On a Regev-Seeman conjecture about \(\mathbb{Z}_2\)-graded tensor products.
If \(A,B\) are superalgebras then, besides the ordinary tensor product \(A\otimes B\), a \(\mathbb{Z}_2\)-graded tensor product \(A\overline\otimes B\) arises. The multiplication in \(A\overline\otimes B\) 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 \(A\overline\otimes B\), where \(A\) and \(B\) are PI-superalgebras. Kemer proved that if \(A,B\) are T-prime algebras then \(A\otimes B\) has the same polynomial identities as a suitable explicitly given T-prime algebra \(C\). Regev and Seeman conjectured that, as in the ordinary case, the graded tensor product of two T-prime PI-algebras in characteristic 0 is PI-equivalent to a T-prime algebra and confirmed their conjecture in several cases. In the paper under review the authors establish results which imply the conjecture of Regev and Seeman. Using techniques for group graded polynomial identities developed in their previous papers, the authors establish that over any infinite field of characteristic \(\neq 2\) the \(\mathbb{Z}_2\)-graded tensor product of two T-prime algebras has the same multilinear polynomial identities as a T-prime algebra. They also give rules for the \(\mathbb{Z}_2\)-graded tensor multiplication of T-prime algebras. It is interesting to mention that in the supercase these rules are different from the rules in the ordinary case. For example, for the Grassmann algebra \(E\) the rules are \(E\overline\otimes E\sim E\) (as shown already by Regev and Seeman) and \(E\otimes E\sim M_{1,1}\), respectively. Clearly, this difference implies other differences in the tensor multiplication, e.g. \(M_m(E)\overline\otimes M_n(E)\sim M_{mn}(E)\) and \(M_m(E)\otimes M_n(E)\sim M_{mn,mn}(E)\), respectively. As the authors have added in the proofs, similar results have been independently obtained by \textit{J. A. Freitas} and \textit{P. Koshlukov} [J. Algebra 321, No. 2, 667-681 (2009; Zbl 1172.16013)].
scientific article; zbMATH DE number 5728341