An analog of the Amitsur-Levitzki theorem for matrix superalgebras.
type: Item
instance of: scholarly article
title: An analog of the Amitsur-Levitzki theorem for matrix superalgebras.
zbMATH Open document ID: 1210.16026
published in: Siberian Mathematical Journal
publication date: 2010-11-15T00:00:00Z
review text: The paper under review studies the algebras \(M_{n,k}(G)\) over a field \(F\) of characteristic 0. Here \(G\) is the infinite dimensional Grassmann algebra equipped with its canonical 2-grading, \(G=G_0\oplus G_1\). Furthermore \(M_{n,k}(G)\) is the subalgebra of \(M_{n+k}(G)\) consisting of all block matrices whose two diagonal blocks are of sizes \(n\times n\) and \(k\times k\), respectively, and whose entries lie in \(G_0\); the two off-diagonal blocks have their entries in \(G_1\). The algebras \(M_{n,k}(G)\) appear in the classification of the T-prime algebras developed by Kemer, and their importance in PI theory is significant. The main result of the paper under review is that \(M_{n,k}(G)\) satisfies identities of degree \(2(nk+n+k)-\min\{n,k\}\). The proof of this result goes through the trace identities of these algebras. The latter were described by \textit{Yu. P. Razmyslov} [Mat. Sb., N. Ser. 128(170), No. 2(10), 194-215 (1985; Zbl 0601.16016); translation in Math. USSR, Sb. 56, 187-206 (1987)] and by the author of the paper, [\textit{L. M. Samoĭlov}, Fundam. Prikl. Mat. 6, No. 4, 1221-1227 (2000; Zbl 1055.16028)]. The identities of the algebras \(M_{n,k}(G)\) are known in very few cases: if \(n\geq k\geq 1\) the only known instance is given by a theorem due to \textit{A. P. Popov} [Algebra Logic 21, 296-316 (1983); translation from Algebra Logika 21, 442-471 (1982; Zbl 0521.16014)]. In particular, the identities of least degree for \(M_{1,1}(G)\) are of degree 5. If \(k=0\) then the well known theorem of Amitsur and Levitzki states that the least degree for a polynomial identity for \(M_n(F)\) equals \(2n\). In the above considered cases the degrees of the identities described in the present paper coincide with the minimal ones. Therefore the author conjectures that the least degree of an identity for \(M_{n,k}(G)\) equals \(2(nk+n+k)-\min\{n,k\}\).
reviewed by: Plamen Koshlukov
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16W55
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 15A75
zbMATH DE Number: 5816264
zbMATH Keywords: minimal degree identities
zbMATH Keywords: Amitsur-Levitzki theorem
zbMATH Keywords: matrix superalgebras
zbMATH Keywords: trace identities
zbMATH Keywords: polynomial identities
zbMATH Keywords: Grassmann algebras
author: L. M. Samoilov
MaRDI profile type: Publication
full work available at URL: s11202-010-0051-2
OpenAlex ID: W2118682897
cites work: Identities of associative algebras
cites work: TRACE IDENTITIES OF FULL MATRIX ALGEBRAS OVER A FIELD OF CHARACTERISTIC ZERO
cites work: TRACE IDENTITIES AND CENTRAL POLYNOMIALS IN THE MATRIX SUPERALGEBRAS $ M_{n,k}$
cites work: The representation theory of the symmetric groups
DOI: 10.1007/S11202-010-0051-2
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description: scientific article; zbMATH DE number 5816264