Graphs and the (co)homology of Lie algebras

TL;DR

This paper introduces a diamond graph theory and applies it to analyze the (co)homology of Lie algebras derived from classical semi-simple Lie algebras, providing new structural insights.

Contribution

It develops a novel diamond graph theory and applies it to compute the weight decomposition and ranks of subgraphs in the (co)homology of Lie algebras.

Findings

Weight decomposition of diamond Lie algebra with Dynkin graph A_{n+1}

Rank calculations for weight subgraphs

Application of graph theory to Lie algebra (co)homology

Abstract

In this paper, we develop a diamond graph theory and apply the theory to the (co)homology of the Lie algebra generated by positive systems of the classical semi-simple Lie algebras over the field of complex numbers. As an application, we give the weight decomposition of the diamond Lie algebra with Dynkin graph $A_{n+1}$ and compute the rank of every weight subgraph of it.