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

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