Combinatorial structures of three vertices and Lie algebras

Abstract

This paper shows a characterization of digraphs of three vertices associated with Lie algebras, as well as determining the list of isomorphism classes for Lie algebras associated with these digraphs. Additionally, we introduce and implement two algorithmic procedures related to this study: the first is devoted to draw, if exists, the digraph associated with a given Lie algebra; whereas the other corresponds to the converse problem and allows us to test if a given digraph is associated or not with a Lie algebra. Finally, we give the complete list of all non-isomorphic combinatorial structures of three vertices associated with Lie algebras and we study the type of Lie algebra associated with each configuration.

Keywords:

• digraph
• combinatorial structure
• Lie algebra
• isomorphism class
• algorithm

2010 AMS Subject Classifications::

• 17B60
• 05C25
• 05C20
• 05C90
• 17B20
• 17B30
• 68W30
• 68R10
• 05C85

Acknowledgements

This work has been partially supported by MTM2010-19336 and FEDER. Additionally, the authors want to thank the referees for their helpful and useful comments and suggestions, which have allowed us to improve the quality of this paper.