ] = ( q n − q − n ) / ( q − q − ), wecan explicitly write the q -dimension as(2.1) dim q V λ = Y α> [ h λ + ρ, α i ][ h ρ, α i ] . In particular, if e is a minimal idempotent in End U ( W ) projecting onto an irreducible sub-module ∼ = V λ of W , we have T r q ( e ) = dim q V λ . The normalized trace tr q is defined by tr q = (1 / dim q W ) T r q .Let A ⊂ B be finite-dimensional semisimple algebras with a nondegenerate normalized trace tr on B such that also its restriction to A is nondegenerate; nondegenerate here means thatthe bilinear form h b , b i = tr ( b b ) is nondegenerate. Then the orthogonal projection from B onto A with respect to this bilinear form is usually called the trace preserving conditionalexpectation ε A . Its values are uniquely determined by tr ( aε ( b )) = tr ( ab ) for all a ∈ A and all b ∈ B .In the setting above, one can define an algebra extension B of B with respect to theinclusion A ⊂ B , Jones’ basic construction, as follows: It is generated by B , acting on itselfvia left multiplication and the projection e A coming from ε A , viewed as a linear operator on B . It is well-known that B is isomorphic as a vector space to Be A B (here we identify B with λ ( B ), the algebra of linear operators on B coming from left-multiplication by elements of B ).Moreover, the multiplication in Be A B is defined by(2.2) ( b e A b )( b e A b ) = b ε A ( b b ) e A b . Assume now that the trivial representation appears in the second tensor power of therepresentation V with multiplicity 1, and it only appears in even tensor powers of V . Decom-posing V ⊗ n as a direct sum of simple U = U q g -modules, we define V ⊗ nold to be the direct sumof those simple modules which already appeared in V ⊗ n − . By semisimplicity and definitionof V ⊗ nold , we have a unique decomposition V ⊗ n = V ⊗ nold ⊕ V ⊗ nnew in these cases. The followingresult has already more or less appeared before in various publications; in the form below, seee.g. [30], Prop. 4.10. Proposition 2.2.
Let C n = End U ( V ⊗ n ) . Then the algebra End U ( V ⊗ n +1 old ) is isomorphic toJones’ basic construction for C n − ⊂ C n . In particular, it is isomorphic as a vector space to ( C n ⊗ p n ( C n ⊗ where p is the projection onto ⊂ V ⊗ , and p n = 1 n − ⊗ p ∈ End U ( V ⊗ n +1 ) . ENTRALIZER ALGEBRAS FOR SPIN REPRESENTATIONS 9
We also need the following well-known property of categorical traces.
Lemma 2.3. If V, W are U -modules with V being irreducible, a ∈ End U ( V ⊗ ) and b ∈ End U ( W ⊗ V ) , then tr (( b ⊗ V )(1 W ⊗ a )) = tr ( b ) tr ( a ) . P roof.
This follows from the categorical definition of tr and is well-known. E.g. the proofof [21], Prop. 1.4(c) can easily be modified to prove the claim.3. Spinors for quantum groups
Roots and weights for orthogonal Lie groups.
For information about roots andweights, see e.g. [10],[14], and about spinor representations, see e.g. [31]. Let { ǫ i , ≤ i ≤ k } be the usual standard basis of R k . We represent the simple roots ( α i ), i = 1 , , ... k of Lietypes B k and D k as usual by α i = ǫ i − ǫ i +1 for i < k , and as α k = ǫ k for Lie type B k and as α k = ǫ k − + ǫ k for Lie type D k . With these notations, the weight lattice is given by Z k ∪ ( Z k + ε ),where ε = (1 / , ... / λ = ( λ i ) which can be explicitly described as the set ofall weights λ satisfying λ ≥ λ ≥ ... ≥ λ k ≥ B k resp. λ ≥ λ ≥ ... ≥ | λ k | fortype D k .3.2. Pin groups.
The unique compact connected and simply connected Lie group corre-sponding to the root systems B k and D k is the spin group Spin ( N ) with N = 2 k + 1 fortype B k and with N = 2 k for type D k . It is a 2-fold covering of the orthogonal group SO ( N ). As usual, we denote the corresponding covering group of O ( N ) by P in ( N ). Weembed g ∈ O (2 k ) → ( g, ǫ ) ∈ O (2 k ) × Z / ⊂ SO (2 k + 1), where the sign is chosen so thatwe obtain determinant one. This embedding carries over to an embedding of P in (2 k ) into Spin (2 k +1), i.e. we can consider P in (2 k ) as a subgroup of Spin (2 k +1). As already indicatedin the previous section, it will be convenient to consider P in (2 k ) instead of Spin (2 k ).Algebraically, P in (2 k ) is a semidirect product of Spin (2 k ) with Z /
2. On the Lie algebralevel, the Z / k − k , i.e.the generators belonging to the endpoints of the D k graph next to its triple vertex. This Z / λ ¯ λ on the weight space determined by permuting theroots α k − and α k , and leaving the other simple roots fixed. It is easy to check that if λ = ( λ , ..., λ k ), then ¯ λ = ( λ , ..., λ k − , − λ k ). The connection between irreducible Spin (2 k )and irreducible P in (2 k )-modules is described easily as follows:- If λ = ¯ λ (i.e. λ k = 0), then there exists a unique irreducible P in (2 k )-module whoserestriction to Spin (2 k ) decomposes as a direct sum of highest weight modules labeled by λ and ¯ λ . We shall denote this P in (2 k )-module by V λ with λ the dominant weight satisfying λ k > λ = ¯ λ , there exist exactly two irreducible nonisomorphic P in (2 k )-modules, denoted by V λ and V λ † whose restriction to Spin (2 k ) is isomorphic to the highest weight module labeledby λ . Observe that in this case λ can be identified with a Young diagram and one takes for λ † the Young diagram with the same columns as λ except that the first one now has 2 k − λ ′
10 HANS WENZL boxes (where λ ′ is the number of boxes in the first column of λ ). For all other dominantweights we define λ † = λ .3.3. Spinors.
