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 $f:\Gamma \to \mathbb{C}$ is non-trivial and $f(0)=0$. More precisely, if $|f(\Gamma )|\ge 4$, then all the transposed Poisson algebra structures on $V(f)$ are mutations of the group algebra structure $(V(f),\cdot )$ on $V(f)$. If $|f(\Gamma )|=3$, then we obtain the direct sum of $3$ subspaces of $V(f)$, corresponding to cosets of ${\Gamma }_0$ in $\Gamma$, with multiplications being different mutations of $\cdot$. The case $|f(\Gamma )|=2$ is more complicated, but also deals with certain mutations of $\cdot$. As a consequence, new Lie algebras that admit non-trivial $\mathrm{Hom}$-Lie algebra structures are found.

References in corpus (3)

• Poisson algebras and symmetric Leibniz bialgebra structures on oscillator Lie algebras
• Transposed Poisson structures on Block Lie algebras and superalgebras
• Poisson Structures on Finitary Incidence Algebras

Cited by in corpus (8)

• A bialgebra theory for transposed Poisson algebras via anti-pre-Lie bialgebras and anti-pre-Lie-Poisson bialgebras
• Non-associative algebraic structures: classification and structure
• Transposed Poisson structures on generalized Witt algebras and Block Lie algebras
• Transposed Poisson structures
• Some generalizations of the variety of transposed Poisson algebras
• Transposed Poisson structures on the Lie algebra of upper triangular matrices
• Transposed Poisson structures on solvable and perfect Lie algebras
• Transposed Poisson structures on Lie incidence algebras