Identities with involution for the matrix algebra of order two in characteristic \(p\). (Q1781945)

The authors study the polynomial identities with involution (or *-polynomial identities) of the \(2\times 2\) matrix algebra \(M_2(K)\) over an infinite field \(K\) of odd characteristic. It is well known that the *-polynomial identities of \(M_2(K)\) are equivalent to those of \(M_2(K,t)\) and \(M_2(K,s)\), where \(t\) and \(s\) are, respectively, the transpose and the symplectic involutions. The authors give bases of the *-polynomial identities in both cases. They prove that all *-identities of \(M_2(K,t)\) follow from the identities \[ [y_1y_2,x]=[y_1,y_2]=[y_1x_1y_2,x_2]-y_1y_2[x_1,x_2]=0, \] \[ [x_1,x_2][x_3,x_4]+[x_1,x_3][x_4,x_2]+[x_1,x_4][x_2,x_3]=0, \] where the \(x\)'s are symmetric and the \(y\)'s are skew-symmetric variables. The basis of *-identities of \(M_2(K,s)\) consists of \([x_1,x_2]=[x,y]=0\). The proofs involve several techniques and use *-proper identities, ordinary polynomial identities of \(M_2(K)\), weak polynomial identities, and the characteristic free approach to invariant theory of classical groups. In characteristic 0, the *-polynomial identities of \(M_2(K)\) were described by \textit{D. V. Levchenko} [Serdica 8, 42-56 (1982; Zbl 0518.16005)].