Let S be the spinor module as constructed via the Clifford algebra in Section1. In the odd-dimensional case, Lie type B k it is the irreducible representation with highestweight Λ k = ε , the fundamental weight dual to α k . In the even-dimensional case, Lie type D k , the module S remains irreducible as a P in (2 k )-module, but decomposes into the directsum of two irreducible Spin (2 k )-modules whose highest weights are the fundamental weightsΛ k − = ε − ǫ k and Λ k = ε .The module S has the following properties: Its weights are given by { ω, ω = P ki =1 ± ǫ i } ,which holds for S being viewed as a Spin (2 k + 1) module as well as a P in (2 k )-module. Thefollowing tensor product rules for spinor groups are well-known and follow easily from generaltheory. More specialized treatments can also be found in e.g. [1], [2] to name but a few. If V λ is an irreducible module with highest weight λ for Lie type B k , then V λ ⊗ S ∼ = M µ V µ , where the summation goes over all dominant weights µ = λ + ω with ω a weight of V . ForLie type D k , we have the following modification: V λ ⊗ S ∼ = V λ † ⊗ V ∼ = M µ V µ ⊕ M µ =¯ µ V µ † , where the summation goes over all dominant weights µ = λ + ω with ω a weight of S andwith µ k ≥ λ † as described above. In particular wewill write [1 r ] for the Young diagram with r boxes in one column. We will need the followingstraightforward examples, which are elevated to the rank of a lemma Lemma 3.1.
We have the following decompositions: (a) S ⊗ ∼ = L ks =0 V [1 k − s ] for Lie type B k , (b) S ⊗ ∼ = L ks = − k V [1 k − s ] for Lie type D k , (c) S ⊗ ∼ = L kr =0 m r V ǫ +[1 k − s ] , where the multiplicity m r is equal to r + 1 for Lie type B k ,and it is equal to r + 1 for Lie type D k , (d) For
P in (2) ∼ = O (2) (see discussion in Section 3.6) and Spin (3) ∼ = SU (2) , S ⊗ nnew consistsof one irreducible representation, except for n = 2 in the P in (2) case, where it is thedirect sum of two irreducible representations. q -Dimensions. Recall that the q -dimension of a representation is given by Eq. 2.1.We need more explicit formulas for certain representations. We use the notation [ n ] q =( q n − q − n ) / ( q − q − ). Let U be equal to U q so N (for N odd) or the semidirect product of ENTRALIZER ALGEBRAS FOR SPIN REPRESENTATIONS 11 U q so N with Z / N even. It is well-known that for the U module V [1 r ] with highest weight (1 , ..., , , ...,
0) (with r q V [1 r ] = d ( r, N ) = (cid:18) N − r (cid:19) q + (cid:18) N − r − (cid:19) q = (cid:18) Nr (cid:19) q q k − r + q r − k q k + q − k , where (cid:0) nm (cid:1) q = [ n ] q ! / [ m ] q ![ n − m ] q !, and where [ n ] q ! = Q ni =1 [ i ] q . This can be derived from thecharacter formulas (see Eq. 2.1) or it can be read of as a special case from the formulas inSection 5 of [28] for λ = [1 r ]. Also observe that the dimension dim q S = [2 k ] qq of the spinormodule S is given by(3.2) [2 k ] qq := ( ( q / + q − / )( q / + q − / ) ... ( q k − / + q / − k ) if N = 2 k + 1 is odd,2( q + q − )( q + q − ) ... ( q k − + q − k ) if N = 2 k is even,3.5. Symmetric representations of C . The calculation of the structure coefficients willbe significantly simplified by the existence of a certain involutive antihomomorphism T on C .In fact, it can be defined for all C n , n ∈ N . It satisfies the involutive property ( c T ) T = c forall c ∈ C n , n ∈ N and the functorial property( c ⊗ c ) T = c T ⊗ c T c ∈ C n , c ∈ C n . The existence of this antihomomorphism is a consequence of Kashiwara’s inner product onmodules of quantum groups (see e.g. [30], Section 1.4 for details).
