Some generalizations of the variety of transposed Poisson algebras

Abstract

It is shown that the variety of transposed Poisson algebras coincides with the variety of Gelfand-Dorfman algebras in which the Novikov multiplication is commutative. The Gr\"obner-Shirshov basis for the transposed Poisson operad is calculated up to degree 4. Furthermore, we demonstrate that every transposed Poisson algebra is F-manifold. We verify that the special identities of GD-algebras hold in transposed Poisson algebras. Finally, we propose a conjecture stating that every transposed Poisson algebra is special, i.e., can be embedded into a differential Poisson algebra.

Maps on the Mirror Heisenberg–Virasoro Algebra

“…Let L be a vector space equipped with two nonzero bilinear operations [18] (see also Section 7.3 [19] and [20] for recent results in this topic). Summarizing results from Theorem 4 and Theorem 26 [17], we have the following corollary.…”

The algebraic classification and degenerations of nilpotent Poisson algebras

“…The notion of transposed Poisson algebras was introduced in [13]. Let us note, that they are related to many other interesting classes of algebras, such as commutative Gelfand-Dorfman algebras and F -manifold algebras [53]. The study of transposed Leibniz-Poisson algebras is motivated by a question posed in [16].…”

Transposed Poisson structures on the Lie algebra of upper triangular matrices

“…Recently, Bai, Bai, Guo, and Wu [1] have introduced a dual notion of the Poisson algebra, 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 [1,3,4,14,15,17]. 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.…”