Standard Lyndon Bases of Lie Algebras and Enveloping Algebras

Abstract

It is well known that the standard bracketings of Lyndon words in an alphabet $A$ form a basis for the free Lie algebra ${\text {Lie}}(A)$ generated by $A$. Suppose that $\mathfrak {g} \cong {\text {Lie}}(A)/J$ 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 $\mathfrak {g}$. We show that a similar approach to the universal enveloping algebra $\mathfrak {g}$ naturally leads to a Poincaré-Birkhoff-Witt type basis of the enveloping algebra of $\mathfrak {g}$. 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 [LR].