Some generalizations of the variety of transposed Poisson algebras

TL;DR

This paper explores the structure and properties of transposed Poisson algebras, establishing their equivalence with certain Gelfand-Dorfman algebras, computing their operad basis, and proposing a conjecture about their embedding into differential Poisson algebras.

Contribution

It demonstrates the equivalence of transposed Poisson algebras with Gelfand-Dorfman algebras with commutative Novikov multiplication, computes their operad basis, and conjectures their embeddability into differential Poisson algebras.

Findings

Transposed Poisson algebras coincide with Gelfand-Dorfman algebras with commutative Novikov multiplication.

The Gr"obner-Shirshov basis for the transposed Poisson operad is computed up to degree 4.

Every transposed Poisson algebra is an F-manifold.

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.