# Gelfand--Dorfman algebras, derived identities, and the Manin product of operads

## TL;DR

This paper explores Gelfand--Dorfman algebras, their identities, and the Manin product of operads, focusing on their relation to differential Poisson algebras and formal calculus of variations.

## Contribution

It provides a general description of identities for operations in differential algebras and connects GD-algebras with the Manin product of operads.

## Key findings

- Identifies identities for operations in differential algebras with derivation d.
- Describes the relation between GD-algebras and differential Poisson algebras.
- Connects Gelfand--Dorfman algebras with the Manin product of operads.

## Abstract

Gelfand--Dorfman bialgebras (GD-algebras) are nonassociative systems with two bilinear operations satisfying a series of identities that express Hamiltonian property of an operator in the formal calculus of variations. The paper is devoted to the study of GD-algebras related with differential Poisson algebras. As a byproduct, we obtain a general description of identities that hold for operations $a\succ b = d(a)b$ and $a\prec b = ad(b)$ on a (non-associative) differential algebra with a derivation~$d$.