Standard Lyndon bases of Lie algebras and enveloping algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 19 May 2015

This is an excerpt of the paper Standard Lyndon bases of Lie algebras and enveloping algebras by Pierre Lalonde Arun Ram.

The first author's research was supported by FCAR and NSERC grants. The second author's research was partially supported by a National Science Foundation Postdoctoral Fellowship.

Abstract

It is well known that the standard bracketings of Lyndon words in an alphabet $A$ form a basis for the free Lie algebra $Lie\left(A\right)$ generated by $A\text{.}$ Suppose that $𝔤\cong \text{Lie}\left(A\right)/J$ is a Lie algebra given by a generating set $A$ and a Lie ideal $J$ of relations. Using a Gröbner basis type approach we define a set of "standard" Lyndon words, a subset of the set Lyndon words, such that the standard bracketings of these words form a basis of the Lie algebra $𝔤\text{.}$ We show that a similar approach to the universal enveloping algebra $𝔤$ naturally leads to a Poincaré-Birkhoff-Witt type basis of the enveloping algebra of $𝔤\text{.}$ We prove that the standard words satisfy the property that any factor of a standard word is again standard. Given root tables, this property is nearly sufficient to determine the standard Lyndon words for the complex finite-dimensional simple Lie algebras. We give an inductive procedure for computing the standard Lyndon words and give a complete list of the standard Lyndon words for the complex finite-dimensional simple Lie algebras. These results were announced in [LRa1993].

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