Null Lagrangians in free Novikov algebras

TL;DR

This paper investigates null Lagrangians in free Novikov algebras by examining symmetrization, embedding into differential algebras, and characterizing the module structure over symmetric groups.

Contribution

It provides a complete description of null Lagrangians and their module structure in free Novikov algebras using embedding techniques and symmetrization.

Findings

Null Lagrangians coincide with elements closed under symmetrized product.

Embedded free Novikov algebra into differential algebra over characteristic zero field.

Complete description of the module structure over symmetric group.

Abstract

We study the symmetrization of the Novikov product. Using the embedding of a free Novikov algebra into a differential algebra over a field of characteristic zero and the Euler operators (variational derivatives), we show that the space of null Lagrangians coincides with the subspace of elements closed under the symmetrized product $a\circ b=ab+ba$. We also completely describe its module structure over symmetric group.