Transposed Poisson structures on Witt type algebras

Abstract

We describe $\frac{1}{2}$-derivations, and hence transposed Poisson algebra structures, on Witt type Lie algebras $V(f)$, where $f:Γ\to\mathbb C$ is non-trivial and $f(0)=0$. More precisely, if $|f(Γ)|\ge 4$, then all the transposed Poisson algebra structures on $V(f)$ are mutations of the group algebra structure $(V(f),\cdot)$ on $V(f)$. If $|f(Γ)|=3$, then we obtain the direct sum of $3$ subspaces of $V(f)$, corresponding to cosets of $Γ_0$ in $Γ$, with multiplications being different mutations of $\cdot$. The case $|f(Γ)|=2$ is more complicated, but also deals with certain mutations of $\cdot$. As a consequence, new Lie algebras that admit non-trivial ${\rm Hom}$-Lie algebra structures are found.