Finely homogeneous computations in free Lie algebras

Abstract

We first give a fast algorithm to compute the maximal Lyndon word (with respect to lexicographic order) of Ly α (A) for every given multidegree alpha in N k. We then give an algorithm to compute all the words living in Ly α (A) for any given α in N k. The best known method for generating Lyndon words is that of Duval [1], which gives a way to go from every Lyndon word of length n to its successor (with respect to lexicographic order by length), in space and worst case time complexity O(n). Finally, we give a simple algorithm which uses Duval's method (the one above) to compute the next standard bracketing of a Lyndon word for lexicographic order by length. We can find an interesting application of this algorithm in control theory, where one wants to compute within the command Lie algebra of a dynamical system (letters are actually vector fields).