A relation between \(\frac{1}{2}\)-derivations of Lie algebras and transposed Poisson algebras has been established. Some non-trivial transposed Poisson algebras with a certain Lie algebra (Witt algebra, the algebra \({\mathcal {W}}(a,-1)\), the thin Lie algebra and a solvable Lie algebra with abelian nilpotent radical) have been done. In particular, we have developed 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 has been proved that there are no non-trivial transposed Poisson algebras with a 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.

已经建立了李代数和转置泊松代数的\(\frac{1}{2}\) - 导数之间的关系。一些具有特定李代数的非平凡转置泊松代数（维特代数，代数\({\mathcal {W}}(a,-1)\)，薄李代数和可解李代数与阿贝尔幂零根）已经完成。特别是，我们开发了一个转置泊松代数的例子，其中结合部分和李部分同构于 Laurent 多项式和 Witt 代数。另一方面，已经证明不存在非平凡转置泊松代数，其李代数部分同构为半简单有限维代数，简单的有限维超代数，Virasoro 代数，\(N=1\ )和\(N=2\)超共形代数，或一个半简单的有限维n -李代数。