https://hal.inria.fr/hal-00955693Andary, PhilippePhilippeAndaryLIFAR - Laboratoire d'informatique fondamentale et appliquée de Rouen - UNIROUEN - Université de Rouen Normandie - NU - Normandie UniversitéFinely homogeneous computations in free Lie algebrasHAL CCSD1997Lie algebrasfinely homogeneous computations[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]Monteil, Alain2014-03-05 09:31:342021-10-19 16:15:072014-03-05 11:23:47enJournal articleshttps://hal.inria.fr/hal-00955693/document10.46298/dmtcs.236application/pdf1We first give a fast algorithm to compute the maximal Lyndon word (with respect to lexicographic order) of \textitLy_α (A) for every given multidegree alpha in \textbfN^k. We then give an algorithm to compute all the words living in \textitLy_α (A) for any given α in \textbfN^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).