Lie algebras associated with labeled directed graphs

Abstract

We construct 2-step nilpotent Lie algebras using labeled directed simple graphs and give a criterion to detect certain ideals and subalgebras by finding special subgraphs. We prove that if a label occurs only once, then reversing its orientation leads to an isomorphic algebra. As a consequence, if every edge is labeled differently, the Lie algebra depends only on the underlying undirected graph. In addition, we construct the graphs of all 2-step nilpotent Lie algebras of dimension $\le 6$ and compute the algebra of strata preserving derivations of the Lie algebra associated with the complete bipartite graph $K_{m,n}$ with two different labelings.

KEYWORDS:

• 2-step nilpotent Lie algebras
• complete bipartite graphs
• degree preserving derivations
• ideals
• labeled directed graph

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B30
• 17B40
• 05C20
• 05C25

Acknowledgments

The second author would like to thank professor María Alejandra Álvarez from Universidad de Antofagasta, Chile, for her hospitality and her comments regarding earlier versions of this paper. Both authors would like to thank professor Andrew Clarke from Universidade Federal do Rio de Janeiro, Brazil, for his hospitality. Finally, we wish to thank the anonymous referee for a very careful reading of the present manuscript and for giving many very helpful suggestions that improved the final version. The results of this paper are part of the Ph.D. thesis of the second author at Universidad de La Frontera, Temuco, Chile.

Additional information

Funding

This research is partially supported by Fondecyt #1181084.