$\frac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras

TL;DR

This paper explores the connection between $rac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras, constructing new examples and proving non-existence results for certain algebraic structures.

Contribution

It establishes a relationship between $rac{1}{2}$-derivations and transposed Poisson algebras, constructs specific examples, and proves non-existence for several classes of Lie algebras.

Findings

Constructed non-trivial transposed Poisson algebras with specific Lie structures.

Proved non-existence of such algebras with semisimple or simple Lie parts.

Provided explicit examples with Laurent polynomials and Witt algebra.

Abstract

A relation between $\frac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras was established. Some non-trivial transposed Poisson algebras with a certain Lie algebra (Witt algebra, algebra $\mathcal{W}(a,-1)$, thin Lie algebra and solvable Lie algebra with abelian nilpotent radical) were constructed. In particular, we constructed an example of the transposed Poisson algebra with associative and Lie parts isomorphic to the Laurent polynomials and the Witt algebra. On the other side, it was proven that there are no non-trivial transposed Poisson algebras with Lie algebra part isomorphic to a semisimple finite-dimensional algebra, a simple finite-dimensional superalgebra, the Virasoro algebra, $N=1$ and $N=2$ superconformal algebras, or a semisimple finite-dimensional $n$-Lie algebra.