L`exponentielle de SLn(C) n`est pas surjective

SLn(C)
GGLn(C)
g={M∈Mn(C)t∈R,exp(tM)∈G}
GLn(C)gln(C) = Mn(C)
det(exp(M)) = exp(tr(M)) SLn(C)
sln(C) = {M∈Mn(C) tr(M) = 0 }
exp: gln(C)→GLn(C)
sln(C)→SLn(C)
DU ×
D+N D
U
U−Id
N n
n D +N
DU D
U= exp(N)
U−Id = N+1
2N2+· · · =N(Id +1 2N+. . .
| {z }
inversible
)
P=D+N DN =ND exp(P) = exp(D) exp(N)D0= exp(D)
U= exp(N)D0U=UD0
N n
N N
Nn−16= 0 x∈CnNn−1(x)6= 0
{1, x, N(x), . . . , Nn−1(x)}Cn
y∈Cny=PaiNi(x)x N
P(N) = PaiNi
C N Q
C(x) = Q(N)(x)C=Q(N)C(y) = Q(N)(y)
y y =P(N)(x)P
C(y) = (C◦P(N))(x)=(P(N)◦C)(x)

C N N
· · · =P(N)(C(x)) = P(N)(Q(N)(x)) = Q(N)(P(N)(x)) = Q(N)(y)
C=Q(N) = q0+q1N+· · · +qdNd≥1
N
C q0
U∈Mn(C)n λ
n M := λU ∈SLn(C)λ6= 1
M
Pexp(P) = M
P=D+N D N DN =ND exp(P) =
exp(D) exp(N)×M=λU
exp(D) = λexp(N) = U
N U n
D µ tr(D) = nµ tr(P) = tr(D)=0
N µ = 0 D= 0 exp(P) = exp(N)
M=λexp(N)
{ } → { }
n
u u u