## 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
$\U0001d524\cong \text{Lie}\left(A\right)/J$ 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 $\U0001d524\text{.}$
We show that a similar approach to the universal enveloping algebra $\U0001d524$ naturally leads to a Poincaré-Birkhoff-Witt type basis of the
enveloping algebra of $\U0001d524\text{.}$ 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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