Transposed Poisson algebras, Novikov-Poisson algebras and 3-Lie algebras

Manin Triples and Bialgebras of Left-Alia Algebras Associated with Invariant Theory

“…Some well-known examples of such structures include Lie bialgebras [14,15], which are closely related to Poisson-Lie groups and play an important role in the infinitesimalization of quantum groups, and antisymmetric infinitesimal bialgebras [16][17][18][19][20] as equivalent structures of double constructions of Frobenius algebras which are widely applied in the 2d topological field and string theory [21,22]. Recently, the notion of anti-pre-Lie bialgebras was studied in [23], which serves as a preliminary to supply a reasonable bialgebra theory for transposed Poisson algebras [24]. The notions of mock-Lie bialgebras [25] and Leibniz bialgebras [26,27] were also introduced with different motivations.…”

Transposed Poisson structures on solvable and perfect Lie algebras

“…One of the natural tasks in the theory of Poisson algebras is the description of all such algebras with a fixed Lie or associative part [2,16,17,38]. Recently, Bai, Bai, Guo, and Wu have introduced a dual notion of the Poisson algebra [3], called a transposed Poisson algebra, by exchanging the roles of the two multiplications in the Leibniz rule defining a Poisson algebra. A transposed Poisson algebra defined this way not only shares some properties of a Poisson algebra, such as the closedness under tensor products and the Koszul self-duality as an operad but also admits a rich class of identities [3,6,12,25,26,33].…”

“…Recently, Bai, Bai, Guo, and Wu have introduced a dual notion of the Poisson algebra [3], called a transposed Poisson algebra, by exchanging the roles of the two multiplications in the Leibniz rule defining a Poisson algebra. A transposed Poisson algebra defined this way not only shares some properties of a Poisson algebra, such as the closedness under tensor products and the Koszul self-duality as an operad but also admits a rich class of identities [3,6,12,25,26,33]. It is important to note that a transposed Poisson algebra naturally arises from a Novikov-Poisson algebra by taking the commutator Lie algebra of its Novikov part [3].…”

“…A transposed Poisson algebra defined this way not only shares some properties of a Poisson algebra, such as the closedness under tensor products and the Koszul self-duality as an operad but also admits a rich class of identities [3,6,12,25,26,33]. It is important to note that a transposed Poisson algebra naturally arises from a Novikov-Poisson algebra by taking the commutator Lie algebra of its Novikov part [3]. Any unital transposed Poisson algebra is a particular case of a 'contact bracket' algebra and a quasi-Poisson algebra [6].…”

“…Let L be a vector space equipped with two nonzero bilinear operations • and [−, −]. The triple (L, •, [−, −]) is called a transposed Poisson algebra if (L, •) is a commutative associative algebra and (L, [−, −]) is a Lie algebra that satisfies the following compatibility condition 2z • [x, y] = [z • x, y] + [x, z • y] .Transposed Poisson algebras were first introduced by Bai et al[3].…”