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

Transposed Poisson structures on Block Lie algebras and superalgebras

“…Summarizing Definitions 1 and 3 we have the following key lemma. Thanks to [13], we have the following useful results.…”

“…More significantly, a transposed Poisson algebra naturally arises from a Novikov-Poisson algebra by taking the commutator Lie algebra of the Novikov algebra. Later, in a recent paper by Ferreira, Kaygorodov, and Lopatkin a relation between 1 2 -derivations of Lie algebras and transposed Poisson algebras have been established [13]. These ideas were used for describing all transposed Poisson structures on the Witt algebra [13], the Virasoro algebra [13], the algebra W(a, b) [13], twisted Heisenberg-Virasoro [29], Schrodinger-Virasoro algebras [29], extended Schrodinger-Virasoro [29] and Block Lie algebras and superalgebras [19].…”

The algebraic and geometric classification of transposed Poisson algebras

“…Some new examples of transposed Poisson algebras are constructed by applying the Kantor product of multiplications on the same vector space [14]. More recently, in a paper by Ferreira, Kaygorodov and Lopatkin, a relation between 1 2 -derivations of Lie algebras and transposed Poisson algebras has been established [17].…”

“…Section 1 is devoted to the complete algebraic classification of non-isomorphic complex 3-dimensional transposed Poisson algebras. To obtain these classification, we will use the algebraic classification of suitable Lie algebras and associative commutative algebras; and the method of describing all transposed Poisson algebra structures on a given Lie algebras (the present method has been developed in [17]).…”

Transposed BiHom-Poisson algebras

“…Transposed Poisson algebras were presented in [1] by exchanging the operations • and [, ] in the compatible condition of Poisson algebras, and at the same time a factor 2 appears on the left-hand side. It was also studied in [8,16,32] recently. A 3-Lie algebra is a vector space A endowed with a ternary skew-symmetric operation satisfying the ternary Jacobi identity [2,4,9].…”