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 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 and irreducible module gives a finite-dimensional Lie algebra with non-trivial -derivations, but without non-trivial transposed Poisson structures.