Identities of *-superalgebras and almost polynomial growth

Abstract

We study the growth of the codimensions of a *-superalgebra over a field of characteristic zero. We classify the ideals of identities of finite dimensional algebras whose corresponding codimensions are of almost polynomial growth. It turns out that these are the ideals of identities of two algebras with distinct involutions and gradings. Along the way, we also classify the finite dimensional simple *-superalgebras over an algebraically closed field of characteristic zero

Classifying Algebras with Graded Involutions or Superinvolutions with Multiplicities of their Cocharacter Bounded by One

“…In fact, a celebrated theorem of Kemer establishes that in case of ordinary polynomial identities, c n (A) is exponentially bounded of grows polynomially (see [11]). Similar results hold also for algebras with graded involution ( [10]) and superinvolution ( [6]).…”

“…So we have to consider three more possibilities. Recall that in addition to (9), we are assuming (10) y…”

On multiplicities of cocharacters for algebras with superinvolution

“…In order to reach a contradiction we need only to show that f is actually the zero polynomial. Since by [10,Theorem 6.3…”

“…Let us now assume by contradiction that m = m λ > N = d(q d ) d1d2d3d4 . If we prove that A satisfies a * -identity of the type (10)…”

Differential identities and varieties of almost polynomial growth

“…Varieties of poylnomial growth were extensively studied in the past years in various settings. We refer the interested reader to [5], [6], [22] for some results about ordinary algebras; to [8], [23], [24], [32] for superalgebras and more generally group graded algebras; to [3], [7], [10], [20], [21], [25] for algebras with involution, graded involution, superinvolution and pseudoinvolution; to [27] for special Jordan algebras.…”