Dualities for spin representations

Abstract

Let $S$ be the spinor representation of $U_q{\mathfrak{so}}_N$, for $N$ odd and $q^2$ not a rooot of unity. We show that the commutant of its action on $S^{\otimes n}$ is given by a representation of the nonstandard quantum group $U_{-q^2}^{\prime }{\mathfrak{so}}_n$. For $N$ even, an analogous statement also holds for $S=S_{+}\oplus S_{-}$ the direct sum of the irreducible spinor representations of $U_q^{\prime }{\mathfrak{so}}_N$, with the commutant given by $U_{-q}^{\prime }{\mathfrak{o}}_n$, a $\mathbb{Z}/2$-extension of $U_{-q}^{\prime }{\mathfrak{so}}_n$. Similar statements also hold for fusion tensor categories with $q$ a root of unity.

Cited by in corpus (1)

• $B_k$ Spin Vertex Models and Quantum Algebras