On Centralizer Algebras for Spin Representations

Abstract

We give a presentation of the centralizer algebras for tensor products of spinor representations 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; in the odd-dimensional case only a certain subalgebra will appear. In the classical case q = 1 the relations boil down to Lie algebra relations. * Supported in part by NSF grants.

SO$(N)_2$ braid group representations are Gaussian

“…In Section 2 we review results about the centralizer algebras End(S ⊗n ) where S is a spinor representation of U q so N respectively U q so N ⋊Z 2 , or the corresponding object in one of the associated fusion categories. Most of these results have already more or less appeared before in [12], [36]. In Section 3, we reprove and extend several results by Klimyk and his coauthors concerning the representation theory of U ′ q so n .…”

“…We will again denote the images of the corresponding tilting modules in U by S (respectivelyS) in the fusion category O(N ) r . We have the following results, most of which were already proved in [36]: P roof. Part (a) follows from Lemma 2.1, using the explicit representations in [36] and the fact that for q not a root of unity the representation theory of Drinfeld-Jimbo quantum groups is essentially the same as for the corresponding Lie algebra.…”

“…For N odd, we similarly get thatS ⊗2 decomposes into the direct sum of two copies of the exterior algebra of C N . It was shown in [36] that the i-th antisymmetrization in S ⊗2 (respectively inS ⊗2 , where it appears with multiplicity 2) is an eigenspace of the so 2 generator B 1 with eigenvalue N/2 − i. This proves that the so 2 character of S ⊗2 (respectively ofS ⊗2 ) is given by Eq 2.3 for N even (respectively by twice the value of Eq 2.3 for N odd).…”

“…By the main result of [36], we have commuting actions of U = U q so N ⋊ Z/2 and U ′ q so n on S ⊗n (for N even) andS ⊗n for N odd. Not surprisingly, the decomposition in the Lemma 2.1 carries over to this setting if q is not a root of unity.…”

“…For N even we consider a semidirect (smash) product U q so N ⋊ Z 2 and we will also denote by S the irreducible U q so N ⋊ Z 2 -module whose restriction to U q so N is S − ⊕ S + . For generic parameter q, the centralizer algebras End U (S ⊗n ) are described ( [36,Theorem 4.8]) in terms of a non-standard deformation U ′ q so n of U so n , for both N odd and even. Although Rep(U) carries a braiding, the image of B n inside End U (S ⊗n ) does not generate these algebras.…”