Polynomial codimension growth of algebras with involutions and superinvolutions

Let A be an associative algebra over a field F of characteristic zero endowed with a graded involution or a superinvolution ⁎ and let cn⁎(A) be its sequence of ⁎-codimensions. In [4,12] it was proved that if A is finite dimensional such sequence is polynomially bounded if and only if A generates a variety not containing a finite number of ⁎-algebras: the group algebra of Z2 and a 4-dimensional subalgebra of the 4×4 upper triangular matrices with suitable graded involutions or superinvolutions. In this paper we focus our attention on such algebras since they are the only finite dimensional ⁎-algebras, up to T2⁎-equivalence, generating varieties of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth. We classify the subvarieties of such varieties by giving a complete list of generating finite dimensional ⁎-algebras. Along the way we classify all minimal varieties of polynomial growth and surprisingly we show that their number is finite for any given growth. Finally we describe the ⁎-algebras whose ⁎-codimensions are bounded by a linear function.