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Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela

A formula for the antipode of the natural Hopf algebra associated to a set operad. Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela. Families of labelled combinatorial structures. 1. 8. 9. 6. 5. 4. 2. 3. 7. 8. 1. 6. 9. Combinatorial Species. Operations.

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Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela

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  1. A formula for the antipode of the natural Hopf algebra associated to a set operad. Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela

  2. Families of labelled combinatorial structures 1 8 9 6 5 4 2 3 7 8 1 6 9

  3. Combinatorial Species

  4. Operations

  5. Product

  6. Disjoint union Product

  7. Substitution Structures inside other structures

  8. Asemblies of structures 2 1 , ( ) 7 1 8 8 3 2 4 3 10 5 6 9 7 5 10 9 6 4 External structure Asembly

  9. -enriched Rooted trees

  10. “ C-monoids, Moebius Species and Coalgebras” M. Mendez Ph. D. thesis, Universidad Central de Venezuela, 1989. “Moebius Species” M. Mendez, J. Yang. Adv. In Math. 1991

  11. Two monoidal categories

  12. c-Monoids Related to associative algebras via the Schur functor.

  13. c-Operads

  14. 9 1 8 1 8 3 3 5 5 6 6 2 2 7 7 4 4 The operad of finite sets 9

  15. Permutative associative operad

  16. 1 1 7 7 2 2 3 5 3 5 6 6 9 9

  17. The operad of -enriched rooted trees f k e j m n f k i e j l m n i l h d d h c b b c a a

  18. 1 1 b d e b d e 5 8 2 5 8 2 c c 7 7 9 6 9 6 4 3 4 3 10 10

  19. Natural extension

  20. 5 6 5 p 6 w p 9 7 l w f l f s b 7 9 4 s d b t d v 4 q 3 t q 2 v 3 2 u 1 u 1

  21. 1 6 1 6 2 2 = 5 5 ≤ 3 3 a b a b d d e e 1 2 e 1 2 e d d 3 3 = ≤ a a 5 7 5 7 6 6

  22. 2 3 4 1 2 3 2 3 2 3 4 4 1 4 1 1 4 2 3 1 3 2 3 2 3 2 3 2 4 1 1 4 1 4 1 4 Ìnterval in the poset

  23. Incidence Coalgebras Reduced incidence Coalgebras

  24. The isomorphism type of a stuctura , denoted by can be thought of as the same structure without its labels. 3 2 d a 4 1

  25. The Natural incidence Hopfalgebra

  26. 2 3 3 2 1 4 1 4 1 1 + 2 + 2 3 2 3 2 3 2 3 2 1 1 + + + + 4 1 1 4 = 2 3 2 3 1 4 1 4 1 4 1 4

  27. Free commutative algebra generated by all the unlabelled trees

  28. “ C-monoids, Moebius Species and Coalgebras” M. Mendez Ph. D. thesis, 1989. “Moebius Species” M. Mendez, J. Yang. Adv. In Math. 1991

  29. Chapoton-Livernet (2007)

  30. Srchöder-Hyparcus M-enriched trees and the antipode formula , ) ( 9 = 3 9 3 7 6 8 4 7 6 8 4 1 5 2 1 11 5 2 10 11 10

  31. 1 2 2 2 5 1 1 3 5 3 3 4 4 5 5 4 1 3 2 4

  32. 2 1 3 1 3 2 3 2 2 1 3 1

  33. Antipode equivalent to cassical Lagrange inversion formula

  34. … Is an epimorphism of Hopf algebras

  35. Empty cut The epimorphism

  36. Open Problem: the Standard reduced Hopf algebra for other Operads, for example: Generalizations of the C-K Hopf algebra

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