Representations of Brauer category and categorification

TL;DR

This paper explores the categorification of certain highest weight modules of Lie algebras using representations of the Brauer category algebra, linking algebraic structures to categorified modules over fields of different characteristics.

Contribution

It constructs a categorification of integrable highest weight modules via representations of the Brauer category algebra, connecting algebraic bases with module categories.

Findings

Grothendieck group categorifies the highest weight module

Standard modules correspond to monomial basis

Projective covers relate to quasi-canonical basis

Abstract

We study representations of the locally unital and locally finite dimensional algebra $B$ associated to the Brauer category $\mathcal B(\delta_0)$ with defining parameter $\delta_0$ over an algebraically closed field $K$ with characteristic $p\neq 2$. The Grothendieck group $K_0(B\text{-mod}^\Delta)$ will be used to categorify the integrable highest weight $\mathfrak {sl}_{K}$-module $ V(\varpi_{\frac{\delta_0-1}{2}})$ with the fundamental weight $\varpi_{\frac{\delta_0-1}{2}}$ as its highest weight, where $B$-mod$^\Delta$ is a subcategory of $B$-lfdmod in which each object has a finite $\Delta$-flag, and $\mathfrak {sl}_{K}$ is either $\mathfrak{sl}_\infty$ or $\hat{\mathfrak{sl}}_p$ depending on whether $p=0$ or $2\nmid p$. As $\mathfrak g$-modules, $\mathbb C\otimes_{\mathbb Z} K_0(B\text{-mod}^\Delta)$ is isomorphic to $ V(\varpi_{\frac{\delta_0-1}{2}})$, where $\mathfrak g$ is a Lie subalgebra of $\mathfrak {sl}_{K}$ (see Definition~4.2). When $p=0$, standard $B$-modules and projective covers of simple $B$-modules correspond to monomial basis and so-called quasi-canonical basis of $V(\varpi_{\frac{\delta_0-1}{2}}) $, respectively.