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- Title
Automorphisms of simple quotients of the Poisson and universal enveloping algebras of sl2.
- Authors
Naurazbekova, Altyngul; Umirbaev, Ualbai
- Abstract
Let P ( sl 2 (K)) be the Poisson enveloping algebra of the Lie algebra sl 2 (K) over an algebraically closed field K of characteristic zero. The quotient algebras P ( sl 2 (K)) / (C P − λ) , where C P is the standard Casimir element of sl 2 (K) in P ( sl 2 (K)) and 0 ≠ λ ∈ K , 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 P ( sl 2 (K)) / (C P − λ) and represent this group as an amalgamated product of its subgroups. Moreover, using similar results by Dixmier [Quotients simples de l'algebre enveloppante de 2 , J. Algebra 24 (1973) 551–564] and O. Fleury [Sur les sous-groupes finis de Aut U ( 2) et Aut U () , J. Algebra 200 (1998) 404–427] for the quotient algebras U ( sl 2 (K)) / (C U − λ) , where C U is the standard Casimir element of sl 2 (K) in the universal enveloping algebra U ( sl 2 (K)) , we prove that the automorphism groups of P ( sl 2 (K)) / (C P − λ) and U ( sl 2 (K)) / (C U − λ) are isomorphic.
- Subjects
UNIVERSAL algebra; POISSON algebras; AUTOMORPHISM groups; GENERATORS of groups; LIE algebras; AUTOMORPHISMS
- Publication
Journal of Algebra & Its Applications, 2023, Vol 22, Issue 7, p1
- ISSN
0219-4988
- Publication type
Article
- DOI
10.1142/S0219498823501517