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On the cohomology of Lie algebras associated with graphs

Marco Aldi, Andrew Butler, Jordan Gardiner, Daniele Grandini, Monica Lichtenwalner, Kevin Pan

TL;DR

The paper studies the cohomology of Lie algebras associated with graphs, focusing on the Dani-Mainkar construction $\mathcal{L}(G)$ and its solvable extensions $\mathcal{L}(G,\Sigma)$. It establishes a canonical decomposition of the Cartan–Chevalley–Eilenberg complex into summands labeled by induced subgraphs, enabling explicit formulas for low-degree cohomology and a complete computation for star graphs. A key contribution is a reduction principle: the cohomology of Grantcharov–Grantcharov–Iliev algebras is isomorphic to that of a suitably constructed graph $\widetilde{G}$, reducing solvable-case computations to the Dani-Mainkar setting. These results yield concrete, closed-form expressions for $b_1$, $b_2$, and $b_3$ in multiple cases and provide topological interpretations via solvmanifolds, supported by the Dixmier exact sequence and essential cohomology. The work deepens connections between graph theory, Lie algebra cohomology, and the topology of nilmanifolds and solvmanifolds.

Abstract

We describe a canonical decomposition of the cohomology of the Dani-Mainkar metabelian Lie algebras associated with graphs. As applications, we obtain explicit formulas for the third cohomology of any Dani-Mainkar Lie algebra and for the cohomology in all degrees of Lie algebras associated with arbitrary star graphs. We also describe a procedure to reduce the calculation of the cohomology of solvable Lie algebras associated with graphs through the Grantcharov-Grantcharov-Iliev construction to the cohomology of Dani-Mainkar Lie algebras.

On the cohomology of Lie algebras associated with graphs

TL;DR

The paper studies the cohomology of Lie algebras associated with graphs, focusing on the Dani-Mainkar construction and its solvable extensions . It establishes a canonical decomposition of the Cartan–Chevalley–Eilenberg complex into summands labeled by induced subgraphs, enabling explicit formulas for low-degree cohomology and a complete computation for star graphs. A key contribution is a reduction principle: the cohomology of Grantcharov–Grantcharov–Iliev algebras is isomorphic to that of a suitably constructed graph , reducing solvable-case computations to the Dani-Mainkar setting. These results yield concrete, closed-form expressions for , , and in multiple cases and provide topological interpretations via solvmanifolds, supported by the Dixmier exact sequence and essential cohomology. The work deepens connections between graph theory, Lie algebra cohomology, and the topology of nilmanifolds and solvmanifolds.

Abstract

We describe a canonical decomposition of the cohomology of the Dani-Mainkar metabelian Lie algebras associated with graphs. As applications, we obtain explicit formulas for the third cohomology of any Dani-Mainkar Lie algebra and for the cohomology in all degrees of Lie algebras associated with arbitrary star graphs. We also describe a procedure to reduce the calculation of the cohomology of solvable Lie algebras associated with graphs through the Grantcharov-Grantcharov-Iliev construction to the cohomology of Dani-Mainkar Lie algebras.
Paper Structure (7 sections, 13 theorems, 58 equations, 4 figures)

This paper contains 7 sections, 13 theorems, 58 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. The Dani-Mainkar Construction
  4. The Grantcharov-Grantcharov-Iliev Construction
  5. Canonical Decomposition of the cohomology of $\mathcal{L}(G)$
  6. Cohomology of Dani-Mainkar Lie Algebras Associated with Star Graphs
  7. The Cohomology of the Grantcharov-Grantcharov-Iliev Lie Algebras

Key Result

Proposition 10

Let $G$ be a finite simple graph and let $\Sigma$ be a collection of cliques of $G$. Then $\mathcal{L}(G,\Sigma)$ is a completely solvable Lie algebra.

Figures (4)

  • Figure 1: The graph $G$ as defined in Example \ref{['ex:4']}.
  • Figure 2: The 23 graphs with non-vanishing third essential cohomology and the corresponding values of $\beta_3(G)$.
  • Figure 3: The graph $G$ as defined in Example \ref{['ex:24']}
  • Figure 4: The graph $G$ as defined in Example \ref{['ex:42']}.

Theorems & Definitions (57)

  • Definition 1
  • Remark 2
  • Example 3
  • Example 4
  • Definition 5
  • Example 6
  • Remark 7
  • Example 8
  • Remark 9
  • Proposition 10
  • ...and 47 more