Lemma 3.2.
For any simple module of
End U ( S ⊗ old ) ⊂ C , we can find a basis ( v r ) of simul-taneous eigenvectors of C ⊗ for which also each element in ⊗ C is given by a symmetricmatrix. In particular, if p ( k ) is the projection onto ⊂ S ⊗ , we can choose the basis such that p ( k )2 = 1 ⊗ p ( k ) is given by the matrix ( q dim q V [1 i ] q dim q V [1 j ] / dim q S ) ij .P roof. The first statement is a special case of e.g. [30], Lemma 1.9. In our special case of C path basis vectors would just be simultaneous eigenvectors for the projections p ( s )1 , with0 ≤ | s | ≤ k , and the argument of that Lemma works for any element A in 1 ⊗ C . Thematrix of p ( k )2 with respect to this basis can be calculated by observing that it coincides withthe projection of Jones’ basic construction for C ⊂ C under the isomorphism stated in Prop.2.2. Indeed, matrices for such projections have been calculated in [24] or [22] in terms of theweight vectors of the trace; for the latter, see Sections 2.3 and 3.4. The second statementfollows from this.3.6. Centralizers.
In the following we denote by U either the Drinfeld-Jimbo q -deformation U q so k +1 of the universal enveloping algebra of so k +1 or the semidirect product of U q so k with Z / group level in the obvious way. So, in particular, the action of t on a weight vector v ω of thespinor representation S or the vector representation V in the usual normalization is given by(3.3) t : v ω v ¯ ω . We define the algebras C (0) n = C and C ( N ) n = End U ( S ⊗ n ), where S is the spinor representa-tion of type B k (for N = 2 k + 1 odd) or D N (for N = 2 k even). We will give an inductiveprocedure how to determine the structure of these algebras. To get the induction started, letus review the cases for P in (2) and
Spin (3), which are well-known:
P in (2): It is easy to check that
P in (2) is isomorphic to the orthogonal group O (2): Thetwo-fold covering of a circle is again a circle, which defines an isomorphism between Spin (2)and SO (2). This isomorphism extends to one between P in (2) and O (2). Moreover, thespinor representation can be identified with the usual two-dimensional representation of O (2)under this isomorphism. It is well-known that in the group case the centralizer algebra C (2) n is isomorphic to a quotient of Brauer’s centralizer algebra D n (2), whose simple componentsare labeled by Young diagrams whose first two columns contain at most two boxes. It is alsowell-known that we do not have a quantum deformation in this case, which was shown on thecategorical level in e.g. [26], Lemma 7.5. Hence we have in general that C (2)2 ∼ = C and C (2) n +1 isisomorphic to a direct sum of Jones’ basic construction for C (2) n − ⊂ C (2) n and a one-dimensionaldirect summand labeled by the Young diagram [ n + 1] (see [5], [29]). Spin (3) ∼ = SU (2): Here the spinor representation S corresponds to the two-dimensionalrepresentation of SU (2). The centralizer algebras are well-understood in this case in theclassical as well as in the quantum case. They are given as quotients of Hecke algebras of type A , which are also known as Temperley-Lieb algebras. Again, also in the type B case, C (3) n +1 can be determined inductively as a direct sum of Jones’ basic construction for C (3) n − ⊂ C (3) n and a one-dimensional direct summand (see [13]).We will need the following notations for the general induction: The element in C ( N )2 whichprojects onto the submodule V [1 k − s ] ⊂ V ⊗ will be denoted by p ( s ) . Observe that p ( k ) is theprojection onto the trivial representation, and that s can be negative for type D (see the endof Section 3.3). Moreover, if a ∈ C ( N )2 , we define a i = 1 i − ⊗ a ⊗ n − i − ∈ End U ( S ⊗ n ) = C ( N ) n . Theorem 3.3.
