Standard Lyndon bases of Lie algebras and enveloping algebras

It is well known that the standard bracketings of Lyndon words in an alphabet A form a basis for the free Lie algebra Lie(A) generated by A . Suppose that g 2 Lie(A)/J is a Lie algebra given by a generating set A and a Lie ideal J of relations. Using a Grobner 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 . We show that a similar approach to the universal enveloping algebra g naturally leads to a Poincare-Birkhoff-Witt type basis of the enveloping algebra of 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]. 1. LYNDONWORDS AND THE FREE LIEALGEBRA In this section we give a short summary of the facts about Lyndon words and the free Lie algebra which we shall use. All of the facts in this section are well known. A comprehensive treatment of free Lie algebras (and Lyndon words) appears in the book by C. Reutenauer [Re]. Let A be an ordered alphabet, and let A* be the set of all words in the alphabet A (the free monoid generated by A) . Let lul denote the length of the word u E A* ,and let u < u denote that the word u is lexicographically smaller than the word u . A word 1 E A* is a Lyndon word if it is lexicographically smaller than all its cyclic rearrangements. Let I be a Lyndon word, and let m ,n be words such that I = mn and n is the longest Lyndon word appearing as a proper right factor of 1. Then m is also a Lyndon word [Lo, Proposition 5.1.31. The standard bracketing of a Lyndon word is given (inductively) by (1.1) b[a] = a , for a E A , b[l] = [b[m],b[n]] , Received by the editors February 23, 1994; originally communicated to the Proceedings of the AMS by Lance W. Small. 1991 Mathematics Subject Classification. Primary 17B01, 17B20, 05E15, 68R15.