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Amitsur’s conjecture for polynomial \(H\)-identities of \(H\)-module Lie algebras. (English) Zbl 1354.17002

Summary: 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.

MSC:

17B01 Identities, free Lie (super)algebras
17B70 Graded Lie (super)algebras
16T05 Hopf algebras and their applications
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