Block-triangular matrix algebras and factorable ideals of graded polynomial identities.
Let \(A\) and \(B\) be PI-algebras and let \(U\) be an \(A\)-\(B\)-bimodule. A theorem of \textit{J. Lewin} [Trans. Am. Math. Soc. 188, 293-308 (1974; Zbl 0343.16002)] gives a condition when the T-ideal \(T(R)\) of the polynomial identities of the algebra \(\left(\begin{smallmatrix} A &U\\ 0& B\end{smallmatrix}\right)\) is equal to the product \(T(A)T(B)\). In the paper under review the authors study a similar problem for the \(G\)-graded polynomial identities of the \(G\)-graded objects \(R\), \(A\), \(B\), and \(U\), where \(G\) is an Abelian group and the base field is infinite. They introduce the notion of \(G\)-regular subalgebra of the matrix algebra, in terms of suitable projections of graded generic algebras. Then the authors prove that \(T_G(R)=T_G(A)T_G(B)\), provided that at least one of the algebras \(A\) and \(B\) is \(G\)-regular. They also give an effective characterization of the \(G\)-gradings which make the full matrix algebra \(G\)-regular. Assuming that the base field is of characteristic 0, and using a result of \textit{E. Formanek} [Proc. AMS-IMS-SIAM Summer Res. Conf., Brunswick/Maine 1984, Contemp. Math. 43, 87-119 (1985; Zbl 0609.16008)] for the Hilbert series of \(T(R)=T(A)T(B)\), the authors express the \(\mathbb{Z}_2\)-graded cocharacters of \(R\) in terms of those of \(A\) and \(B\), generalizing a result of \textit{A. Berele} amd \textit{A. Regev} [Isr. J. Math. 103, 17-28 (1998; Zbl 0919.16017)]. This result is illustrated on the concrete example of the \(\mathbb{Z}_2\)-graded algebra \(R\) constructed from \(A=M_{1,0}\), \(B=M_{1,1}\), and \(U=M_{1\times 2}\). (Also submitted to MR.)
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