Hans Wenzl (University of California - San Diego)

Centralizer algebras for spin representations

Let $$U$$ be the semi-direct product of the quantum group $$U_q(\mathfrak{so}_{2k})$$ with $$\mathbb{Z}_2$$, where the $$\mathbb{Z}_2$$ action is given via the graph automorphism on $$D_k$$. Let $$S$$ be the spinor representation of $$U$$. Then there exist actions of $$U$$ and of the non-standard $$q$$-deformation of $$\mathfrak{so}_n$$ on $$S^{\otimes n}$$ which generate each others commutants. A similar statement also holds for $$U_q(so_N)$$ with $$N$$ odd.