An analog of the Amitsur-Levitzki theorem for matrix superalgebras.
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 \(2(nk+n+k)-\min\{n,k\}\).
scientific article; zbMATH DE number 5816264
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