The structure of C ( N ) n , as defined above, is determined for n = 2 by Lemma3.1, and it is determined inductively for n > and N > by C ( N ) n +1 ∼ = C ( N − n +1 ⊕ C ( N ) n p ( N ) n C ( N ) n , as a direct sum of algebras, with the multiplicative structure as in Eq 2.2. In particular, C ( N ) n is generated by the elements a i , i = 1 , , ... n − , with a ∈ End U ( S ⊗ ) . P roof . We shall prove this theorem by induction on both N and n . Observe that thetheorem follows from the discussion above for N = 2 and N = 3. Also observe that the trivial ENTRALIZER ALGEBRAS FOR SPIN REPRESENTATIONS 13 representation appears in the second tensor power of the spinor representation S in generalfor arbitrary N . Hence, by Prop. 2.2, End U ( S ⊗ n +1 old ) ∼ = C ( N ) n p ( N ) n C ( N ) n . So it suffices to showthat End U ( S ⊗ n +1 new ) ∼ = C ( N − n +1 . The statement about the generators follows by induction on n and N .By Lemma 2.1(a), the commutant of the action of the quantum group U ′ of type B k − resp.type D k − on S ⊗ n +1 [0] is isomorphic to C ( N − n +1 . Hence it suffices to show that the U -modulegenerated by V ⊗ n +1 [0] is equal to V ⊗ n +1 new , where now U is the quantum group of type B k resp. type D k ; indeed as every irreducible U ′ submodule in S ⊗ n +1 [0] generates an irreducible U module, we have C ( N − n +1 ∼ = End U ′ ( S ⊗ n +1 [0]) ∼ = End U ( S ⊗ n +1 new ).As | ω i | ≤ / ω of S , it follows from the tensor product rules for S (see Section3.3) by induction that also | λ | ≤ n/ λ in S ⊗ n . On the other hand,it is easy to check by induction, that any module V λ resp V λ † labeled by a dominant weight λ with | λ | ≤ n/ S ⊗ n : the claim is obviously true for n = 1, and givena dominant weight λ with λ ≥
1, we can always find a weight ω of S such that λ ′ = λ − ω isdominant. As V λ ′ ⊂ V ⊗ ( n − by induction assumption, we obtain V λ ⊂ V λ ′ ⊗ S ⊂ S ⊗ n . Hence S ⊗ nnew is a direct sum of highest weight modules V λ such that λ = n/
2, and any irreduciblesubmodule of S ⊗ n with such a highest weight is contained in S ⊗ nnew .As n/ n h Λ , λ i , it follows that all these highest weight vectors are contained in S ⊗ n [0].Hence S ⊗ nnew is contained in the U -module generated by S ⊗ n [0]. The other inclusion followsfrom the fact that any highest weight vector in V ⊗ n [0] has a weight λ satisfying h λ, Λ i = n/ S ⊗ nnew . This finishes the proof. Corollary 3.4. If N = 2 k is even, the algebra End
P in (2 k ) ( S ⊗ l ) is a quotient of U ( k, l ) , asdefined in Def. 1.9. If N = 2 k + 1 , the algebra End
Spin (2 k +1) ( S ⊗ l ) is a quotient of U o ( k, l ) . P roof.
We have seen that the element C has N + 1 distinct eigenvalues, and, for N odd, theelement C has ( N + 1) / C resp C generates End P in ( N ) ( S ⊗ )for N even, resp End Spin ( N ) ( S ⊗ ) for N odd. The rest follows from Theorem 3.3.4. Structure Coefficients
Invariant elements, Clifford approach.
We have seen in Section 1 that the element2 C = P e i ⊗ e i ∈ Cl ⊗ ∼ = End( S ⊗ ) generates the commutant of the action of P in (2 k ), where( e i ) i is an orthonormal basis. For the odd-dimensional case, it is convenient to consider therestriction of the action on the module in the last sentence to P in ( N − P in ( N −
1) module by ˜ S . It decomposes into a direct sum of irreducible P in ( N − S + ⊕ S − , which are isomorphic as Spin ( N − C for the inner product of an N − V . If wetake the usual weight vectors as before, we can express C in the form(4.1) C ( v λ ⊗ v µ ) = 12 α ( N/ λ,µ v ¯ λ ⊗ v ¯ µ + X ≤ j 2. It is not hard to calculate the coefficients α ( j ) λ,µ which are equalto ± q -Clifford algebra, which has already been studied (see [9], [6]). However, in this context,one would also have to deform the multiplication of the second tensor power of the q -Cliffordalgebra in a nontrivial way. This makes calculations cumbersome. Instead, we shall producethe q -analog of the invariant element C by a straightforward calculation of the coefficients α ( j ) λ,µ in the quantum case.4.2. Invariant elements, direct approach. As motivated in the previous subsection, wenow determine a special element C ∈ End U ( S ⊗ ) by finding suitable coefficients for theexpression in Eq. 4.1. Proposition 4.1. Let U = U q so k ⋊ Z / . The element C ∈ End( S ⊗ ) , defined by C ( v µ ⊗ v ν ) = X j δ µ j , − ν j ( − q ) { ν − µ } j − v ¯ µ j ⊗ v ¯ ν j commutes with the action of U . Here { γ } j − = P j − i =1 γ i for any γ ∈ R k . Moreover ¯ γ j isdefined to coincide with γ except for a sign change in the j -th coordinate. P roof. It is easy to check that C leaves invariant the weight spaces. So it does commutewith the generators K i of U . Also, as the action of C on v µ ⊗ v ν only depends on the lastcoordinates of µ and ν as far as whether they are equal or not, C also commutes with thegenerator t of Z / U (see Eq 3.3). It remains to check the equation(4.2) C ∆( X j )( v µ ⊗ v ν ) = ∆( X j ) C ( v µ ⊗ v ν ) , for X j = E j , F j and 1 ≤ j ≤ k . Recall that the coproduct is defined by∆( X j ) = K / j ⊗ X j + X j ⊗ K − / j . It will be convenient to use the following notations for the matrix coefficients of C :(4.3) a ( j ) µ,ν = C ¯ µ j , ¯ ν j µ,ν = δ µ j , − ν j ( − q ) { ν − µ } j − . Comparing the coefficients of the vector v µ + ǫ j ⊗ v ν − ǫ j +1 in 4.10 for X j = E j , we obtain(4.4) q −h ν,α j i / a ( j +1) µ + α j ,ν + q h µ,α j i / a ( j ) µ,ν + α j = q −h ν − ǫ j +1 ,α j i / a ( j +1) µ,ν + q h µ + ǫ j ,α r i / a ( j ) µ,ν . A similar equation follows if we consider the coefficients of the vector v µ − ǫ j +1 ,ν + ǫ j .Let us first consider the case U = U q so . We can write the weights and basis vectors of S as pairs of signs, e.g. ε = (++); similarly, the basis vectors of S ⊗ S are written as a vector ENTRALIZER ALGEBRAS FOR SPIN REPRESENTATIONS 15 with two such pairs, such as e.g. (++ , ++). Then we deduce the following identities from Eq4.10: a ( j +1)(+ − , − +) = − q − a ( j )( − + , + − ) , a ( j +1)( − + , + − ) = − q − a ( j )(+ − , − +) ,a ( j +1)(++ , −− ) = − q − a ( j )( −− , ++) , a ( j +1)( −− , ++) = − q − a ( j )(++ , −− ) , (4.5) a ( j +1)(+ − , ++) = a ( j )( − + , ++) , a ( j +1)(++ , − +) = a ( j )(++ , + − ) . Indeed, e.g. the equations in the first line follow from Eq. 4.10 for the vector ( − + , − +)by comparing the coefficents of the vectors (++ , −− ) and ( −− , ++), see also Eq 4.4. Theother equations can be derived similarly, using generators E , E , F , F . Essentially, theseare calculations within U q sl , applied to tensor products of vectors of weights ± U = U q so k , with k > h ǫ j , α r i 6 = 0, i.e. j ∈ { j, j + 1 } for j < k , only the j -th and ( j + 1)-stcoordinates of µ and ν are relevant for checking Eq. 4.10, up to a common multiple for bothsides. We again get equations as in 4.5 from which we can determine the coefficients a ( j ) µ,ν byinduction on j , starting with a (1) µ,ν = δ µ , − ν . Lemma 4.2. (a) If N = 2 k even, the eigenvalues of the map C are [ j ] = ( q j − q − j ) / ( q − q − ) for − k ≤ j ≤ k . The corresponding eigenspaces are V [1 k − j ] ⊂ S ⊗ , see Lemma 3.1.(b) If N = 2 k + 1 is odd, C has the eigenvalues [ j + 1 / for − k − ≤ j ≤ k . P roof. Let v = P λ ( − q ) h ε − λ,ρ i v λ ⊗ v − λ , where ρ = ( k − i ) i and ε is the highest weightvector of S . Then the v λ ⊗ v − λ coordinate of C v is given by X j a ( j )¯ λ j , − ¯ λ j ( − q ) h ε − ¯ λ j ,ρ i = ( − q ) h ε − λ,ρ i X j q h ¯ λ j − λ,ρ i−{ λ } j − , where we used the fact that h ¯ λ j − λ, ρ i + j − v is an eigenvector of C if we canshow that the set of exponents of q , namely { λ j ( k − j ) − { λ } j − , ≤ j ≤ k } coincides withthe set of numbers k + 1 − r , 1 ≤ r ≤ k . This is easily shown by induction on k , by usingthe induction assumption for the weight µ = ( λ , λ , ...., λ k ) and observing that for j = 1 weget ± ( k − 1) depending on the sign of λ . This shows that [ k ] is an eigenvalue. Changing thesign of the coefficient of v λ ⊗ v − λ for which λ is a weight in S − , one also sees that − [ k ] is aneigenvalue.To prove the claim for the other values, observe that C leaves invariant the span S ⊗ r spanned by vectors v µ ⊗ v µ for which µ j = ν j = + if j > r . Moreover, the action onto thissubspace coincides with the one of the element C for so r . Hence we also have the eigenvalues ± [ r ] for any 0 ≤ r < k . As C ∈ End U ( S ⊗ ) can have at most 2 k + 1 distinct eigenvalues, theclaim follows.Part (b) can be shown similarly.4.3. Odd-dimensional case. The same method also works in the odd-dimensional case. Weshall do the case O (3) in detail. We shall consider a faithful representation of Cl (3) on a simple module of Cl (4). We again use notation (++) , (+ − ) ... for the basis vectors. Then it is easyto see that the maps E : ( −− ) (++) , ( − +) (+ − ) , and F being the transposed of E with respect to this basis define a representation of U q sl . Itis the direct sum of two simple