Solvability of Poisson algebras

Let P be a Poisson algebra with a Lie bracket {,} over a field F of characteristic p≥0. In this paper, the Lie structure of P is investigated. In particular, if P is solvable with respect to its Lie bracket, then we prove that the Poisson ideal J of P generated by all elements {{{x1,x2},{x3,x4}},x5}...

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Bibliographic Details
Published inJournal of algebra Vol. 568; pp. 349 - 361
Main Authors Siciliano, Salvatore, Usefi, Hamid
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.02.2021
Subjects
Lie identity
Nilpotent ideal
Nilpotent Lie algebra
Poisson algebra
Poisson ideal
Solvable Lie algebra
Symmetric Poisson algebra
16R10
17B01
17B63
17B30
Symmetric Poisson algebra
Solvable Lie algebra
Nilpotent Lie algebra
17B50
Poisson algebra
Nilpotent ideal
Lie identity
Poisson ideal
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Summary:Let P be a Poisson algebra with a Lie bracket {,} over a field F of characteristic p≥0. In this paper, the Lie structure of P is investigated. In particular, if P is solvable with respect to its Lie bracket, then we prove that the Poisson ideal J of P generated by all elements {{{x1,x2},{x3,x4}},x5} with x1,…,x5∈P is associative nilpotent of index bounded by a function of the derived length of P. We use this result to further prove that if P is solvable and p≠2, then the Poisson ideal {P,P}P is nil.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2020.10.012