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

Henning Haahr Andersen, Catharina Stroppel and Daniel Tubbenhauer

Vol. 292 (2018), No. 1, 21–59

DOI: 10.2140/pjm.2018.292.21

Abstract

We use the theory of $U_q$-tilting modules to construct cellular bases for centralizer algebras. Our methods are quite general and work for any quantum group $U_q$ attached to a Cartan matrix and include the nonsemisimple 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 semisimplicity criterion for centralizer algebras, and we recover the cellularity of several known algebras (with partially new cellular bases) which all fit into our general setup.

Keywords

cellular algebras, cellular bases, tilting modules, quantum enveloping algebras

Mathematical Subject Classification 2010

Primary: 17B10, 17B37, 20G05

Secondary: 16S50, 20C08, 20G42

Milestones

Received: 1 October 2016

Revised: 22 February 2017

Accepted: 2 May 2017

Published: 22 September 2017