On the *-polynomial identities of \(M_{1,1}(E)\).
Let \(F\) be a field of characteristic 0 and let \(E\) be the Grassmann algebra over an infinite dimensional vector space \(L\). If \(\alpha\) and \(\beta\) are, respectively, involutions of the matrix algebra \(M_n(F)\) and on \(E\), then they induce an involution \(\gamma=\gamma_{\alpha,\beta}\) on the algebra \(M_n(E)\). Let \(F\langle Y,Z\rangle\) be the free associative algebra with involution, where \(Y\) and \(Z\) are the free generators which are, respectively, symmetric and skew-symmetric with respect to the involution. The first result of the paper under review is that if \(\beta(L)=L\), then every *-polynomial identity of \(M_n(E)\) is obtained from an ordinary polynomial identity: If \(f(y_1,\dots,y_l,z_1,\dots,z_m)=0\) is a *-polynomial identity for \(M_n(E)\), then \(f(x_1,\dots,x_l,x_{l+1},\dots,x_{l+m})=0\) is an ordinary polynomial identity for \(M_n(E)\). A similar result holds for the algebra \(M_{k,l}(E)\) assuming that \(\alpha\) keeps invariant the \(\mathbb Z_2\)-graded components of \(M_{k+l}(F)\) in the definition of \(M_{k,l}(E)\) and \(\beta(L)=L\). Further, the authors study the *-polynomial identities of the algebra \(M_{1,1}(E)\) equipped with the involution \((ae_{11}+be_{12}+ce_{21}+de_{22})^*=(de_{11}+be_{12}-ce_{21}+ae_{22})\), \(a,d\in E_0\), \(b,c\in E_1\). This involution is induced by the transposition superinvolution of the superalgebra \(M_{1,1}(F)\). In the main result of the paper the authors present an explicit set of 10 polynomials in \(F\langle Y,Z\rangle\) which forms a basis of the *-polynomial identities of \(M_{1,1}(E)\). The basis consists of the polynomials \([y_1,y_2]\), \(z_1z_2z_3-z_3z_2z_1\), \([z_1,z_2][z_3,z_4]\) and 7 more polynomials of degree 4, 5 and 6.
scientific article; zbMATH DE number 5833471
scientific article