Automorphisms of simple quotients of the Poisson and universal enveloping algebras of sl2

Altyngul Naurazbekova, Ualbai Umirbaev

2022

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DOI 10.1142/s0219498823501517
Data confidence 70 medium
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Abstract

Let [Formula: see text] be the Poisson enveloping algebra of the Lie algebra [Formula: see text] over an algebraically closed field [Formula: see text] of characteristic zero. The quotient algebras [Formula: see text] [Formula: see text], where [Formula: see text] is the standard Casimir element of [Formula: see text] in [Formula: see text] and [Formula: see text], are proven to be simple in [U. Umirbaev and V. Zhelyabin, A Dixmier theorem for Poisson enveloping algebras, J. Algebra 568 (2021) 576–600]. Using a result by Makar–Limanov [22], we describe generators of the automorphism group of [Formula: see text] and represent this group as an amalgamated product of its subgroups. Moreover, using similar results by Dixmier [Quotients simples de l’algebre enveloppante de [Formula: see text], J. Algebra 24 (1973) 551–564] and O. Fleury [Sur les sous-groupes finis de [Formula: see text] et [Formula: see text], J. Algebra 200 (1998) 404–427] for the quotient algebras [Formula: see text], where [Formula: see text] is the standard Casimir element of [Formula: see text] in the universal enveloping algebra [Formula: see text], we prove that the automorphism groups of [Formula: see text] and [Formula: see text] are isomorphic.