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.