Polynomial growth of the codimensions sequence of algebras with group graded involution (Q6588722)

This paper is a continuation of a research project by the authors' group, which focuses on algebras with additional structures satisfying polynomial identities and on classifying varieties of such algebras based on the growth of their codimension sequences.\N\NIf \( G \) is a group, a \( G \)-graded algebra \( A \), equipped with an involution \( * \), is called a \((G,*)\)-algebra if the involution \( * \) is a graded map, i.e., if for any \( g \in G \), \( * \) maps the \( g \)-component of \( A \) onto itself.\N\NThe authors study the case where \( G \) is a finite abelian group. Based on the results of [\textit{L. M. C. Oliveira}, et al., Algebras and Representation Theory 26, 663--677 (2023; Zbl 1530.16023)], which provide a characterization of varieties of \((G,*)\)-algebras of polynomial growth generated by finite-dimensional algebras, the paper classifies subvarieties of \((G,*)\)-algebras with almost polynomial growth, also generated by finite-dimensional \((G,*)\)-algebras.\N\NFurthermore, the authors present a complete list of \((G,*)\)-algebras whose sequences of \((G,*)\)-codimensions are bounded by a linear function. They also provide a characterization of varieties with polynomial growth generated by finite-dimensional algebras. This classification depends on the structure of such algebras, particularly on their simple components and the Jacobson radical in the Wedderburn-Malcev decomposition.\N\NThe proofs employ standard techniques from the theory of PI-algebras.