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.

Families of labelled combinatorial structures

71 68

9

6 91 8

5

24

3

Combinatorial Species

•

•

•

•

Operations

Product

Disjoint union

Product

Substitution

Structures inside other structures

1

23

4

5

6

7 8

9

10

Asemblies of structures

Asembly

( ,1 2

7 83

10 4

9 6 5

)r

External structure

-enriched Rooted trees

“ 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

Two monoidal categories

c-Monoids

Related to associative algebras via the Schur functor.

c-Operads

159 8

3

2 46

7

915

83

2 46

7

The operad of finite sets

Permutative associative operad

1

2 3

69

7

5

1

2

35

7

69

The operad of -enriched rooted trees

a

bc

d

e f k m i j n l

h

a

b c

d

e f

k m

i j

n l

h

1

25 8

9

3 4

10

6

7

b d e

c

1

25 8

9

3 4

10

6

7

b d e

c

Natural extension

1

23

4

79

d

f l

5 6

u

v t q

s

w p

b

1

23

d

f l

5 6

4

79

u

vt

q

bs

w p

1

2

3

6

5

ab

de

≤

1

2

3

6

5

a

b

de

=

1 23

5 76

a

d e

≤=

1 23 d e

5 76

a

1

2 3

4

1 2 3 4

1

2 3

4 1

2 3

4 1

2 3

4

Ìnterval in the poset

1

2 3

4

2 3

1 4

23

1 4

2 3

1 4

Incidence Coalgebras

Reduced incidence Coalgebras

23

4a d

1

The isomorphism type of a stuctura , denoted by can be thought of as the same structure without its labels.

Mm )(m

The Natural incidence Hopf algebra

= + 2

+ +

1

1

+ 2

+ +

1

1

1

2 3

4

2 3

1 4

23

1 4

2 3

1 4

2 3

1 4

2 3

1 4

2 3

1 4

3

1 4

2

Free commutative algebra generated by all the unlabelled trees

“ C-monoids, Moebius Species and Coalgebras”

M. Mendez Ph. D. thesis, 1989.

“Moebius Species” M. Mendez, J. Yang.Adv. In Math. 1991

Chapoton-Livernet (2007)

2

3

5

7

9

1

4 68

10 11

1m

2m 3m 4m

5m6m

}}]6,2,5{},8,4{},9{},11,10,1,7,3[{{M

=(2

3

5

7

9

1

4 68

10 11

1m

2m 3m 4m

5m6m

, 1m)

Srchöder-Hyparcus M-enriched trees and the antipode formula

1

3

45

2 1

3 4 5

2 1

3

4

52

4

3

51

2

12

3

1 2

3

1 3

2

3 2

1

(S )

Antipode equivalent to cassical Lagrange inversion formula

…

Is an epimorphism of Hopf algebras

The epimorphism

Empty cut

Open Problem: the Standard reduced Hopf algebra for other Operads, for example:

Generalizations of the C-K Hopf algebra