miguel méndez (ivic-ucv) jean carlos liendo (ucv) caracas, venezuela
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/1.jpg)
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.
![Page 2: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/2.jpg)
Families of labelled combinatorial structures
71 68
9
6 91 8
5
24
3
![Page 3: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/3.jpg)
Combinatorial Species
•
•
•
•
![Page 4: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/4.jpg)
Operations
![Page 5: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/5.jpg)
Product
![Page 6: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/6.jpg)
Disjoint union
Product
![Page 7: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/7.jpg)
![Page 8: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/8.jpg)
Substitution
Structures inside other structures
![Page 9: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/9.jpg)
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
![Page 10: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/10.jpg)
![Page 11: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/11.jpg)
-enriched Rooted trees
![Page 12: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/12.jpg)
“ 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
![Page 13: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/13.jpg)
Two monoidal categories
![Page 14: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/14.jpg)
c-Monoids
Related to associative algebras via the Schur functor.
![Page 15: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/15.jpg)
c-Operads
![Page 16: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/16.jpg)
159 8
3
2 46
7
915
83
2 46
7
The operad of finite sets
![Page 17: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/17.jpg)
Permutative associative operad
![Page 18: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/18.jpg)
![Page 19: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/19.jpg)
1
2 3
69
7
5
1
2
35
7
69
![Page 20: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/20.jpg)
![Page 21: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/21.jpg)
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
![Page 22: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/22.jpg)
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
![Page 23: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/23.jpg)
![Page 24: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/24.jpg)
Natural extension
![Page 25: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/25.jpg)
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
![Page 26: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/26.jpg)
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
![Page 27: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/27.jpg)
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
![Page 28: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/28.jpg)
![Page 29: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/29.jpg)
Incidence Coalgebras
Reduced incidence Coalgebras
![Page 30: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/30.jpg)
![Page 31: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/31.jpg)
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
![Page 32: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/32.jpg)
The Natural incidence Hopf algebra
![Page 33: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/33.jpg)
= + 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
![Page 34: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/34.jpg)
Free commutative algebra generated by all the unlabelled trees
![Page 35: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/35.jpg)
“ C-monoids, Moebius Species and Coalgebras”
M. Mendez Ph. D. thesis, 1989.
“Moebius Species” M. Mendez, J. Yang.Adv. In Math. 1991
![Page 36: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/36.jpg)
Chapoton-Livernet (2007)
![Page 37: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/37.jpg)
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
![Page 38: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/38.jpg)
1
3
45
2 1
3 4 5
2 1
3
4
52
4
3
51
2
![Page 39: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/39.jpg)
![Page 40: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/40.jpg)
12
3
1 2
3
1 3
2
3 2
1
(S )
![Page 41: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/41.jpg)
![Page 42: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/42.jpg)
Antipode equivalent to cassical Lagrange inversion formula
![Page 43: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/43.jpg)
…
Is an epimorphism of Hopf algebras
![Page 44: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/44.jpg)
The epimorphism
Empty cut
![Page 45: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/45.jpg)
![Page 46: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/46.jpg)
![Page 47: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/47.jpg)
![Page 48: Miguel Méndez (IVIC-UCV) Jean Carlos Liendo (UCV) Caracas, Venezuela](https://reader036.vdocuments.net/reader036/viewer/2022081506/56813fdd550346895daac543/html5/thumbnails/48.jpg)
Open Problem: the Standard reduced Hopf algebra for other Operads, for example:
Generalizations of the C-K Hopf algebra