The varieties generated by \(M_{1,1}(E)\) or \(M_2(K)\) (Q2762971)

An associative algebra \(A\) over a field \(K\) of characteristic zero is said to be a superalgebra, if it is equal to the sum (not necessarily direct) of two vector subspaces \(A_0\) and \(A_1\), such that \(A_i\cdot A_j\subseteq A_{i+j}\bmod 2\). The paper under review announces without detailed proofs (they will appear elsewhere) a description of the \(T_2\)-ideals of the subvarieties of the variety of unitary superalgebras generated by the matrix superalgebra \(M_2(K)\) (with respect to the \(\mathbb{Z}_2\)-grading \(M_2(K)=(K\cdot e_{11}\oplus K\cdot e_{22})\oplus(K\cdot e_{12}\oplus K\cdot e_{21})\)). These ideals play the same role in the study of superalgebras with \(\mathbb{Z}_2\)-graded polynomial identities as T-ideals do in the theory of associative PI-algebras. The problem considered by the authors is motivated by \textit{A. R. Kemer}'s PI-theory and, especially, by his results on the generating identities of the so-called prime varieties [see Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 5, 1042-1059 (1984; Zbl 0586.16010)]. Their starting point is the fact that every variety of associative algebras is generated by the Grassmann envelope \(G(A)=(A_0\otimes E_0)\oplus(A_1\otimes E_1)\) of a suitably chosen finite dimensional superalgebra \(A=A_0+A_1\), where \(E=E_0\oplus E_1\) is the Grassmann algebra of an infinite dimensional \(K\)-vector space (with respect to the natural \(\mathbb{Z}_2\)-grading). They also rely on the observation that \(\mathbb{Z}_2\)-graded identities of any superalgebra \(A\) are consequences of the so-called even-proper identities (defined in the paper). The authors call two \(T_2\)-ideals asymptotically equal, if they have the same even-proper identities of any sufficiently large degree \(n\). In addition, they associate with each pair \((p,q)\) of positive integers \(K\)-superalgebras \(R_p\) and \(S_q\). With these notations, the main result of the paper states that a \(T_2\)-ideal of a proper subvariety of the variety generated by the superalgebra \(M_2(K)\) is asymptotically equal to the intersection \(T_2(R_p)\cap T_2(S_q)\), for suitable \(p\) and \(q\).