Pseudocompact algebras and highest weight categories

TL;DR

This paper introduces pseudocompact algebras to study highest weight categories, defining tilting modules and Ringel duals, and illustrates the theory with an example involving the supergroup $GL(1|1)$.

Contribution

It develops a new framework using pseudocompact algebras for highest weight categories, including the construction of tilting modules and Ringel duals, with applications to supergroups.

Findings

Defined ascending and descending quasi-hereditary pseudocompact algebras.

Investigated tilting modules and Ringel duals in this context.

Provided an explicit example with the supergroup $GL(1|1)$.

Abstract

We develop a new approach to highest weight categories $\cal{C}$ with good (and cogood) posets of weights via pseudocompact algebras by introducing ascending (and descending) quasi-hereditary pseudocompact algebras. For $\cal{C}$ admitting a Chevalley duality, we define and investigate tilting modules and Ringel duals of the corresponding pseudocompact algebras. Finally, we illustrate all these concepts on an explicit example of the general linear supergroup $GL(1|1)$.