Cell algebra structures on monoid and twisted monoid algebras

TL;DR

This paper explores the structure of monoid algebras with a focus on cell algebra properties, conditions for quasi-heredity and semi-simplicity, and extends results to twisted monoid algebras.

Contribution

It establishes that monoid algebras with Schutzenberger group cell algebra structures also have a standard cell algebra structure, and extends this to twisted monoid algebras.

Findings

Monoid algebras have a standard cell algebra structure under certain conditions.

Conditions for monoid algebra to be quasi-hereditary are identified.

Semi-simplicity of monoid algebras is characterized for inverse semi-groups over fields.

Abstract

In this paper we study finite monoids M such that the group algebras over a domain R for all Schutzenberger groups of M are cell algebras. We show that for any such M the monoid algebra A over R has a standard cell algebra structure. Using properties of cell algebras we then find conditions for A to be quasi-hereditary and we show that if such an M is an inverse semi-group and R is a field k, then A is semi-simple if and only if the group algebras over k for all maximal subgroups of M are semi-simple. Finally, we show that for any "compatible" twisting of M into R the twisted monoid algebra is also a cell algebra and can thus be analyzed using cell algebra properties.