Automorphisms of simple quotients of the Poisson and universal enveloping algebras of sl2
Naurazbekova A. Umirbaev U.
1 July 2023World Scientific
Journal of Algebra and its Applications
2023#22Issue 7
Let P(sl2(K)) be the Poisson enveloping algebra of the Lie algebra sl2(K) over an algebraically closed field K of characteristic zero. The quotient algebras P(sl2(K))/(CP −λ), where CP is the standard Casimir element of sl2(K) in P(sl2(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(sl2(K))/(CP − 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 sl2, J. Algebra 24 (1973) 551–564] and O. Fleury [Sur les sous-groupes finis de AutU(sl2) et AutU(h), J. Algebra 200 (1998) 404–427] for the quotient algebras U(sl2(K))/(CU −λ), where CU is the standard Casimir element of sl2(K) in the universal enveloping algebra U(sl2(K)), we prove that the automorphism groups of P(sl2(K))/(CP −λ) and U(sl2(K))/(CU −λ) are isomorphic.
automorphism , Casimir element , free product , Poisson enveloping algebra , Universal enveloping algebra
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Department of Mechanics and Mathematics L. N. Gumilyov Eurasian National University, Nur-Sultan, Kazakhstan
Department of Mathematics, Wayne State University, Detroit, 48202, MI, United States
Department of Mathematics, Al-Farabi Kazakh National University, Almaty, 050040, Kazakhstan
Institute of Mathematics and Mathematical Modeling, Almaty, 050010, Kazakhstan
Department of Mechanics and Mathematics L. N. Gumilyov Eurasian National University
Department of Mathematics
Department of Mathematics
Institute of Mathematics and Mathematical Modeling
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