Endomorphismes semi-simples

E k n
u∈L(E)F⊂E u
u
u
u
u k
u µu=Pα1
1. . . P αr
r
Ei= ker(Pαi
i(u))
F⊂E u F =⊕F∩Ei
F∩Ei
F x ∈F x =x1+· · · +xrxi∈Ei
πi:E→Eix7→ xiu F u
πixi=πi(x)∈F
⇐⇒ ⇐⇒
⇒u µu
µu=P2Q(P Q)(u)=0
u P 2Q F = ker(P(u)) S u F
a= (P Q)(u)
•a F a = (QP )(u) = Q(u)◦P(u)
•a S y ∈S a(y)∈F P (u)[a(y)] = (P2Q)(u)(y) = 0
a(y)∈S S u a u
a(y)∈F∩S a(y) = 0 F∩S= 0
a= 0 µu
⇐µu=P1. . . PrPi
F Ei= ker(Pi(u))
F=⊕F∩Eii
F∩EiEiF E
µu|Ei=Piµu=P
F=E G = 0 x∈E−F
k ϕ:k[X]→E
Q7→ Q(u)(x)

Gx={Q(u)(x), Q ∈k[X]}Px
F∩Gx= 0 Gx0, Gx00
Gx⊕Gx0⊕Gx00 ...
µu=P Px|P P
y∈F∩Gxy=Q(u)(x)P|Q
y= 0 F∩Gx= 0 P Q
UP +V Q = 1 ϕ
E
U(u)[P(u)(x)
| {z }
=0
] + V(u)[Q(u)(x)
| {z }
y
] = x
V(u)(y)∈F F u x ∈E−F
u=d+n
car(k) = 0 car(k) = p > 0
x7→ xpu∈L(E) (s, n)
u=s+n
s n
sn =ns
k=RMn(R)
Mn(C)a
a
u=d+nC
u=u u =d+n d =d
n=n d n Rd
K µu
k k G K k
EL(E⊗K) (n, n)
K G L(E⊗K)
k=KGL(E⊗K)
GL(E)
K u =d+n d n L(E⊗K)
σ∈G uσ=u u ∈L(E)uσ=dσ+nσ
dσnσd+n
dσ=d nσ=n d n G L(E)
s=d K k
u=s+n