Generating Hypergraphs, Decomposability and Classification of Two-Step Nilpotent Lie Algebras
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 . In 1999, Galitski and Timashev continued this approach to classify two-step nilpotent Lie algebras of dimension . 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 . 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 . We give a direct proof that the five two-step nilpotent Lie algebras for dimension , classified by Ren and Zhu in 2011, are all indecomposable. We also introduce a conventional nomenclature for two-step nilpotent Lie algebras of dimension , classified by Ren, Zhu, Yan and Deng, etc.
Cite
@article{arxiv.1905.08000,
title = {Generating Hypergraphs, Decomposability and Classification of Two-Step Nilpotent Lie Algebras},
author = {Shenglong Hu and Liqun Qi and Honglian Zhang},
journal= {arXiv preprint arXiv:1905.08000},
year = {2019}
}