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

$\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 $Sp(N) \subset SL(N)$.

Keywords

module categories

2010 Mathematics Subject Classification

Primary 17B20. Secondary 46L65.

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Received 2 January 2022

Received revised 15 August 2022

Accepted 18 September 2022

Published 30 January 2024