Algèbres de Hopf combinatoires

G×G→G
A
m:A⊗A→A
. . . m
m◦(m⊗I) = m◦(I⊗m) : A⊗A⊗A→A
C
∆ : C→C⊗C
(∆ ⊗I)◦∆=(I⊗∆) ◦∆ : C→C⊗C⊗C
H
∆ : H→H⊗H m :
H⊗H→H

H n K A =K[x1, . . . , xn]
∆(xi) = 1 ⊗xi+xi⊗1
∆ : A→A⊗A∼K[x0
1, . . . , x0 n, x00 1, . . . , x00 n]
∆P(x0
1, . . . , x0 n, x00 1, . . . , x00 n) = P(x0 1+x00 1, . . . , x0 n+x00 n)
H∗⊗H∗⊂(H⊗H)∗
m∗:H∗→(H⊗H)∗H∗⊗H∗
∆(x) = 1 ⊗x+x⊗1
x
x
ξ1, . . . , ξn
ξ1, . . . , ξn
ξ1, . . . , ξn
(x1. . . xp)(y1. . . yq) = x1. . . xpy1. . . yq
∆(ξi) = 1 ⊗ξi+ξi⊗1
x1. . . xp
∆(x1. . . xp) = Xxi1. . . xim⊗xim+1 . . . xip

i1. . . ip[1, p]
ξ1, . . . , ξn
∆(ω) = X
ω=ω0ω00
ω0⊗ω00
x1. . . xpxp+1 . . . xp+q=X
σ∈S(p)
p+q
xσ(1) . . . xσ(p+q)
σ p
f(x) = x+c2x2+c3x3+. . .
f g f ◦g(x) =
f(g(x))
F B =K[c2, c3, . . .]ci
K[c2, c3, . . .]
∆(ck) = ck(f◦g)
f(x) = x+a2x2+a3x3+. . . , g(x) = x+b2x2+b3x3+. . .
ai=ci⊗I, bi=I⊗ci∆(ck)∈F B ⊗F B
c
Pc(t)Rc(t)
∆(t) = e⊗t+t⊗e+X
c
Pc(t)⊗Rc(t)

t
ss
sssss
s s ss
s
s
AAAA
AA
QQQ
s s
s s
s
s
AA⊗ s s
s
s
s
sAA AA
J
J
Γ
γΓ//γ
Γγ
∆(Γ) = X
γ
(Γ//γ)⊗γ
ζ(s) = Pn1
nsζ(n)
ζ(3,2) = P0<m<n 1
m3n2ζ(n1, . . . , nk)