On module categories related to $Sp(N-1) \subset Sl(N)$

Abstract

$\def\End{\operatorname{End}}$$\def\Rep{\operatorname{Rep}}$$\def\sl{\mathfrak{sl}}$Let $V = \mathbb{C}^N$ with $N$ odd.We construct a $q$-deformation of $\End_{Sp(N-1)}(V^{\otimes n})$ which contains $\End_{U_q \sl_N} (V^{\otimes n})$. It is a quotient of an abstract two-variable algebra which is defined by adding one more generator to the generators of the Hecke algebras $H_n$. These results suggest the existence of module categories of $\Rep(U_q \sl_N)$ which may not come from already known coideal subalgebras of $ U_q \sl_N$. We moreover indicate how this can be used to construct module categories of the associated fusion tensor categories as well as subfactors, along the lines of previous work for inclusions $Sp(N) \subset SL(N)$.