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.
It is well known that the standard bracketings of Lyndon words in an alphabet form a basis for the free Lie algebra generated by Suppose that is a Lie algebra given by a generating set and a Lie ideal 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 We show that a similar approach to the universal enveloping algebra naturally leads to a Poincaré-Birkhoff-Witt type basis of the enveloping algebra of 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].