At first glance the notion of an algebra with a generalized $H$-action may appear too general, however it enables to work with algebras endowed with various kinds of additional structures (e.g. Hopf (co)module algebras, graded algebras, algebras with an action of a (semi)group 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 simple with respect to a generalized $H$-action over a field of characteristic $0$, then there exists $\lim_{n\to\infty}\sqrt[n]{c_n^H(A)} \in \mathbb R_+$ where $\left(c_n^H(A)\right)_{n=1}^\infty$ 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 $\lim_{n\to\infty}\sqrt[n]{c_n^{\mathrm{gr}}(A)} \in \mathbb R_+$ where $\left(c_n^{\mathrm{gr}}(A)\right)_{n=1}^\infty$ is the sequence of co 查看全文>>