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