Central polynomials for \(\mathbb{Z}_2\)-graded algebras and for algebras with involution.
The authors describe the central polynomials of three important graded algebras (with their natural \(\mathbb{Z}_2\)-grading) over an infinite field \(K\) of characteristic different from 2: the \(2\times 2\) matrix algebra \(M_2(K)\), the algebra \(M_{1,1}(E)\), and the tensor square \(E\otimes E\) of the Grassmann algebra \(E\). If \(x_i,y_i\) are, respectively, even and odd variables from the free algebra \(K\langle X\cup Y\rangle\), and \(z_i\) is any variable from \(X\cup Y\), then the central polynomials of \(M_2(K)\) belong to the \(T_2\)-space generated by the polynomials (i.e. are linear combinations of evaluations of these polynomials) \(y_1^2\), \(z_1[x_1,x_2]z_2\), \(z_1(y_1y_2y_3-y_3y_2y_1)z_2\). The \(T_2\)-space of central polynomials of the algebra \(M_{1,1}(E)\) is generated by \([y_1,y_2]\), \(z_1[x_1,x_2]z_2\), \(z_1(y_1y_2y_3+y_3y_2y_1)z_2\). For \(E\otimes E\), the \(T_2\)-space of central polynomials is generated by the list for \(M_{1,1}(E)\) together with \(x_1^p\), \(z_1[x_1^p,y_1]z_2\) if \(K\) is of odd characteristic \(p\). Further, the authors consider the \(2\times 2\) matrix algebra with involution and solve a similar problem for the central polynomials in both cases of transpose and symplectic involution. Again, explicit lists of generators of the corresponding *-invariant T-spaces are given.
scientific article; zbMATH DE number 5083167