Poisson algebras and symmetric Leibniz bialgebra structures on oscillator Lie algebras

Transposed Poisson structures on Block Lie algebras and superalgebras

“…Observe that 1 2 -derivations are a particular case of δ-derivations introduced by Filippov in [8] (see also [11,20] and references therein). The space of all 1 2 -derivations of an algebra L will be denoted by ∆(L).…”

“…Poisson algebras arose from the study of Poisson geometry in the 1970s and have appeared in an extremely wide range of areas in mathematics and physics, such as Poisson manifolds, algebraic geometry, operads, quantization theory, quantum groups, and classical and quantum mechanics. The study of all possible Poisson algebra structures with a certain Lie or associative part is an important problem in the theory of Poisson algebras [4,15,18,28]. Recently, a dual notion of the Poisson algebra (transposed Poisson algebra) by exchanging the roles of the two binary operations in the Leibniz rule defining the Poisson algebra has been introduced in the paper of Bai, Bai, Guo, and Wu [6].…”

“…Poisson algebras originated from the Poisson geometry in the 1970s and have shown their importance in several areas of mathematics and physics, such as Poisson manifolds, algebraic geometry, operads, quantization theory, quantum groups, and classical and quantum mechanics. One of the popular topics in the theory of Poisson algebras is the study of all possible Poisson algebra structures with fixed Lie or associative part [1,9,12,17]. Recently, Bai, Bai, Guo, and Wu [2] have introduced a dual notion of the Poisson algebra, called transposed Poisson algebra, by exchanging the roles of the two binary operations in the Leibniz rule defining the Poisson algebra.…”

“…Thanks to [4], any unital transposed Poisson algebra is a particular case of a "contact bracket" algebra and a quasi-Poisson algebra. Later, in a recent paper by Ferreira, Kaygorodov, and Lopatkin a relation between 1 2 -derivations of Lie algebras and transposed Poisson algebras has been established [7]. These ideas were used to describe all transposed Poisson structures on Witt and Virasoro algebras in [7]; on twisted Heisenberg-Virasoro, Schrödinger-Virasoro and extended Schrödinger-Virasoro algebras in [19]; on oscillator algebras in [4].…”

Classification of five-dimensional symmetric Leibniz algebras

“…Symmetric Leibniz algebras are related to Lie racks [1] and every symmetric Leibniz algebra is flexible, power-associative and nil-algebra with nilindex 3 [15]. A symmetric Leibniz algebra under commutator and anticommutator multiplications gives a Poisson algebra [3].…”