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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 \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).
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Philippe Andary. Finely homogeneous computations in free Lie algebras. Discrete Mathematics and Theoretical Computer Science, DMTCS, 1997, Vol. 1, pp.101-114. ⟨10.46298/dmtcs.236⟩. ⟨hal-00955693⟩



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