Minimal supervarieties with factorable ideal of graded polynomial identities.
A well known result of Giambruno and Zaicev states that a variety \(\mathcal V\) of PI algebras of finite basic rank (i.e., generated by a finite dimensional algebra) is minimal if, and only if, it is generated by an upper block triangular matrix algebra \(UT(m_1,\ldots,m_n)\). The T-ideal (the ideal of identities) of such algebras factors as the product of the ideals of the main blocks. In the superalgebras setting, one replaces upper block triangular matrix algebras with minimal superalgebras but the factorability property for their \(T_{\mathbb Z_2}\)-ideals (ideals of their \(\mathbb Z_2\)-graded identities) fails. If \(A\) is a minimal superalgebra let \(A=A_{ss}+J\) be its Wedderburn-Malcev decomposition and \(A_{ss}=A_1\oplus\cdots\oplus A_n\) be the decomposition of \(A_{ss}\) as a sum of graded simple algebras. In the paper under review, the authors investigate the question of when the ideal of graded identities of such algebra \(A\) factors as the product of the ideals of the graded simple components of \(A_{ss}\). They provide a positive answer for the case in which the algebras \(A_1,\ldots,A_n\) are non-simple graded simple except for at most one between \(A_1\) and \(A_n\). If more than one of the first components are simple and the other are non-simple (or conversely), they prove that such ideal factorizes in a weaker sense, also as a product of \(T_{\mathbb Z_2}\)-ideals of subalgebras of \(A\). In [\textit{O. M. Di Vincenzo} and \textit{E. Spinelli}, J. Algebra 368, 182-198 (2012; Zbl 1279.16021)] it has been stated that any supervariety of superexponent \(\geq 2\) is generated by one of the so called minimal superalgebras. But the problem of which minimal superalgebras generate a minimal supervariety of a fixed graded exponent is still open and according to the authors of the paper under review, ``its possible solution seems to be more involved than that of the ungraded case''. In the mentioned paper it is proved that if \(A\) is a minimal superalgebra whose semisimple part decomposes as a direct sum of graded simple components that are simple, then \(A\) generates a minimal supervariety. In the paper under review the authors prove that if there exists \(1\leq h\leq n\) such that \(A_1,\ldots,A_h\) are simple graded simple and \(A_{h+1},\ldots,A_n\) are non-simple graded simple (or conversely), then the superalgebra \(A\) generates a minimal supervariety of fixed superexponent \(d=\dim_F(A_1\oplus\cdots\oplus A_n)\).
scientific article; zbMATH DE number 6517746