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


Author: A. S. Gordienko
Journal: Trans. Amer. Math. Soc. 367 (2015), 313-354
MSC (2010): Primary 17B01; Secondary 17B40, 17B70, 16T05, 20C30, 14L17
DOI: https://doi.org/10.1090/S0002-9947-2014-06059-5
Published electronically: September 16, 2014
MathSciNet review: 3271263
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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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Additional Information

A. S. Gordienko
Affiliation: Department of Mathematics, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada
Email: asgordienko@mun.ca

DOI: https://doi.org/10.1090/S0002-9947-2014-06059-5
Keywords: Lie algebra, polynomial identity, grading, Hopf algebra, Hopf algebra action, $H$-module algebra, codimension, cocharacter, symmetric group, Young diagram, affine algebraic group.
Received by editor(s): July 6, 2012
Received by editor(s) in revised form: December 18, 2012
Published electronically: September 16, 2014
Additional Notes: This work was supported by postdoctoral fellowships from the Atlantic Association for Research in Mathematical Sciences (AARMS), the Atlantic Algebra Centre (AAC), the Memorial University of Newfoundland (MUN), and the Natural Sciences and Engineering Research Council of Canada (NSERC)
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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