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

Gordienko, A.

doi:10.1090/S0002-9947-2014-06059-5

Amitsur’s conjecture for polynomial $H$-identities of $H$-module Lie algebras
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Trans. Amer. Math. Soc. 367 (2015), 313-354 Request permission

Abstract:

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 behaviour 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.

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