Cellular structures using $\textbf{U}_q$-tilting modules

TL;DR

This paper develops a general framework using $ extbf{U}_q$-tilting modules to construct cellular bases for centralizer algebras, applicable in non-semisimple cases and broad categories, with applications to semisimplicity criteria and known algebra cellularity.

Contribution

It introduces a unified method for constructing cellular bases of centralizer algebras via $ extbf{U}_q$-tilting modules, extending to non-semisimple and infinite-dimensional cases.

Findings

Constructed cellular bases for various centralizer algebras.

Provided a new criterion for semisimplicity of these algebras.

Reproduced cellularity of known algebras within a unified framework.

Abstract

We use the theory of $\textbf{U}_q$-tilting modules to construct cellular bases for centralizer algebras. Our methods are quite general and work for any quantum group $\textbf{U}_q$ attached to a Cartan matrix and include the non-semisimple cases for $q$ being a root of unity and ground fields of positive characteristic. Our approach also generalizes to certain categories containing infinite-dimensional modules. As applications, we give a new semisimplicty criterion for centralizer algebras, and recover the cellularity of several known algebras (with partially new cellular bases) which all fit into our general setup.