Weak polynomial identities for \(M_{1,1}(E)\) (Q2761694)

The superalgebras \(M_{k,l}\) play a very important rôle in the structure theory of PI algebras developed by A. Kemer. Their T-ideals are among the ``building blocks'' for the T-ideals of associative algebras over fields of characteristic 0. That is why collecting information about the polynomial identities in \(M_{k,l}\) is an important task in PI theory. The only nontrivial case where the identities of \(M_{k,l}\) are known is \(k=l=1\). Since \(E\otimes E\) and \(M_{1,1}\) satisfy the same polynomial identities, the description of the identities in \(M_{1,1}\) is given by \textit{A. P. Popov} [Algebra Logic 21, 296-316 (1983); translation from Algebra Logika 21, 442-471 (1982; Zbl 0521.16014)]. Here \(E\) stands for the infinite dimensional Grassmann algebra over a field of characteristic 0. Note that the identities of \(E\otimes E\) are unknown in positive characteristic.NEWLINENEWLINENEWLINEThe paper under review deals with the weak identities satisfied by \(M_{1,1}\). The authors prove that they admit a basis consisting of two identities, namely \([[x_1,x_2],x_3]\) and \([x_1,x_2][x_1,x_3][x_1,x_4]\). This is achieved by means of a careful study of the multilinear weak identities and applying the theory of representations of the symmetric group. The cocharacter sequence and the generating function of the codimension sequence for the weak identities of \(M_{1,1}\) are computed as well.NEWLINENEWLINENEWLINEIf one considers the algebra of \(2\times 2\) matrices over an infinite field \(K\) of characteristic \(\neq 2\), the problem of finding a basis of their identities was resolved by using the properties of the weak identities. That is why the authors believe their results may be helpful in obtaining alternative descriptions of the polynomial identities in \(M_{1,1}\) (and my feeling is that they may be helpful in the case of positive characteristic of the base field as well).