Convolution of prime varieties of associative algebras
by
Leonid Mihailovich Samoilov
Ulyanovsk State University, Russia

We study the \gamma-classical varieties of associative algebras with trace which were introduced by A.R. Kemer. In the case of characteristic zero these varieties are generated by the matrix superalgebras M_{n, k}. In the case of characteristic p > 0 there exist the other varieties (\gamma in { 0, 1, 2, ..., p-1}).

We prove that in the case of characteristic p > 0 there exists only a finite number of minimal \gamma-classical varieties. The bases of trace identities of these varieties are described.

Theorem. Let char F=p > 0, [V\tilde] - a minimal \gamma-classical varieties with ideal of trace identities [(\Gamma)\tilde], \gamma in { 0, 1, 2, ..., p-1}.

1) If \gamma = 1 (\gamma = p-1) then [(\Gamma)\tilde] is generated by the polinomials Tr(1)-1 and x-Tr(x) (Tr(1)+1 and x+Tr(x));

2) If char F=2 and \gamma = 0 then [(\Gamma)\tilde] is generated by the polinomials Tr(1), xy+yx+Tr(x)Tr(y), xTr(y)+yTr(x)+Tr(xy);

3) If char F=p > 2 and \gamma = 0 then [(\Gamma)\tilde] is ideal of trace identities of matrix superalgebras M_{1, 1};

4) If char F=p > 5 and \gamma in {2, 3, ..., p-2} then [(\Gamma)\tilde] is generated by the Caley-Hamilton identity of degree \gamma, the symmetric Caley-Hamilton identity of degree p-\gamma and polinomial Tr(1)- \gamma.

We also consruct some new examples of prime varieties (of ordinary algebras) using a new construction of convolution.

Date received: June 28, 2000