On the radical of a Hecke-Kiselman algebra

TL;DR

This paper investigates the structure of Hecke-Kiselman algebras derived from finite oriented graphs, characterizing their radicals and showing how they relate to matrix algebras and their centers.

Contribution

It provides a detailed description of the radical of Hecke-Kiselman algebras, especially for PI-algebras, and explores their structural properties related to centers and localizations.

Findings

Hecke-Kiselman algebra of an oriented cycle is semiprime.

The algebra's central localization is a finite product of matrix algebras over $K(x)$.

The radical is explicitly described via a congruence on the monoid.

Abstract

The Hecke-Kiselman algebra of a finite oriented graph $\Theta$ over a field $K$ is studied. If $\Theta$ is an oriented cycle, it is shown that the algebra is semiprime and its central localization is a finite direct product of matrix algebras over the field of rational functions $K(x)$. More generally, the radical is described in the case of PI-algebras, and it is shown that it comes from an explicitly described congruence on the underlying Hecke-Kiselman monoid. Moreover, the algebra modulo the radical is again a Hecke-Kiselman algebra and it is a finite module over its center.