On the Radical of a Hecke–Kiselman Algebra

Abstract

The Hecke-Kiselman algebra of a finite oriented graph Θ over a field K is studied. If Θ 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.

Irreducible representations of Hecke-Kiselman monoids

“…We construct two types of representations of K[C n ]: those coming from the representations of K[M i ], and those related to the idempotents in C n . It is known that sandwich matrices P i are invertible as matrices in M n i (K(s i )), see [12], and thus K 0 [M i ] are almost Munn algebras. Simple modules over such algebras can be described, see Chapter 5 in [11].…”

“…The radical of the algebra K[HK Θ ] in this case was described in Theorem 2 in [12]. Assume that Θ ′ is the subgraph of Θ obtained by deleting all arrows x → y that are not contained in any cyclic subgraph of Θ.…”

“…The investigation of ring theoretic properties of the infinite dimensional Hecke-Kiselman algebras was started in [10]. Results obtained therein and in [12,13] suggest that the structure of the Hecke-Kiselman monoid associated to the oriented cycle of length n plays a crucial role in understanding the structure of the algebras K[HK Θ ], over a field K, for arbitrary oriented graphs Θ.…”

“…We denote these words by x n q i , or s i , and by x n q i we understand the cyclic monoid generated by x n q i . We recall the result characterizing the structure of the Hecke-Kiselman monoid C n , that will be crucial in our approach to representation theory of K[C n ], proved in [13], Corollary 3.25 and [12], proof of Theorem 1.1.…”