Algebras simple with respect to a Taft algebra action

Abstract

Abstract. Algebras simple with respect to an action of a Taft algebra H m 2 (ζ) deliver an interesting example of H-module algebras that are H-simple but not necessarily semisimple. We describe finite dimensional H m 2 (ζ)-simple algebras and prove the analog of Amitsur's conjecture for codimensions of their polynomial H m 2 (ζ)-identities. In particular, we show that the Hopf PI-exponent of an H m 2 (ζ)-simple algebra A over an algebraically closed field of characteristic 0 equals dim A. The groups of automorphisms preserving the structure of an H m 2 (ζ)-module algebra are studied as well.The notion of an H-(co)module algebra is a natural generalization of the notion of a graded algebra, an algebra with an action of a group by automorphisms, and an algebra with an action of a Lie algebra by derivations. In particular, if H m 2 (ζ) is the m 2 -dimensional Taft algebra, an H m 2 (ζ)-module algebra is an algebra endowed both with an action of the cyclic group of order m and with a skew-derivation satisfying certain conditions. The Taft algebraThe theory of gradings on matrix algebras and simple Lie algebras is a well developed area [3,6]. Quaternion H 4 (−1)-extensions and related crossed products were considered in [9]. In [14], the author classified all finite dimensional H 4 (−1)-simple algebras. Here we classify finite dimensional H m 2 (ζ)-simple algebras over an algebraically closed field (Sections 2-3).Amitsur's conjecture on asymptotic behaviour of codimensions of ordinary polynomial identities was proved by A. Giambruno and M. V. Zaicev [10, Theorem 6.5.2] in 1999.Suppose an algebra is endowed with a grading, an action of a group G by automorphisms and anti-automorphisms, an action of a Lie algebra by derivations or a structure of an Hmodule algebra for some Hopf algebra H. Then it is natural to consider, respectively, graded, G-, differential or H-identities [1,2,4,7,15].The analog of Amitsur's conjecture for polynomial H-identities was proved under wide conditions by the author in [12,13]. However, in those results the H-invariance of the Jacobson radical was required. Until now the algebras simple with respect to an action of H 4 (−1) were the only example where the analog of Amitsur's conjecture was proved for an H-simple non-semisimple algebra [14]. In this article we prove the analog of Amitsur's conjecture for all finite dimensional H m 2 (ζ)-simple algebras not necessarily semisimple (Section 4) assuming that the base field is algebraically closed and of characteristic 0.

Equivalences of (co)module algebra structures over Hopf algebras

“…d = 1 and (5.8) follows from [16,Theorem 1]. In the last case F [x]/(x 2 ) is an H 4 -simple algebra (see [18,19]), i.e. d = 0 and (5.…”

Lie algebras simple with respect to a Taft algebra action

“…Furthermore, anti-commutativity of the commutator in a Lie algebra is a strong restriction on the H m 2 (ζ)-action and we get much less possible parameters to describe H m 2 (ζ)-simple Lie algebras. In addition, every finite dimensional H m 2 (ζ)-module Lie algebra simple in the ordinary sense, is just a Z m -graded Lie algebra with the zero skew-derivation (Theorem 4.11), which is in contrast to the associative case [14,Theorem 1].…”

Actions of Ore extensions and growth of polynomialH-identities

“…As we have mentioned above, the proofs in all previous papers [1,2,17,21,23,24,38] worked only in the case when the Jacobson radical J(A) was an H-submodule or A was H-simple itself. In the current article we do not assume that the Jacobson radical of A is H-invariant, replacing the Wedderburn -Mal'cev theorem by its weak analog (Lemma 2.6) which still makes it possible to transfer the computations to H-simple algebras.…”

“…Property (*) implies that the exponents of growth of H-identities of the corresponding H-simple algebras are integer and equal their dimensions. Property (*) holds for all finite dimensional semisimple (in ordinary sense) H-simple algebras [20,Theorem 7] (see also the remark in Example 6.1 below) and for all finite dimensional algebras simple with respect to an action of a Taft algebra H m 2 (ζ) [24,Lemma 7]. However, there exist finite dimensional algebras with a generalized H-action, which again are semigroup graded algebras, where the H-PI-exponent is non-integer and Property (*) does not hold [25].…”

“…Exact module categories over the category Rep(H m 2 (ζ)) were studied in [15,Theorem 4.10]. Finite dimensional H m 2 (ζ)-simple H m 2 (ζ)-module algebras were classified in [23,24]. H m 2 (ζ)-actions on path algebras of quivers were studied in [28].…”