From Lie algebra crossed modules to tensor hierarchies

TL;DR

This paper establishes a mathematical framework connecting Lie algebra crossed modules to tensor hierarchies via Lie-Leibniz triples, providing a clearer construction and suggesting new Leibniz gauge theories.

Contribution

It introduces a new perspective linking Lie-Leibniz triples to tensor hierarchies, generalizing Lie algebra crossed modules with a more straightforward construction.

Findings

Lie-Leibniz triples induce differential graded Lie algebras

Tensor hierarchies generalize Lie algebra crossed modules

Construction suggests potential for Leibniz gauge theories

Abstract

The present paper, though inspired by the use of tensor hierarchies in theoretical physics, establishes their mathematical credentials, especially as genetically related to Lie algebra crossed modules. Gauging procedures in supergravity rely on a pairing - the embedding tensor - between a Leibniz algebra and a Lie algebra. Two such algebras, together with their embedding tensor, form a triple called a Lie-Leibniz triple, of which Lie algebra crossed modules are particular cases. This paper is devoted to showing that any Lie-Leibniz triple induces a differential graded Lie algebra - its associated tensor hierarchy - whose restriction to the category of Lie algebra crossed modules is the canonical assignment associating to any Lie algebra crossed module its corresponding unique 2-term differential graded Lie algebra. This shows that Lie-Leibniz triples form natural generalizations of Lie algebra crossed modules and that their associated tensor hierarchies can be considered as some kind of 'lie-ization' of the former. We deem the present construction of such tensor hierarchies clearer and more straightforward than previous derivations. We stress that such a construction suggests the existence of further well-defined Leibniz gauge theories.