On HH-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. (co)module algebras over Hopf 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 over a field of characteristic 0 and A is simple with respect to a generalized H-action, then there exists an exponent of the codimension growth of polynomial H-identities of A. In particular, if A is a finite dimensional (not necessarily group graded) graded-simple algebra, then there exists an exponent of the codimension growth of graded polynomial identities of A. In addition, we study the free-forgetful adjunctions corresponding to (not necessarily group) gradings and generalized H-actions.Comment: 16 pages. The classification of Lie algebras simple with respect to a Taft algebra action has been moved to arXiv:1705.05809. The sections related to gradings by sets and free-forgetful adjunctions have been added. The misprints found by the referee have been corrected and several clarifications have been made. (To appear in Journal of Algebra and Its Applications.

Lie algebras simple with respect to a Taft algebra action

We classify finite dimensional $H_{m^2}(\zeta)$-simple $H_{m^2}(\zeta)$-module Lie algebras $L$ over an algebraically closed field of characteristic $0$ where $H_{m^2}(\zeta)$ is the $m$th Taft algebra. As an application, we show that despite the fact that $L$ can be non-semisimple in ordinary sense, $\lim_{n\to\infty}\sqrt[n]{c_n^{H_{m^2}(\zeta)}(L)} = \dim L$ where $c_n^{H_{m^2}(\zeta)}(L)$ is the codimension sequence of polynomial $H_{m^2}(\zeta)$-identities of $L$. In particular, the analog of Amitsur's conjecture holds for $c_n^{H_{m^2}(\zeta)}(L)$.Comment: 20 pages. This text was previously a part of arXiv:1508.03764. A gap in the proof has been filled, many misprints have been corrected. (To appear in J.Algebra.

Semigroup graded algebras and codimension growth of graded polynomial identities

We show that if $T$ is any of four semigroups of two elements that are not groups, there exists a finite dimensional associative $T$-graded algebra over a field of characteristic $0$ such that the codimensions of its graded polynomial identities have a non-integer exponent of growth. In particular, we provide an example of a finite dimensional graded-simple semigroup graded algebra over an algebraically closed field of characteristic $0$ with a non-integer graded PI-exponent, which is strictly less than the dimension of the algebra. However, if $T$ is a left or right zero band and the $T$-graded algebra is unital, or $T$ is a cancellative semigroup, then the $T$-graded algebra satisfies the graded analog of Amitsur's conjecture, i.e. there exists an integer graded PI-exponent. Moreover, in the first case it turns out that the ordinary and the graded PI-exponents coincide. In addition, we consider related problems on the structure of semigroup graded algebras.Comment: 21 pages; Minor misprints are correcte

Amitsur's conjecture for polynomial H-identities of H-module Lie algebras

Consider a finite dimensional H-module Lie algebra L over a field of characteristic 0 where H is a Hopf algebra. We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial H-identities of L under some assumptions on H. In particular, the conjecture holds when H is finite dimensional semisimple. As a consequence, we obtain the analog of Amitsur's conjecture for graded codimensions of any finite dimensional Lie algebra graded by an arbitrary group and for G-codimensions of any finite dimensional Lie algebra with a rational action of a reductive affine algebraic group G by automorphisms and anti-automorphisms.Comment: 38 pages; this article is a generalization of M.V. Zaicev's paper (Izv. Math, 2002) and the author's arXiv:1112.6245; the outline of the proof of the main result is the same as in those articles, however we have to deal with new phenomena that appear in H-module algebras; the introductory part, where we give a survey of what has already been done, overlaps with arXiv:1203.538