Representations of cyclotomic oriented Brauer categories

TL;DR

This paper constructs a categorification of tensor products of certain Lie algebra representations using modules over a cyclotomic oriented Brauer category, extending previous results to arbitrary characteristic fields.

Contribution

It introduces a new algebraic framework for tensor product categorification of integrable Lie algebra modules over fields of any characteristic.

Findings

Provides a categorification for tensor products of Lie algebra representations

Extends previous characteristic zero results to positive characteristic

Establishes a new algebraic structure linking Brauer categories and Lie algebra representations

Abstract

Let $A$ be the locally unital algebra associated to a cyclotomic oriented Brauer category over an arbitrary algebraically closed field $\Bbbk$ of characteristic $p\ge 0$. The category of locally finite dimensional representations of $A $ is used to give the tensor product categorification (in the general sense of Losev and Webster) for an integrable lowest weight with an integrable highest weight representation of the same level for the Lie algebra $\mathfrak g$, where $\mathfrak g$ is a direct sum of copies of $\mathfrak {sl}_\infty$ (resp., $ \hat{\mathfrak {sl}}_p$ ) if $p=0$ (resp., $p>0$). Such a result was expected in [3] when $\Bbbk=\mathbb C$ and proved previously by Brundan in [2] when the level is $1$.