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 Lie(A) generated by A. Suppose that g ≅ 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 aé-Birkhoff-Witt type basis of the enveloping algebra of 0. 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-dim..