Distinguishing simple algebras by means of polynomial identities
type: Item
instance of: scholarly article
title: Distinguishing simple algebras by means of polynomial identities
zbMATH Open document ID: 1435.17007
published in: São Paulo Journal of Mathematical Sciences
publication date: 2019-06-28T00:00:00Z
review text: Let \(A\) and \(B\) be two algebras over a field \(FK\). If \(A\) and \(B\) are isomorphic then they clearly satisfy the same polynomial identities. The paper under review addresses the opposite problem. Suppose \(A\) and \(B\) are not necessarily associative and satisfy the same polynomial identities. Then what conditions one has to impose on \(A\) and \(B\), and on \(F\), in order to guarantee \(A\cong B\)? A bit of thought suffices to see that \(A\) and \(B\) must be finite dimensional, central simple and over an algebraically closed field. If \(A\) and \(B\) are associative then it follows from the well known theorem due to Amitsur and Levitzki that \(A\) and \(B\) are isomorphic if and only if they satisfy the same identities. For Lie algebras (and for irreducible representations of Lie algebras) the same was established by \textit{A. Kh. Kushkuleĭ} and \textit{Yu. P. Razmyslov} [Vestn. Mosk. Univ., Ser. I 1983, No. 5, 4--7 (1983; Zbl 0516.17007)]. The case of Jordan algebras was solved by \textit{V. S. Drensky} and \textit{M. L. Racine} [Commun. Algebra 20, No. 2, 309--327 (1992; Zbl 0745.17027)]. \textit{E. Neher} [J. Algebra 211, No. 1, 206--224 (1999; Zbl 0920.17014)] settled the case of Jordan pairs. Similar results were obtained for algebras graded by finite groups. It should be noted that \textit{Yu. P. Razmyslov} [Identities of algebras and their representations. Transl. from the Russian. Providence, RI: AMS (1994; Zbl 0827.17001)], in Section 1.5, obtained a more general result, in the context of universal algebras. The authors first give a detailed proof of Razmyslov's theorem. Recall that in it one considers \(\Omega\)-algebras where \(\Omega\) is a fixed set of operations (not only binary), and proves that if \(A\) and \(B\) are finite dimensional prime and satisfy the same identities then they are isomorphic provided the base field is algebraically closed. Then the authors proceed to the case of centrally closed prime algebras \(A\) and \(B\) over …
Mathematics Subject Classification ID: 17A42
Mathematics Subject Classification ID: 08B20
Mathematics Subject Classification ID: 16R50
zbMATH DE Number: 7074578
zbMATH Keywords: graded algebra
zbMATH Keywords: polynomial identity
zbMATH Keywords: universal algebra
author: Yuri Bahturin
author: Felipe Yukihide Yasumura
reviewed by: Plamen Koshlukov
MaRDI profile type: Publication
full work available at URL: s40863-019-00126-7
OpenAlex ID: W2933580915
Wikidata QID: Q107720026
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DOI: 10.1007/S40863-019-00126-7
description: scientific article; zbMATH DE number 7074578