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Jean Carlos Liendo – One of the best experts on this subject based on the ideXlab platform.

  • An Antipode formula for the natural Hopf algebra of a set operad
    Advances in Applied Mathematics, 2014
    Co-Authors: Miguel A. Méndez, Jean Carlos Liendo

    Abstract:

    A symmetric set operad is a monoid in the category of combinatorial species with respect to the operation of substitution. From a symmetric set operad, we give here a simple construction of a commutative and non-co-commutative Hopf algebra, that we call the natural Hopf algebra of the operad. We obtain a combinatorial formula for its Antipode in terms of Schroder trees, generalizing the Haiman-Schmitt formula for the Faa di Bruno Hopf algebra. From there we derive Antipode formulas for specific operads. The classical Lagrange inversion formula is obtained in this way from the set operad of pointed sets. We also derive Antipodes formulas for the natural Hopf algebra corresponding to the operads of connected graphs, the NAP operad, and for its generalization, the set operad of trees enriched with a monoid. When the set operad is left cancellative, we can construct a family of posets. The natural Hopf algebra is then obtained as a reduced incidence Hopf algebra, by taking a suitable equivalence relation over the intervals of that family of posets. We also present a simple combinatorial construction of an epimorphism from the natural Hopf algebra corresponding to the NAP operad, to the Connes and Kreimer Hopf algebra. For non-symmetric operads a similar construction leads to the world of non-commutative Hopf algebras. We recover from our general formula the Novelli-Thibon combinatorial form of the Antipode for the non-commutative Hopf algebra of formal diffeomorphisms.

  • An Antipode formula for the natural Hopf algebra of a set operad
    arXiv: Quantum Algebra, 2013
    Co-Authors: Miguel A. Méndez, Jean Carlos Liendo

    Abstract:

    A set-operad is a monoid in the category of combinatorial species with respect to the operation of substitution. From a set-operad, we give here a simple construction of a Hopf algebra that we call {\em the natural Hopf algebra} of the operad. We obtain a combinatorial formula for its Antipode in terms of Shr\”oder trees, generalizing the Hayman-Schmitt formula for the Fa\’a di Bruno Hopf algebra. From there we derive more readable formulas for specific operads. The classical Lagrange inversion formula is obtained in this way from the set-operad of pointed sets. We also derive Antipodes formulas for the natural Hopf algebra corresponding to the operads of connected graphs, the NAP operad, and for its generalization, the set-operad of trees enriched with a monoid. When the set operad is left cancellative, we can construct a family of posets. The natural Hopf algebra is then obtained as an incidence reduced Hopf algebra, by taking a suitable equivalence relation over the intervals of that family of posets. We also present a simple combinatorial construction of an epimorphism from the natural Hopf algebra corresponding to the NAP operad, to the Connes and Kreimer Hopf algebra.

Kyriacos C Nicolaou – One of the best experts on this subject based on the ideXlab platform.

  • total synthesis of calicheamicin gamma 1i 2 development of an enantioselective route to calicheamicinone
    Journal of the American Chemical Society, 1993
    Co-Authors: Adrian L. Smith, Emmanouil N Pitsinos, C.k. Hwang, Hiroyuki Saimoto, Gerard R. Scarlato, T Suzuki, Kyriacos C Nicolaou

    Abstract:

    The first enantioselective total synthesis of (-)-calicheamicinone (3), the naturally occurring Antipode of the calicheamicin aglycon, has been achieved

Mitja Mastnak – One of the best experts on this subject based on the ideXlab platform.

  • The primitives and Antipode in the Hopf algebra of symmetric functions in noncommuting variables
    Advances in Applied Mathematics, 2011
    Co-Authors: Aaron Lauve, Mitja Mastnak

    Abstract:

    We identify a collection of primitive elements generating the Hopf algebra NCSym of symmetric functions in noncommuting variables and give a combinatorial formula for the Antipode.