Relative cellular algebras

TL;DR

This paper introduces relative cellular algebras, a generalization of cellular algebras with varied partial orderings, providing classification of simple modules and examples including quantum groups and arc algebras.

Contribution

It defines and studies relative cellular algebras, extending cellular algebra theory and providing new examples such as restricted enveloping algebras and arc algebras.

Findings

Classification of simple modules for relative cellular algebras

Identification of algebras that are relative cellular but not cellular

Examples include restricted enveloping algebra and small quantum group for sl2

Abstract

In this paper we generalize cellular algebras by allowing different partial orderings relative to fixed idempotents. For these relative cellular algebras we classify and construct simple modules, and we obtain other characterizations in analogy to cellular algebras. We also give several examples of algebras that are relative cellular, but not cellular. Most prominently, the restricted enveloping algebra and the small quantum group for $\mathfrak{sl}_{2}$, and an annular version of arc algebras.