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Antipode
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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, 2014CoAuthors: Miguel A. Méndez, Jean Carlos LiendoAbstract: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 noncocommutative 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 HaimanSchmitt 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 nonsymmetric operads a similar construction leads to the world of noncommutative Hopf algebras. We recover from our general formula the NovelliThibon combinatorial form of the Antipode for the noncommutative Hopf algebra of formal diffeomorphisms.

An Antipode formula for the natural Hopf algebra of a set operad
arXiv: Quantum Algebra, 2013CoAuthors: Miguel A. Méndez, Jean Carlos LiendoAbstract:A setoperad is a monoid in the category of combinatorial species with respect to the operation of substitution. From a setoperad, 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 HaymanSchmitt 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 setoperad 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 setoperad 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, 1993CoAuthors: Adrian L. Smith, Emmanouil N Pitsinos, C.k. Hwang, Hiroyuki Saimoto, Gerard R. Scarlato, T Suzuki, Kyriacos C NicolaouAbstract: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, 2011CoAuthors: Aaron Lauve, Mitja MastnakAbstract: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.