Graded algebras with homogeneous involution and varieties of almost polynomial growth

An important aspect in the theory of algebras with polynomial identities is the study of the asymptotic behavior of the codimension sequence $c_n(A),\, n\geq 1,$ which measures the growth of polynomial identities of a given algebra $A$. In this context, graded identities naturally arise as prominent tools, since ordinary polynomial identities can be viewed as a particular case of graded identities. Moreover, as an involution does not necessarily preserve the homogeneous components of a grading, it is natural to consider the notion of a homogeneous involution. In this work, we investigate the behavior of the codimension sequence in the setting of $G$-graded algebras endowed with a homogeneous involution. More specifically, we characterize the varieties of polynomial growth in terms of the exclusion of a list of algebras from the variety. As a consequence, we provide the classification of the varieties with almost polynomial growth in this setting.