Standard Lyndon bases of Lie algebras and enveloping algebras

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