2-step Nilpotent $L_\infty$-algebras and Hypergraphs

TL;DR

This paper establishes a correspondence between simple hypergraphs and nilpotent strong homotopy Lie algebras, providing algebraic characterizations of hypergraph isomorphisms and properties.

Contribution

It introduces a novel procedure to associate strong homotopy Lie algebras to hypergraphs and characterizes hypergraph isomorphism via algebra isomorphism.

Findings

Hypergraph isomorphism corresponds to strong homotopy Lie algebra isomorphism.

Characterization of hypergraphs with systems of distinct representatives using symplectic forms.

Description of low-degree cohomology of the associated Lie algebras.

Abstract

We describe a procedure to attach a nilpotent strong homotopy Lie algebra to every simple hypergraph and prove that two hypergraphs are isomorphic if and only if the corresponding strong homotopy Lie algebras are isomorphic. As an application, we characterize hypergraphs admitting a system of distinct representatives in terms of symplectic forms on the corresponding strong homotopy Lie algebra. We conclude with a combinatorial description of the cohomology of these strong homotopy Lie algebras in low degree.