On the radical of a Hecke-Kiselman algebra. (English) Zbl 1504.16046
Summary: The Hecke-Kiselman algebra of a finite oriented graph \(\varTheta\) over a field \(K\) is studied. If \(\varTheta\) 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.
MSC:
| 16S15 | Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) |
| 16S36 | Ordinary and skew polynomial rings and semigroup rings |
| 16P40 | Noetherian rings and modules (associative rings and algebras) |
| 16N20 | Jacobson radical, quasimultiplication |
| 16R20 | Semiprime p.i. rings, rings embeddable in matrices over commutative rings |
| 20M05 | Free semigroups, generators and relations, word problems |
| 20M25 | Semigroup rings, multiplicative semigroups of rings |
| 20C08 | Hecke algebras and their representations |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
Keywords:
Hecke-Kiselman algebra; monoid; simple graph; reduced words; algebra of matrix type; Noetherian algebra; PI algebra; Jacobson radicalReferences:
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