Polynomial identities for graded tensor products of algebras.
Due to the pioneer work of \textit{A. R. Kemer} [see Ideals of identities of associative algebras. Transl. Math. Monogr. 87. Providence: AMS (1991; Zbl 0732.16001), for the account], nowadays the study of \(\mathbb{Z}_2\)-graded polynomial identities is an important part of the theory of PI-algebras. The multiplication in the \(\mathbb{Z}_2\)-graded analogue of the tensor product \(A\overline\otimes B\) of two \(\mathbb{Z}_2\)-graded algebras is defined by \((a_1\overline\otimes b_1)(a_2\overline\otimes b_2)=(-1)^{|b_1||a_2|}a_1a_2\overline\otimes b_1b_2\), where \(a_i\in A\) and \(b_i\in B\) are homogeneous elements of degree \(|a_i|\) and \(|b_i|\), respectively. \textit{A. Regev} and \textit{T. Seeman} [J. Algebra 291, No. 1, 274-296 (2005; Zbl 1083.16017)] established an analogue of the \(A\otimes B\)-theorem of Regev for the \(\mathbb{Z}_2\)-graded tensor product of two PI-algebras. They also conjectured that the graded tensor product of two T-prime PI-algebras in characteristic 0 is PI-equivalent to a T-prime algebra, as in the ordinary case. Regev and Seeman confirmed their conjecture in several cases. (The T-prime algebras are the matrix algebras \(M_n(K)\) and \(M_n(E)\) over the base field \(K\) and over the Grassmann algebra \(E\) and the subalgebras \(M_{k,l}\) of \(M_{k+l}(E)\).) The first main result of the paper under review is to prove completely the conjecture. In the remaining open cases, the authors show the PI-equivalences \(M_{k,l}\overline\otimes E\sim M_{k+l}(E)\) and \(M_{p,q}\overline\otimes M_{r,s}\sim M_{pr+qs,ps+qr}\), as in the ordinary case. (As the authors have added in proof, similar results have been independently obtained by \textit{O. M. Di Vincenzo} and \textit{V. Nardozza} [accepted for publication in Isr. J. Math.].) On the other hand, the paper shows that over an infinite field of characteristic \(p>2\) the conjecture of Regev and Seeman does not hold: the equivalence \(M_{1,1}\overline\otimes E\sim M_2(E)\) in characteristic 0 is replaced by the strict inclusion of T-ideals \(T(M_{1,1}\overline\otimes E)\subset T(M_2(E))\). Further, the authors consider polynomial identities of \(\beta\)-tensor products (also called commutation factors) \(A\otimes_\beta B\) of two PI-algebras \(A\) and \(B\) graded by a finite Abelian group \(G\), where \(\beta\) is a skew-symmetric bicharacter of \(G\). They establish the exponential bound for the codimension sequence \(c_n(A\otimes_\beta B)\leq |G|^{2n}c_n(A)c_n(B)\) which implies a \(\beta\)-analogue of the \(A\otimes B\)-theorem of Regev. Finally, the authors show the PI-equivalence of \(M_n(E)\) and \(M_n(K)\otimes_\beta E\).
scientific article; zbMATH DE number 5545853