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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aragona, R., Andrea, A.D.: Hecke-Kiselman monoids of small cardinality. Semigroup Forum 86, 32–40 (2013)
Bergman, G.M.: The diamond lemma for ring theory. Adv. Math. 29, 178–218 (1978)
Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups, vol. 1. Amer. Math. Soc., Providence (1964)
Denton, T.: Algebra, Excursions into Combinatorics at q = 0, PhD thesis University of California, Davis. arXiv:1108.4379 (2011)
Forsberg, L.: Effective representations of Hecke-Kiselman monoids of type A. arXiv:1205.0676v4 (2012)
Fountain, J., Petrich, M.: Completely 0-simple semigroups of quotients. J Algebra 101, 365–402 (1986)
Ganyushkin, O., Mazorchuk, V.: On Kiselman quotients of 0–Hecke monoids. Int. Electron. J. Algebra 10, 174–191 (2011)
Krause, G.R., Lenagan, T.H.: Growth of Algebras and Gelfand-Kirillov Dimension Revised Ed., Amer. Math. Soc., Providence (2000)
Kudryavtseva, G., Mazorchuk, V.: On Kiselman’s semigroup. Yokohama. Math. J. 55, 21–46 (2009)
Mȩcel, A., Okniński, J.: Growth alternative for Hecke–Kiselman monoids. Publicacions Matemàtiques 63, 219–240 (2019)
Mȩcel, A., Okniński, J.: Gröbner basis and the automaton property of Hecke–Kiselman algebras. Semigroup Forum 99, 447–464 (2019)
Okniński, J.: Semigroup Algebras Monographs and Textbooks in Pure and Applied Mathematics, 138, Marcel Dekker, Inc., New York (1991)
Okniński, J., Wiertel, M.: Combinatorics and structure of Hecke–Kiselman algebras, Communications in Contemporary Mathematics 22, 2050022 (2020)
Rowen, L.H.: Polynomial Identities in Ring Theory, Academic Press, New York (1980)
Rowen, L.H.: Ring Theory, 2, Academic Press, New York (1988)
Small, L.W., Stafford, J.T., Warfield, R.B.: Affine algebras of Gelfand-Kirillov dimension one are PI. Math. Proc. Cambridge Philos. Soc. 97, 407–414 (1985)
Acknowledgements
This work was supported by grant 2016/23/B/ST1/01045 of the National Science Centre (Poland).
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by: Kenneth Goodearl
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Okniński, J., Wiertel, M. On the Radical of a Hecke–Kiselman Algebra. Algebr Represent Theor 24, 1431–1440 (2021). https://doi.org/10.1007/s10468-020-09997-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-020-09997-3
Keywords
- Hecke–Kiselman algebra
- Monoid
- Simple graph
- Reduced words
- Algebra of matrix type
- Noetherian algebra
- PI algebra
- Jacobson radical