The disoriented skein and iquantum Brauer categories

TL;DR

This paper introduces a diagrammatic framework using the disoriented skein category to study the representation theory of quantum symmetric pairs related to orthosymplectic Lie superalgebras, connecting it to the iquantum Brauer category.

Contribution

It establishes an equivalence between the disoriented skein category and the iquantum Brauer category, providing new insights and explicit bases for their morphism spaces.

Findings

Disoriented skein category admits full functors to modules over iquantum enveloping algebras.

An equivalence of module categories is established between the disoriented skein and iquantum Brauer categories.

Explicit bases for morphism spaces are constructed for both categories.

Abstract

We develop a diagrammatic approach to the representation theory of the quantum symmetric pairs corresponding to orthosymplectic Lie superalgebras inside general linear Lie superalgebras. Our approach is based on the disoriented skein category, which we define as a module category over the framed HOMFLYPT skein category. The disoriented skein category admits full incarnation functors to the categories of modules over the iquantum enveloping algebras corresponding to the quantum symmetric pairs, and it can be viewed as an interpolating category for these categories of modules. We define an equivalence of module categories between the disoriented skein category and the iquantum Brauer category (also known as the $q$-Brauer category), after endowing the latter with the structure of a module category over the framed HOMFLYPT skein category. The disoriented skein category has some advantages over the iquantum Brauer category, possessing duality structure and allowing the incarnation functors to be strict morphisms of module categories. Finally, we construct explicit bases for the morphism spaces of the disoriented skein and iquantum Brauer categories.