Distinguishing simple algebras by means of polynomial identities
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 an arbitrary infinite field. In it there is an extension \(L\) of the base field \(K\) such that \(A\) and \(B\) are \(K\)-forms of the same algebra. As applications the authors deduce that algebras graded by a semigroup can be considered in the setting of universal algebras. Thus one applies the general theorem and deduces that if the algebras \(A\) and \(B\) are finite dimensional over an algebraically closed field, graded by a semigroup \(G\), and are graded simple then \(A\cong B\) as graded algebras if and only if they satisfy the same graded identities. Furthermore the authors show that analogous results hold for algebras with involution, superalgebras with involution and with superinvolution, for colour Lie superalgebras and \(p\)-superalgebras, algebras with trace and algebras with actions of a Hopf algebra or a generalized action. The paper is very well written, settles important problems and will be of interest to specialists in PI theory.
scientific article; zbMATH DE number 7074578
scientific article