Continual Lie algebras determined by chain complexes

TL;DR

This paper establishes a fundamental link between chain complexes and continual Lie algebras, providing new examples and explicit relations, especially through the cech-de Rham complex in foliation theory.

Contribution

It introduces a novel construction of continual Lie algebras from chain complexes with specific algebraic properties, expanding the sources of such algebras.

Findings

Chain complexes with Leibniz-product and Jacobi identity induce continual Lie algebras.

Explicit commutation relations derived for cech-de Rham complex.

Provides new examples of continual Lie algebras from geometric contexts.

Abstract

Continual Lie algebras are infinite-dimensional generalizations of Lie algebras with discrete root system by considering continual root systems. In this paper we establish the general relation between chain complexes and continual Lie algebras. The natural orthogonality condition with respect to a product among elements of a chain complex $\mathcal C$ spaces brings about to $\mathcal C$ the structure of a graded algebra with differential relations. We prove the main result of this paper: a chain complex endowed with an appropriate Leibniz-property product of elements of its spaces and the Jacobi identity brings about the structure of a continual Lie algebra with the root space determined by parameters for the complex. That provides a new source of examples of continual Lie algebras. Finally, as an example, we consider the case of \v{C}ech-de Rham complex associated to a foliation of a smooth manifold. In a particular case of this chain complex, we derive explicitly the commutation relations for the corresponding continual Lie algebra.