Transposed Poisson structures on Witt type algebras

Abstract

We describe 12-derivations, and hence transposed Poisson algebra structures, on Witt type Lie algebras V(f), where f:Γ→C is non-trivial and f(0)=0. More precisely, if |f(Γ)|≥4, then all the transposed Poisson algebra structures on V(f) are mutations of the group algebra structure (V(f),⋅) on V(f). If |f(Γ)|=3, then we obtain the direct sum of 3 subspaces of V(f), corresponding to cosets of Γ0 in Γ, with multiplications being different mutations of ⋅. The case |f(Γ)|=2 is more complicated, but also deals with certain mutations of ⋅. As a consequence, new Lie algebras that admit non-trivial Hom-Lie algebra structures are found.