# Generating Hypergraphs, Decomposability and Classification of Two-Step Nilpotent Lie Algebras

## TL;DR

This paper introduces a hypergraph-based method to classify two-step nilpotent Lie algebras, identifying decomposable cases and confirming indecomposability of known classifications in dimensions 8 and 9.

## Contribution

It defines generating hypergraphs for these algebras and uses them to determine decomposability, advancing classification techniques for dimensions 8 and 9.

## Key findings

- Decomposability corresponds to disconnected hypergraphs.
- Identified some decomposable algebras in dimension 9.
- Confirmed indecomposability of all classified algebras in dimension 8.

## Abstract

In 1973, Gauger proposed a generator-relation method and a duality theory for two-step nilpotent Lie algebras. Based upon these, he classified two-step nilpotent Lie algebras of dimension $8$. In 1999, Galitski and Timashev continued this approach to classify two-step nilpotent Lie algebras of dimension $9$. Their results were partially improved by Ren and Zhu in 2011, Yan and Deng in 2013. Some decomposable two-step nilpotent Lie algebras were excluded in the case of dimension $8$. In this paper, we define generating hypergraph for a two-step nilpotent Lie algebra. The two-step nilpotent Lie algebra is decomposable if and only if its generating hypergraph is not connected under certain bases. Using this result, we identify some decomposable two-step nilpotent Lie algebras in dimension $9$. We give a direct proof that the five two-step nilpotent Lie algebras for dimension $8$, classified by Ren and Zhu in 2011, are all indecomposable. We also introduce a conventional nomenclature for two-step nilpotent Lie algebras of dimension $n = 8, 9$, classified by Ren, Zhu, Yan and Deng, etc.