On the simple transposed Poisson algebras and Jordan superalgebras

Abstract

We prove that a transposed Poisson algebra is simple if and only if its associated Lie bracket is simple. Consequently, any simple finite-dimensional transposed Poisson algebra over an algebraically closed field of characteristic zero is trivial. Similar results are obtained for transposed Poisson superalgebras. An example of a non-trivial simple finite-dimensional transposed Poisson algebra is constructed by studying the transposed Poisson structures on the modular Witt algebra. Furthermore, we show that the Kantor double of a transposed Poisson algebra is a Jordan superalgebra, that is, we prove that transposed Poisson algebras are Jordan brackets. Additionally, a simplicity criterion for the Kantor double of a transposed Poisson algebra is obtained.