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

TL;DR

This paper constructs a new q-deformation of endomorphism algebras related to Sp(N-1) within SL(N), revealing potential new module categories for quantum group representations and their applications in subfactor theory.

Contribution

It introduces a novel q-deformation of endomorphism algebras that extends known structures, suggesting new module categories for quantum groups beyond existing coideal subalgebras.

Findings

Constructed a q-deformation containing known endomorphism algebras.

Proposed existence of new module categories for $Rep(U_q\sl_N)$.

Indicated applications to subfactors and fusion categories.

Abstract

Let $V=\C^N$ 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 $Sp(N)\subset SL(N)$.