Algebras simple with respect to a Sweedler's algebra action

Journal of Pure and Applied Algebra

2015

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.

mentioning

confidence: 86%

“…The case m = 2 is worked out in detail in [14]. Below we list several examples that are consequences of Theorems 1 and 2.…”

Section: −1mentioning

confidence: 99%

See 1 more Smart Citation

Algebras simple with respect to a Taft algebra action

Journal of Pure and Applied Algebra

2015

“…Exact module categories over the category Rep(H m 2 (ζ)) were studied in [6,Theorem 4.10]. Finite dimensional associative H m 2 (ζ)simple H m 2 (ζ)-module algebras were classified in [13,14]. H m 2 (ζ)-actions on path algebras of quivers were studied in [17].…”

Section: Introductionmentioning

confidence: 99%

Lie algebras simple with respect to a Taft algebra action

2019

Journal of Algebra

We classify finite dimensional H m 2 (ζ)-simple H m 2 (ζ)-module Lie algebras L over an algebraically closed field of characteristic 0 where H m 2 (ζ) is the mth Taft algebra. As an application, we show that despite the fact that L can be non-semisimple in ordinary sense, lim n→∞ n cis the codimension sequence of polynomial H m 2 (ζ)-identities of L. In particular, the analog of Amitsur's conjecture holds for cas H m 2 (ζ)-module Lie algebras for α = 0. In Theorem 5.1 we prove that every non-semisimple H m 2 (ζ)-simple Lie algebra is

“…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.…”

Section: Introductionmentioning

confidence: 99%

Actions of Ore extensions and growth of polynomialH-identities

Communications in Algebra

2017

Abstract. We show that if A is a finite dimensional associative H-module algebra for an arbitrary Hopf algebra H, then the proof of the analog of Amitsur's conjecture for Hcodimensions of A can be reduced to the case when A is H-simple. (Here we do not require that the Jacobson radical of A is an H-submodule.) As an application, we prove that if A is a finite dimensional associative H-module algebra where H is a Hopf algebra H over a field of characteristic 0 such that H is constructed by an iterated Ore extension of a finite dimensional semisimple Hopf algebra by skew-primitive elements (e.g. H is a Taft algebra), then there exists integer PIexp H (A). In order to prove this, we study the structure of algebras simple with respect to an action of an Ore extension.