Leibniz-Hopf algebra and quasi-symmetric functions

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Let $\mathcal{M}$ be the graded dual of the Leibniz–Hopf algebra over the integers. The strong Ditters conjecture states that $\mathcal{M}$ is a free commutative algebra with as generators the concatenation powers of elementary Lyndon words. This conjecture is still open (as of 2001); the initial proof contains mistakes (so the assertion of its proof in Leibniz–Hopf algebra is incorrect), and so does a later version [a1] of it. Meanwhile, the weak Ditters conjecture, which states that $\mathcal{M}$ is free over the integers without giving a concrete set of generators, has been proved; see Quasi-symmetric function and [a2].


[a1] E.J. Ditters, A.C.J. Scholtens, "Free polynomial generators for the Hopf algebra $\mathit{Qsym}$ of quasi-symmetric functions" J. Pure Appl. Algebra , 144 (1999) pp. 213–227
[a2] M. Hazewinkel, "Quasi-symmetric functions" D. Krob (ed.) A.A. Mikhalev (ed.) A.V. Mikhalev (ed.) , Formal Power Series and Algebraic Combinatorics (Moscow 2000) , Springer (2000) pp. 30–44
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Leibniz-Hopf algebra and quasi-symmetric functions. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article