Graded identities for T-prime algebras over fields of positive characteristic
The general structure theory of T-ideals, as developed by Kemer, is based on a careful study of T-semiprime and T-prime T-ideals and heavily depends on the investigation of both ordinary and super polynomial identities of finite dimensional \(\mathbb{Z}_2\)-graded algebras and their Grassmann enveloping algebras. In the case of characteristic 0, the only cases when the polynomial identities of T-prime algebras are completely understood are the Grassmann algebra \(G\), the \(2\times 2\) matrix algebra \(M_2(K)\) with entries from the base field \(K\), the tensor square \(G\otimes G\) of the Grassmann algebra and the algebra \(M_{1,1}(G)\) of all \(2\times 2\) matrices with Grassmann entries which are even on the main diagonal and odd on the other diagonal. It is known that in characteristic 0 the algebras \(G\otimes G\) and \(M_{1,1}(G)\) have the same identities. The study of super (or \(\mathbb{Z}_2\)-graded) identities is both a problem of independent interest and an attempt to understand better the ordinary polynomial identities of PI-algebras. The algebras \(G\), \(M_2(K)\), \(G\otimes G\) and \(M_{1,1}(G)\) have natural \(\mathbb{Z}_2\)-gradings and their \(\mathbb{Z}_2\)-graded identities in characteristic 0 are known. The purpose of the paper under review is to solve similar problems over infinite fields of odd characteristic \(p\). The main results are the following. The graded polynomial identities of \(M_2(K)\) (where the matrix units \(e_{11},e_{22}\) are even and \(e_{12},e_{21}\) are odd) follow from the known identities \(y_1y_2-y_2y_1\) and \(z_1z_2z_3-z_3z_2z_1\), where \(y_i\) are even and \(z_j\) are odd variables. The graded identities of \(M_{1,1}(G)\) (the even and the odd components are at the same places as those of \(M_2(K)\)) are \(y_1y_2-y_2y_1\) and \(z_1z_2z_3+z_3z_2z_1\). In order to obtain the graded identities of \(G\otimes G\) (the even component is \(G_0\otimes G_0\oplus G_1\otimes G_1\) and the odd one is \(G_0\otimes G_1\oplus G_1\otimes G_0\)) one has to add to the graded identities of \(M_{1,1}(G)\) the identity \(y_1^pz_1-z_1y_1^p\). The proofs use various combinatorial techniques and a graded version of the Specht reduction of the polynomial identities of unitary algebras to the so called proper identities. As consequences the authors obtain proofs of some known results on graded identities in characteristic 0. The paper concludes with a list of open problems with interesting comments on relations with other topics in PI-theory.
graded polynomial identities of matrices
scientific article; zbMATH DE number 1819984