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 $(2n+1)$-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 trivial multiplication. We also answered one question on transposed Poisson algebras early posted in a paper by Beites, Ferreira and Kaygorodov. Namely, we found that the semidirect product of ${\mathfrak sl}_2$ and irreducible module gives a finite-dimensional Lie algebra with non-trivial $\frac{1}{2}$-derivations, but without non-trivial transposed Poisson structures.