A Groupoid Approach to Discrete Inverse Semigroup Algebras

TL;DR

This paper develops a topological framework using groupoid theory to analyze inverse semigroup algebras, generalizing previous results and providing new methods for constructing and classifying their irreducible representations.

Contribution

It introduces a groupoid-based approach to describe inverse semigroup algebras, extending earlier work and enabling explicit construction of irreducible representations over arbitrary fields.

Findings

Describes the semigroup algebra as a convolution algebra of functions on an associated étale groupoid.

Provides a method to construct finite dimensional irreducible representations via induced representations from groups.

Characterizes irreducible representations satisfying a specific finiteness condition, including those with primitive idempotents.

Abstract

Let $K$ be a commutative ring with unit and $S$ an inverse semigroup. We show that the semigroup algebra $KS$ can be described as a convolution algebra of functions on the universal \'etale groupoid associated to $S$ by Paterson. This result is a simultaneous generalization of the author's earlier work on finite inverse semigroups and Paterson's theorem for the universal $C^*$-algebra. It provides a convenient topological framework for understanding the structure of $KS$, including the center and when it has a unit. In this theory, the role of Gelfand duality is replaced by Stone duality. Using this approach we are able to construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups, generalizing the well-studied case of an inverse semigroup with finitely many idempotents. More generally, we describe the irreducible representations of an inverse semigroup $S$ that can be induced from associated groups as precisely those satisfying a certain "finiteness condition". This "finiteness condition" is satisfied, for instance, by all representations of an inverse semigroup whose image contains a primitive idempotent.