Bidendriform bialgebras, trees, and free quasi-symmetric functions
نویسنده
چکیده
we introduce bidendriform bialgebras, which are bialgebras such that both product and coproduct can be split into two parts satisfying good compatibilities. For example, the Malvenuto-Reutenauer Hopf algebra and the non-commutative Connes-Kreimer Hopf algebras of planar decorated rooted trees are bidendriform bialgebras. We prove that all connected bidendriform bialgebras are generated by their primitive elements as a dendriform algebra (bidendriformMilnor-Moore theorem) and then is isomorphic to a Connes-Kreimer Hopf algebra. As a corollary, the Hopf algebra of Malvenuto-Reutenauer is isomorphic to the Connes-kreimer Hopf algebra of planar rooted trees decorated by a certain set. We deduce that the Lie algebra of its primitive elements is free in characteristic zero (G. Duchamp, F. Hivert and J.-Y. Thibon conjecture). RESUME : nous introduisons les bigèbres bidendriformes, qui sont des bigèbres dont le produit et le coproduit peuvent être scindés en deux avec de bonnes compatibilités. Par exemple, l’algèbre de Hopf de Malvenuto-Reutenauer et les algèbres de Hopf non-commutative de ConnesKreimer sur les arbres plans enracinés décorés sont des bigèbres bidendriformes. Nous montrons que toute bigèbre bidendriforme connexe est engendrée par ses éléments totalement primitifs comme algèbre dendriforme (version bidendriforme du théorème de Milnor-Moore) et qu’elle est alors isomorphe à une algèbre de Hopf de Connes-Kreimer. En conséquence, l’algèbre de Hopf de Malvenuto-Reutenauer est isomorphe à l’algèbre de Connes-Kreimer des arbres plans enracinés décorés par un certain ensemble. On en déduit que l’algèbre de Lie de ses éléments primitifs est libre en caractéristique zéro (conjecture de G. Duchamp, F. Hivert et J.-Y. Thibon).
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Some Properties of Certain Subclasses of Close-to-Convex and Quasi-convex Functions with Respect to 2k-Symmetric Conjugate Points
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