two-dimensional representations with highest weight vectors(++) and (+ − ). Using the coproduct as in the proof of Prop. 4.1, one can determine acommuting operator C as in Eq. 4.1. If we set α (1) λ,µ = 1 for any λ, µ with λ = − µ , we candetermine α (2) λ,µ = 1 / [2] if both v λ and v µ are highest weight vectors, and α (2)(++ , −− ) = α (2)(+ − , − +) = − q − / [2] , α (2)( −− , ++) = α (2)( − + , + − ) = − q/ [2] , where [2] = q / + q − / . Moreover, the coefficients for any tensor product of two basis vectorsare one of the above, where it only depends whether the tensor factors are a highest or a lowestweight vector. Proposition 4.3. If N = 2 k + 1 is odd, we can determine coefficients α ( j ) λ,µ , ≤ j ≤ k + 1 for C as in Eq. 4.1 such that C commutes with U = U q so N on ˜ S ⊗ , and that C has theeigenvalues [ j + 1 / 2] = ( q j +1 / − q − j − / ) / ( q − q − ) for − k − ≤ j ≤ k . P roof. If j ≤ k , we take for α ( j ) λ,µ the value as in Prop. 4.1. For α ( k +1) λ,µ , we define ε, κ tobe the ’vectors’ consisting of the k -th and ( k + 1)-st components of λ and µ respectively. If α ( k ) λ,µ = 0, we multiply it by a (2) ε,κ = − q ± / [2] as in the O (3)-case to get α ( k +1) λ,µ . If α ( k ) λ,µ = 0, weset α ( k +1) λ,µ = α ( k )¯ λ k ,µ / [2], where ¯ λ k coincides with λ except for the k -th coordinate. The claimabout the coefficients of C now follows from Prop. 4.1 and the calculations for the O (3) case.The claim about the eigenvalues is shown as in Lemma 4.2, where we now pick as eigenvector v = P ( − q ) h ε − λ,ρ i v λ ⊗ v − λ , where ρ = ( k + 1 / − i ) i and ε is the highest weight vector of S .4.4. Action in third tensor power. The main result of this subsection is listed in Lemma4.4. It is elementary. We will first deal with the slightly easier case N even. Recall that the i -th antisymmetrization V i V of the vector representation V of O ( N ) appears with multiplicity1 in S ⊗ and that V i V ⊗ S contains a unique summand which is isomorphic to S , which wedenote by S i , i.e. we have a direct summand S i ⊂ S ⊗ defined by S ∼ = S i ⊂ i ^ V ⊗ S ⊂ S ⊗ , ≤ i ≤ N. Let now ( v i ) be an orthonormal basis of highest weight vectors v i ∈ S i of weight ǫ . Their spanis a module of the commutant of the U action. Let(4.6) v i = X α ( i ) µ ,µ ,µ v µ ⊗ v µ ⊗ v µ , with v µ j a weight vector of S for all indices j . We extend the partial order of weights to tensorproducts of weight vectors in alphabetic order, i.e. the order structure is determined by thefirst factor for which the weights are not the same. Let Λ i be the highest weight of V i V for ENTRALIZER ALGEBRAS FOR SPIN REPRESENTATIONS 17 i = N/ U q so N module). Then v ǫ ⊗ v Λ i − ǫ ⊗ v ǫ − Λ i and v ¯ ǫ ⊗ v Λ i − ¯ ǫ ⊗ v ǫ − Λ i are two maximal vectors with nonzero coefficients in the linear combinationof v i and v N − i for i < N/ 2. Hence (1 ⊗ C ) v i and (1 ⊗ C ) v N − i are linear combinations of thevectors v j with j ≤ i + 1 or j ≥ N − i − S = S + ⊕ S − as a U q so N module. Then V i V is contained in S ⊗ ⊕ S ⊗ − if N/ − i is even, and in S + ⊗ S − ⊕ S − ⊗ S + if N/ − i is odd. Hence we also have that (1 ⊗ C ) v i is a linear combination of vectors v j such that j − i is odd. We have set up everything for N even for the following Lemma. Lemma 4.4. Let C be the linear map in End U ( S ⊗ ) (for N even) resp. in End U ( ˜ S ⊗ ) asdefined in the previous sections. The vector (1 ⊗ C ) v i is a linear combination of v i − and v i +1 . P roof. It follows from the definitions that t has to map a highest weight vector v λ of an so N module to a highest weight vector. As V i V remains irreducible as an so N module for i = N/ t = 1, we have tv λ = ± v λ in this case. One can now check directly for the highestweight vectors of V i V that it is +1 for i < N/ − i > N/ 2. Hence we can normalizethe vectors v i and v N − i such that in their basis expansions the vector v ǫ ⊗ v Λ i − ǫ ⊗ v ǫ − Λ i hasthe same coefficient, and the coefficients of the vector v ¯ ǫ ⊗ v Λ i − ¯ ǫ ⊗ v ǫ − Λ i differ by a sign. Itfollows from the definition of C that (1 ⊗ C ) v i is a linear combination of the vectors v j with j ≤ i + 1 and with j − i odd, for i < N/ 2. A similar result also holds for (1 ⊗ C ) v N − i .Finally, we use Kashiwara’s inner product (see e.g. [19], Section 3.5 or also [30], Section 1.4)for representations of quantum groups. We actually only need the fact that it is multiplicativefor tensor products, and that its adjoint maps End U q g ( W ) into itself for any U q g module W .As the weight spaces are mutually orthogonal and End U ( S ⊗ ) is abelian, one deduces that C T = C , and hence also (1 ⊗ C ) is self-adjoint. Hence, after suitably normalizing the mutuallyorthogonal vectors v i , we can assume C to be symmetric. The claim for N even follows fromthis and the statements at the end of the last paragraph. The proof for N odd goes exactlythe same way, after the notation is set up the right way. This will be done in the remainderof this subsection.If N is odd, we consider the module ˜ S ∼ = ˜ S + ⊕ ˜ S − , where ˜ S ± ∼ = S with highest weightvectors ε + = ε and ε − = ¯ ε ; this is exactly the same decomposition of the corresponding P in ( N + 1)-module into a direct sum of irreducible Spin ( N + 1)-modules. Then we have asbefore that as O ( N )-modules, S ⊗ ∼ = ⊕ ki =0 V i V and S + ⊗ S − ∼ = ⊕ ki =0 V N − i V . If ( N + 1) isdivisible by 4, also v − ε ∈ ˜ S + . In this case, we define v i to be a higest weight vector of theunique module S ∼ = S i ⊂ V i V ⊗ ˜ S + ⊂ ˜ S ⊗ , and we define v N − i to be a higest weight vectorof the unique module S ∼ = S N − i ⊂ V N − i V ⊗ ˜ S + ⊂ ˜ S + ⊗ ˜ S ⊗ − . If ( N + 1) is not divisible by4, we define the vectors v i ∈ ˜ S ⊗ ⊗ ˜ S − and v N − i ∈ ˜ S + ⊗ ˜ S − ⊗ ˜ S + . Using the fact that C maps ˜ S ⊗ to ˜ S ⊗ − and ˜ S + ⊗ ˜ S − to ˜ S − ⊗ ˜ S + , it should now be no problem for the reader toadapt the proof of Lemma 4.4 for the case N odd. Technical lemma. Let { i } = q i + q − i and define b i up to a sign by b i = (cid:18) Ni (cid:19) q { N/ − i }{ N/ } . Then we have the following lemma. Lemma 4.5. Let A be a symmetric ( N + 1) × ( N + 1) matrix with a ij = 0 unless | i − j | = 1 .Then the entries of A are completely determined by one eigenvalue λ and its correspondingeigenvector b via Eq. 4.7. In particular, if A has the eigenvalue [ N/ with eigenvector b = ( b i ) , where b i is as defined above, then a i,i +1 = a i +1 ,i = [ i + 1][ N − i ] { N/ − i }{ N/ − i − } .P roof. It is straightforward to show by induction on i , using the equation A b = λ b , that(4.7) a i,i +1 = λb i b i +1 ( b i − b i − + b i − ... ) . Now let λ = [ N/ 2] and let its eigenvector b be given as above. Then we also have b i = (cid:18) N − i (cid:19) q + (cid:18) N − i − (cid:19) q , from which we get a i,i +1 = [ N/ (cid:0) N − i (cid:1) q b i b i +1 . Using the definitions, it now is straightforward to show the claim.Let H, E, K ± be the usual generators of the Drinfeld-Jimbo quantum group U q sl , andlet V N be its ( N + 1)-dimensional simple representation. It can be defined via a basis { e o , e , ... e N } of eigenvectors of K which satisfies(4.8) E.e r = ([ N − r + 1][ r ]) / e r − , K.e r = q N − r e r , F.e r = ([ N − r ][ r + 1]) / e r +1 . Corollary 4.6. The element ⊗ C ∈ End U ( S ⊗ ) is given with respect to the basis ( v r ) inLemma 3.2 by A = ( K / + K − / ) − / ( E + F )( K / + K − / ) − / , where the elements on the right hand side stand for the matrices representing these quantumgroup elements in V N . P roof. Let N be even. It follows from Lemmas 3.2, 4.2 and 4.4 that the matrix A rep-resenting 1 ⊗ C with respect to the basis ( v r ) satisfies the conditions of the lemma; for thestatement about b observe that p ( k )2 is an eigenprojection of C in the notation of Lemma 3.2.For N odd, we get the appropriate eigenvectors for C , restricted to the basis vectors in ˜ S ⊗ ,as well as for the eigenvectors in ˜ S + ⊗ ˜ S ⊗ − , by Lemma 3.2. The claim can now be shown alsofor C , using its block structure with respect to our basis. ENTRALIZER ALGEBRAS FOR SPIN REPRESENTATIONS 19 Relations for centralizer algebras. We will need the following nonstandard q -defor-mation of the universal enveloping algebra of so N . It was defined by Gavrilik and Klimyk (seee.g. [7]) and by Noumi and Sugitani [20]. It is also a special case of the co-ideal subalgebrasof U q sl N defined by Letzter for Θ = id , see[18], Remark 2.4. It is not isomorphic to the usualDrinfeld-Jimbo quantum group, see [18], Remark 2.3. Definition 4.7. (a) The algebra U ′ q so l is defined via generators B , B , ... B l − and relations B i B j = B j B i for | i − j | > B i B i ± − ( q + q − ) B i B i ± B i + B i ± B i = B i ± . (b) The algebra U q ( l, k ) is the quotient of U ′ q so l defined via the additional relation Q kj = − k ( B i − [ j ]) = 0.