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Some generalizations of the variety of transposed Poisson algebras

B. K. Sartayev

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 Gröbner-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.

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 Gröbner-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.
Paper Structure (4 sections, 34 equations)

This paper contains 4 sections, 34 equations.

Table of Contents

  1. Introduction
  2. Gröbner base of commutative GD-operad
  3. Examples of transposed Poisson algebras
  4. Special identities of GD-algebras on transposed Poisson algebras

Theorems & Definitions (2)

  • proof
  • proof