Minimal varieties of PI-algebras with graded involution (Q6615434)

In the paper under review the authors study varieties of \(G\)-graded algebras endowed with a graded involution \(*\), where \(G\) is a cyclic group of prime order. Every algebra is considered over an algebraically closed field \(F\) of characteristic 0. More precisely, the authors consider the class of finite dimensional \((G,*)\)-algebras and they characterize the varieties generated by algebras belonging to this class that are minimal with respect to the \((G,*)\)-exponent.\N\NIn Theorem 2 the authors do classify finite dimensional \((G,*)\)-simple algebra over \(F\) showing a list of four types of graded simple algebras turning out to be called \emph{classical} if \(F\) is not algebraically closed.\N\NIn my opinion, the main results of the paper are the following: 1) (see Theorem 4 in the text) the authors prove any minimal variety of \((G,*)\)-algebras of \((G,*)\)-exponent \(d\) is generated by a suitable upper block triangular matrix algebra \(UT_G^*(A_1,\ldots,A_m)\), where \(A_1,\ldots, A_m\) are classical finite dimensional \((G,*)\)-simple algebras such that \(\dim_F(A_1\oplus\cdots\oplus A_m) = d\); 2) (see Theorem 8 in the text) any variety of \((G,*)\)-algebras generated by a finite dimensional \((G,*)\)-algebra is minimal of \((G,*)\)-exponent \(d\) if and only if it is generated by \(UT_G^*(A_1,\ldots,A_m)\), where \(A_1,\ldots, A_m\) are classical finite dimensional \((G,*)\)-simple algebras such that \(\dim_F(A_1\oplus\cdots\oplus A_m) = d\).\N\NThe paper is well written and very well detailed and can be used to furnish a complete background to a Ph.D. student approaching such a kind of research themes.