On H-simple not necessarily associative algebras

An algebra A with a generalized H-action is a generalization of an H-module algebra where H is just an associative algebra with 1 and a relaxed compatibility condition between the multiplication in A and the H-action on A holds. At first glance, this notion may appear too general, however, it enables to work with algebras endowed with various kinds of additional structures (e.g. comodule algebras over Hopf algebras, graded algebras, algebras with an action of a semigroup by anti-endomorphisms). This approach proves to be especially fruitful in the theory of polynomial identities. We show that if A is a finite dimensional (not necessarily associative) algebra over a field of characteristic 0 and A is simple with respect to a generalized H-action, then there exists lim(n ->infinity) (n)root c(n)(H)(A) is an element of R+ where (c(n)(H) (A))(n=1)(infinity) is the sequence of codimensions of polynomial H-identities of A. In particular, if A is a finite dimensional (not necessarily group graded) graded-simple algebra, then there exists lim(n ->infinity) (n)root c(n)(gr) (A) is an element of R+ where (c(n)(gr) (A))(n=1)(infinity) is the sequence of codimensions of graded polynomial identities of A. In addition, we study the free-forgetful adjunctions corresponding to (not necessarily group) gradings and generalized H-actions.