Representations of weakly triangular categories

TL;DR

This paper introduces a new class of locally unital, locally finite dimensional algebras with weakly triangular decompositions, explores their module categories, and connects them to categorifications of Lie algebra representations.

Contribution

It defines and studies weakly triangular decompositions for a broad class of algebras, linking them to stratified categories and categorifications of Lie algebra modules.

Findings

The category of locally finite dimensional modules is a fully stratified category.

Semisimplicity of the algebra is characterized by centralizer subalgebras.

Categorifications of Lie algebra representations are achieved via these algebraic structures.

Abstract

A new class of locally unital and locally finite dimensional algebras $A$ over an arbitrary algebraically closed field is discovered. Each of them admits an upper finite weakly triangular decomposition, a generalization of an upper finite triangular decomposition. Any locally unital algebra which admits an upper finite Cartan decomposition is Morita equivalent to some special locally unital algebra $A$ which admits an upper finite weakly triangular decomposition. It is established that the category $A$-lfdmod of locally finite dimensional left $A$-modules is an upper finite fully stratified category in the sense of Brundan-Stroppel. Moreover, $A$ is semisimple if and only if its centralizer subalgebras associated to certain idempotent elements are semisimple. Furthermore, certain endofunctors are defined and give categorical actions of some Lie algebras on the subcategory of $A$-lfdmod consisting of all objects which have a finite standard filtration. In the case $A$ is the locally unital algebra associated to one of cyclotomic oriented Brauer categories, cyclotomic Brauer categories and cyclotomic Kauffman categories, $A$ admits an upper finite weakly triangular decomposition. This leads to categorifications of representations of the classical limits of coideal algebras, which come from all integrable highest weight modules of $\mathfrak {sl}_\infty$ or $\hat {\mathfrak{sl}}_e$. Finally, we study representations of $A$ associated to either cyclotomic Brauer categories or cyclotomic Kauffman categories in details, including explicit criteria on the semisimplicity of $A$ over an arbitrary field, and on $A$-lfdmod being upper finite highest weight category in the sense of Brundan-Stroppel, and on Morita equivalence between $A$ and direct sum of infinitely many (degenerate) cyclotomic Hecke algebras.