Lie algebras simple with respect to a Taft algebra action

Abstract

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 $c_n^{H_{m^2}(\zeta )}(L)$.

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.)