Null Lagrangians in free Novikov algebras

TL;DR

The paper analyzes the symmetric elements and null Lagrangians in free Novikov algebras by embedding into a differential polynomial framework and employing Euler (variational) derivatives. It proves that the space of null Lagrangians coincides with the subspace closed under the symmetrized product $a∘b=ab+ba$ and provides an explicit basis for these symmetric elements via differential polynomials. A practical criterion using Euler operators links symmetry to null Lagrangians, supported by the generalized Gel'fand–Dikii transform. Finally, it gives a complete $S_n$-module decomposition of the multilinear symmetric space in terms of induced permutation modules and Kostka numbers, clarifying the representation-theoretic structure of these elements.

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.

Theorems & Definitions (17)

• Theorem 2.1
• Lemma 3.1
• Theorem 3.2
• proof
• Corollary 3.3
• proof
• Theorem 4.1: GelʹfandDikii1975, OlverShakiban1978
• Theorem 4.2: Shakiban1981
• Theorem 4.3: Shakiban1981, Thm. 4
• Lemma 4.4: Shakiban1981, Lemma 5(iii)