Combinatorial structures of three vertices and Lie algebras

Linear Algebra and its Applications, 2012

In this paper, we study the structure and properties of those n-dimensional Lie algebras associated with either summed structures of complete graphs or some families of digraphs, having into consideration that all these combinatorial structures are made up of n vertices. Our main goal is to obtain criteria determining when a Lie algebra is associated with some of combinatorial structures considered in this paper, as well as to study the properties of those structures in order to use them as a tool for classifying the types of Lie algebras associated with them.

International Journal of Computer Mathematics, 2013

>IJH=?J This paper deals with several operations on graphs and combinatorial structures linking them with their associated Lie algebras. More concretely, our main goal is to obtain some criteria to determine when there exists a Lie algebra associated with a combinatorial structure arising from those operations. Additionally, we show an algorithmic method for one of those operations.

2004

Given a Lie algebra of finite dimension, with a selected basis of it, we show in this paper that it is possible to associate it with a combinatorial structure, of dimension 2, in general. In some particular cases, this structure is reduced to a weighted graph. We characterize such graphs, according to they have 3-cycles or not.

Mathematical Methods in the Applied Sciences, 2018

This paper studies the link between isomorphic digraphs and isomorphic Leibniz algebras, determining in detail this fact when using (psuedo)digraphs of 2 and 3 vertices associated with Leibniz algebras according to their isomorphism classes. Moreover, we give the complete list with all the combinatorial structures of 3 vertices associated with Leibniz algebras, studying their isomorphism classes. We also compare our results with the current classifications of 2-and 3-dimensional Leibniz algebras. Finally, we introduce and implement the algorithmic procedure used for our goals and devoted to decide if a given combinatorial structure is associated or not with a Leibniz algebra.

Linear Algebra and its Applications, 2005

A new class of Lie algebras of finite dimension, those which are associated with a certain combinatorial configuration made up by triangles of weighted and non-directed edges, is introduced and a characterization theorem for them is proved. Moreover, some subclasses of such Lie algebras are classified.

2010

We give an algebraic construction of the topological graph-tree configuration pairing of Sinha and Walter beginning with the classical presentation of Lie coalgebras via coefficients of words in the associative Lie polynomial. Our work moves from associative algebras to preLie algebras to the graph complexes of Sinha and Walter, justifying the use of graph generators for Lie coalgebras by iteratively expanding the set of generators until the set of relations collapses to two simple local expressions. Our focus is on new computational methods allowed by this framework and the efficiency of the graph presentation in proofs and calculus involving free Lie algebras and coalgebras. This outlines a new way of understanding and calculating with Lie algebras arising from the graph presentation of Lie coalgebras.

Mathematics and Computers in Simulation, 2014

In this paper, we introduce an algorithmic process to associate Leibniz algebras with combinatorial structures. More concretely, we have designed an algorithm to automatize this method and to obtain the restrictions over the structure coefficients for the law of the Leibniz algebra and so determine its associated combinatorial structure. This algorithm has been implemented with the symbolic computation package Maple. Moreover, we also present another algorithm (and its implementation) to draw the combinatorial structure associated with a given Leibniz algebra, when such a structure is a (pseudo)digraph. As application of these algorithms, we have studied what (pseudo)digraphs are associated with low-dimensional Leibniz algebras by determination of the restrictions over edge weights (i.e. structure coefficients) in the corresponding combinatorial structures. c

Applied Mathematics Letters, 2012

This work shows how to associate the Lie algebra h n , of upper triangular matrices, with a specific combinatorial structure of dimension 2, for n ∈ N. The properties of this structure are analyzed and characterized. Additionally, the results obtained here are applied to obtain faithful representations of solvable Lie algebras.

Advances in Applied Mathematics, 2003

This paper shows how to uniformly associate Lie algebras to the simply-laced Dynkin diagrams excluding E 8 by constructing explicit combinatorial models of minuscule representations using only graph-theoretic ideas. This involves defining raising and lowering operators in a space of ideals of certain distributive lattices associated to sequences of vertices of the Dynkin diagram.

Involve, 2016

We construct a basis for free Lie algebras via a left greedy bracketing algorithm on Lyndon-Shirshov words. We use a new tool-the configuration pairing between Lie brackets and graphs of Sinha and Walter-to show that the left greedy brackets form a basis. Our constructions further equip the left greedy brackets with a dual monomial Lie coalgebra basis of star graphs. We end with a brief example using the dual basis of star graphs in a Lie algebra computation.