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

Marco Aldi, Samuel Bevins

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.

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

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.
Paper Structure (4 sections, 9 theorems, 37 equations)

This paper contains 4 sections, 9 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. The 2-step nilpotent $L_\infty$-algebra of a hypergraph
  3. System of Distinct Representatives and Symplectic Forms
  4. Cohomology

Key Result

Theorem 14

Two finite simple hypergraphs $G$ and $G'$ are isomorphic if and only if $\mathcal{L}(G)$ and $\mathcal{L}(G')$ are isomorphic as $L_\infty$-algebras.

Theorems & Definitions (50)

  • Definition 1
  • Remark 2
  • Definition 3
  • Remark 4
  • Definition 5
  • Definition 6
  • Remark 7
  • Definition 8
  • Example 9
  • Definition 10
  • ...and 40 more