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Transposed Poisson structures on solvable and perfect Lie algebras

Ivan Kaygorodov, Abror Khudoyberdiyev

Abstract

We described all transposed Poisson algebra structures on oscillator Lie algebras, i.e., on one-dimensional solvable extensions of the $(2n+1)$-dimensional Heisenberg algebra; on solvable Lie algebras with naturally graded filiform nilpotent radical; on $(n+1)$-dimensional solvable extensions of the $(2n+1)$-dimensional Heisenberg algebra; and on $n$-dimensional solvable extensions of the $n$-dimensional algebra with the trivial multiplication. We also gave an answer to one question on transposed Poisson algebras early posted in a paper by Beites, Ferreira, and Kaygorodov. Namely, we found a finite-dimensional Lie algebra with non-trivial $\frac{1}{2}$-derivations, but without non-trivial transposed Poisson algebra structures.

Transposed Poisson structures on solvable and perfect Lie algebras

Abstract

We described all transposed Poisson algebra structures on oscillator Lie algebras, i.e., on one-dimensional solvable extensions of the -dimensional Heisenberg algebra; on solvable Lie algebras with naturally graded filiform nilpotent radical; on -dimensional solvable extensions of the -dimensional Heisenberg algebra; and on -dimensional solvable extensions of the -dimensional algebra with the trivial multiplication. We also gave an answer to one question on transposed Poisson algebras early posted in a paper by Beites, Ferreira, and Kaygorodov. Namely, we found a finite-dimensional Lie algebra with non-trivial -derivations, but without non-trivial transposed Poisson algebra structures.
Paper Structure (7 sections, 16 theorems, 29 equations)

This paper contains 7 sections, 16 theorems, 29 equations.

Table of Contents

  1. Preliminaries
  2. Transposed Poisson algebra structures on oscillator Lie algebras
  3. Transposed Poisson structures on finite-dimensional solvable Lie algebras
  4. Transposed Poisson structures on solvable Lie algebras with naturally graded filiform nilpotent radical
  5. Transposed Poisson structures on solvable Lie algebras with Heisenberg nilpotent radical
  6. Transposed Poisson structures on solvable Lie algebras with abelian nilpotent radical
  7. Transposed Poisson structure on perfect Lie algebras

Key Result

Lemma 4

Let $({\mathfrak L},[-,-])$ be a Lie algebra and $\cdot$ a new binary (bilinear) operation on ${\mathfrak L}$. Then $({\mathfrak L},\cdot,[-,-])$ is a transposed Poisson algebra if and only if $\cdot$ is commutative and associative and for every $z\in{\mathfrak L}$ the multiplication by $z$ in $({\m

Theorems & Definitions (31)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 4
  • Theorem 5
  • Definition 6
  • Proposition 7
  • proof
  • Corollary 8
  • Theorem 9
  • ...and 21 more