Varieties of algebras with pseudoinvolution: Codimensions, cocharacters and colengths

Let A be a finitely generated superalgebra with pseudoinvolution ⁎ over an algebraically closed field F of characteristic zero. In this paper we develop a theory of polynomial identities for this kind of algebras. In particular, we shall consider three sequences that can be attached to Id⁎(A), the T2⁎-ideal of identities of A: the sequence of ⁎-codimensions cn⁎(A), the sequence of ⁎-cocharacter χ〈n〉⁎(A) and the ⁎-colength sequence ln⁎(A). Our purpose is threefold. First we shall prove that the ⁎-codimension sequence is eventually non-decreasing, i.e., cn⁎(A)≤cn+1⁎(A), for n large enough. Secondly, we study superalgebras with pseudoinvolution having the multiplicities of their ⁎-cocharacter bounded by a constant. Among them, we characterize the ones with multiplicities bounded by 1. Finally, we classify superalgebras with pseudoinvolution A such that ln⁎(A) is bounded by 3. In the last section we relate the ⁎-colengths with the polynomial growth of the ⁎-codimensions: we show that ln⁎(A) is bounded by a constant if and only if cn⁎(A) grows at most polynomially.