Polynomial growth of the codimensions sequence of algebras with group graded involution

An algebra graded by a group G and endowed with a graded involution * is called a (G, *)-algebra. Here we consider G a finite abelian group and classify the subvarieties of the varieties of almost polynomial growth generated by finite-dimensional (G, *)-algebras. Also, we present, up to equivalence, the complete list of (G, *)-algebras generating varieties of at most linear growth. Along the way, we give a new characterization of varieties of polynomial growth generated by finite-dimensional (G, *)-algebras by considering the structure of the generating algebra.