Transposed Poisson structures on solvable and perfect Lie algebras

TL;DR

This paper classifies all transposed Poisson algebra structures on various classes of solvable and perfect Lie algebras, providing new insights and answering an open question about their existence.

Contribution

It offers a complete description of transposed Poisson structures on specific solvable Lie algebras and resolves an open problem regarding their non-triviality.

Findings

Classified transposed Poisson structures on oscillator Lie algebras

Described structures on solvable Lie algebras with graded filiform nilpotent radical

Provided an example of a Lie algebra with non-trivial 1/2-derivations but no non-trivial transposed Poisson structures

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.