Examen de M024: Groupes et Algèbres de Lie - IMJ-PRG

R[t]
R[t]∂(fg) = ∂(f)g+f∂g
f, g ∈R[t]H=1 0
0−1, X =0 1 0 0, Y =0 0 1 0
sl2(R)
P1, P2∈R[t]∂1:= P1(t)d
dt ∂2:= P2(t)d dt [∂1, ∂2] = ∂1◦∂2−
∂2◦∂1
n= 0,1,2DnP(t)d
dt P
6n
[., .]Dn
D0D1
∂2=t2d
dt , ∂1= 2td dt , ∂0=−d dt D2[∂i, ∂j
D2sl2(R)
sl2(R)
SO(2,1) 2(R)G=SL2(R)
g:= sl2(R)
T=xX +yY +zH, (x, y, z)∈R3gad :g→gl(g)
g
x, y, z ad(T)
g{H, X, Y }
q(T) = −T r(ad(T))2q(T)x, y, z
Ad :G→GL(g)G
g∈G, X ∈gq(Ad(g)(X)) = q(X)
2(R)/±I2
SO(2,1)
SO(2,1)
SO(2,1) 2(R)/±I2
SO4(R)H=R.1⊕R.i ⊕R.j ⊕R.k
R
i·j=k , j ·k=i , k ·i=j , i2=j2=k2=−1,∀x∈H,1·x=x·1 = x

R.1'Rx=a+bi +cj +dk x :=
a−bi −cj −dk ∈H
x·x=x·x=a2+b2+c2+d2
N(x) = √x·xH
x, y ∈Hx·y=y·x N(x·y) = N(x)N(y)
x∈Hφ(x) : H→Hx
v φ(x)(v) := x·v
φHL(H/R)'
M4(R)H
f∈L(H/R)Hf φ(H)
∀v, w ∈Hf(v·w) = f(v)·w
h={z∈H, z +z= 0}=R.i ⊕R.j ⊕R.k
[z1, z2] = z1·z2−z2·z1h
R
h=Dh h
S3={x∈H/ N(x) = 1}H
H
S3
σ φ S3x∈S3σ(x)∈O(4)
σ:S3→O(4) σ(S3)
SO4(R)
σ S3
σ(S3)
Lie(σ(S3)) = φ(h)
hS3
ρ(x) : H→H
v ρ(x)(v) := x·v·x−1SO3(R)
S3/{±1}SO3(R)
SO3(R) SO4(R)
fHf(1) = 1 (x1, x2)∈S3×S3q(x1, x2) : H→H
q(x1, x2)(v) := x1·v·x−1
2
q S3×S3SO4(R)
q
SO4(R)S3×S3/{±(1,1)}
SO3(R)×SO3(R)'SO4(R)/{±Id}.