A basis for the graded identities of the matrix algebra of order two over a finite field of characteristic \(p\neq 2\) (Q1867474)

Studying the polynomial identities of the \(2\times 2\) matrix algebra \(M_2(K)\) over a field \(K\) one has to consider three cases and apply three completely different methods. When the base field is of characteristic 0 it is sufficient to consider multilinear identities only. It is possible to apply methods of the representation theory of symmetric and general linear groups. When the base field is infinite but of positive characteristic, one tries to mimic the approach in characteristic 0 but has to consider all multihomogeneous identities, replacing group representation theory with characteristic free invariant theory. Finally, for algebras over finite fields, the combinatorial methods in the above two cases of infinite fields are replaced by methods of ring structure theory. A basis of the polynomial identities of \(M_2(K)\) when \(K\) is the field with \(q\) elements was obtained by \textit{Yu. N. Mal'tsev} and \textit{E. N. Kuz'min} [Algebra Logika 17, 28--32 (1978); translation in Algebra Logic 17, 18--21 (1978; Zbl 0395.16014)]. The main results of the paper under review are the following. First the authors determine up to isomorphism all \(\mathbb Z_2\)-gradings of \(M_n(K)\) when \(K\) is a finite field with \(q\) elements, \(q\) odd. They show that there are two different gradings: the standard one \(\Omega=\Omega_0\oplus\Omega_1\), where \[ \Omega_0=\left\{\begin{pmatrix} a&0\\ 0&d\end{pmatrix}\Bigg|\;a,d\in K\right\},\quad\Omega_1=\left\{\begin{pmatrix} 0&b\\ c&0\end{pmatrix}\Bigg|\;b,c\in K\right\}; \] and one more, \(\Omega^\alpha\), where \(\alpha\in K\) is not a square in \(K\), and \[ \Omega^\alpha_0=\left\{\begin{pmatrix} a&d\\ \alpha d&a\end{pmatrix}\Bigg|\;a,d\in K\right\},\quad\Omega^\alpha_1=\left\{\begin{pmatrix} b&c\\ -\alpha c&-b\end{pmatrix}\Bigg|\;b,c\in K\right\}. \] The second result gives the bases of the graded identities of \(M_2(K)\) with respect to these gradings. The basis of the identities for the grading \(\Omega\) consists of the identity \(y_1^q=y_1\) and \[ ((y_1+z_1)-(y_1+z_1)^q)((y_2+z_2)-(y_2+z_2)^{q^2})(1-[y_1+z_1,y_2+z_2]^{q-1})=0, \] which is a graded analogue of one of the identities of Mal'tsev and Kuz'min, where \(y_i\) and \(z_i\) are even and odd variables, respectively, and \([x_1,x_2]=x_1x_2-x_2x_1\) is the usual commutator. In the case of \(\Omega^\alpha\) the basis consists of the identities \(y_1^{q^2}=y_1\) and \(z_1^{2q-1}=z_1\) and the graded identity of Mal'tsev and Kuz'min. In particular, the gradings of \(M_2(K)\) can be distinguished in terms of graded polynomial identities.