(c) The algebra U o q ( l, k ) is the quotient of the subalgebra of U ′ q so l generated by B i , 1 ≤ i < l defined via the additional relation Q kj =1 ( B i − [ j − / ) = 0.It is clear that for q = 1 we obtain the relations for the universal enveloping algebra of theLie algebra so l , see e.g. the remarks before Lemma 1.3. Theorem 4.8. (a) If N = 2 k is even, we have representations of U = U q so N ⋊ Z / and U ′ q so l on S ⊗ l which are each others commutant. Moreover, the image of U ′ q so l factors through U q ( l, k ) .(b) If N = 2 k + 1 is odd, we have actions of U = U q so N and U o q ( l, k ) on S ⊗ l which areeach others commutant. P roof. In case (a), it follows from Lemma 4.2 that C has N + 1 eigenvalues, and hencegenerates End U ( S ⊗ ). It follows from Theorem 3.3 that End U ( S ⊗ l ) is generated by theelements C i , 1 ≤ i < l .It can easily be seen that the claim will follow if the commutation relations between C and C are checked, using Lemma 4.2. Now observe that by Lemma 4.2 and by Cor. 4.6 theelements C and C act on the span of the vectors ( v r ) ⊂ S ⊗ via the matrices representing theelements D = ( k / − k − / ) / ( q − q − ) and A in U q sl . To check the commutation relationsbetween C and C , observe that[ k, E ] = ( q − Ek = (1 − q − ) kE, [ k − , E ] = (1 − q ) k − E = ( q − − Ek − . The relations involving F are obtained from above by substituting E by F and q by q − .Setting D = ( k / − k − / ) / ( q − q − ), we obtain[ D , E ] = q / Ek / + q − / Ek − / = E ( q / k / + q − / k − / ) , and a similar expression with E replaced by F . Hence[ D , [ D , E + F ]] = E ( q / k / + q − / k − / ) + F ( q − / k / + q / k − / ) = ( q / + q − / ) ( E + F ) + ( k / − k − / )( E + F )( k / − k − / ) . This can be rewritten as D ( E + F ) − ( q + q − ) D ( E + F ) D + ( E + F ) D = ( E + F ) . As ( k / + k − / ) − / commutes with D , and obviously also C i commutes with C j provided | i − j | > 1, we have proved the relation in Def. 4.7 for i and i + 1 for i = 1. Proving therelation with B and B interchanged can again be done via a calculation in U q sl , or onechecks that one gets the same matrices in our set-up if we interchange C and C . This showsthat the relation holds for the summand C ( N )2 p ( N )2 C ( N )2 of C ( N )3 in Theorem 3.3. The generalclaim follows from that theorem and induction by N , observing that the projection p ( N ) n is aneigenprojection of C n .Again, the proof for the case N odd goes along the same lines. Remark . We have stated our results here only for q generic over the ground field ofthe complex numbers. It is not hard to generalize them to more general fields; e.g. eventhough square roots appear in our proofs, we do not expect them to be essential. Moreoverthe definition of the algebra U ′ q so l only involves elements in the ring Z [ q, q − ]. So a similarresult should also hold over that ring; indeed, by general results of Lusztig’s also centralizeralgebras of suitably defined quantum groups have canonical bases defined over Z [ q, q − ], see[19], Section 27.3. Remark . In previous work [16], [25] tensor categories were classified whose Grothendiecksemiring was the one of a unitary, orthogonal or symplectic group. There an intrinsic descrip-tion of centralizer algebras via braid group representations played a crucial role. For spinorrepresentations, it is more difficult to describe these braid representations as the standardgenerators have too many eigenvalues. This made it necessary to consider a new descriptionwhich generalizes the braid relations. Indeed, in many ways, the algebra U ′ q so l can be consid-ered an algebraic object of type A l − . They should be useful in proving similar classificationresults for spinor groups. Remark . In the recent publication [23] Rowell and Wang study certain representationsof braid groups which they call Gaussian representations. They are of particular interest intheir studies motivated by quantum computing. They conjecture that these representationsare related to the centralizer algebras of quantum groups for spinor representations for certainroots of unity. Again, as already noted in Remark 4.10, it is diffcult to determine these braidrepresentations because of the increasing number of eigenvalues. The detailed analysis ofrepresentations of C in this paper could be useful in solving this problem, see e.g. the proofof Theorem 4.8. Remark . It is not hard to deduce from Theorem 3.3 that for N odd the R -matricesgenerate End U ( S ⊗ l ) for all l . 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