Cours introductif de M2 Groupes et Algèbres de Lie - IMJ-PRG

n(R)
sl2(R)

C∞
C∞
G
m:G×G−→ G, (x, y)7→
xy i :G−→ G, x 7→ x−1
C∞
m i
C∞m i
(R,+)
(R×,·) (C×,·) (S1,·)
G=n(R)C∞Mn(R)'Rn2
(A, B)7→ AB A B C∞
A7→ A−1=1
Acom(A)C∞
VR(V) := AutR(V)
n V V (V)∼ −→
n(R)
n(C)VC
(V) := C(V)
G g ∈G Lg:G−→ G, x 7→ gx
g G
deLg:TeG∼ −→ TgG
G
Gϕ −→ G0

C∞
:n(R)−→ R×
(R,+) exp −→ (R×
>0,·)
θ∈(R,+) 7→ exp(2iπθ)∈S1
θ∈(R,+) 7→ (exp(2iπθ),exp(2iπθα)) ∈
S1×S1α∈R\Q
λ: (R,+) −→ G
ρ:G−→ (V)G
V V ρ
ϕ dhϕ=
de0Lϕ(h)◦deϕ◦(deLh)−1ϕ
H G
H →G
H=ZG=R
H=S1G=C×
i:H →G
α
S1×S1(R,+)
H G
H H h H
UG H ∩ U U
G H
H g H H G
Ue G U ∩ HU
U U ∩ U−1U=U−1U

gUg gU ∩ H6=∅h
g∈hU−1∩H=hU ∩ H hUhU ∩ H
hU ∩ H hU
hU ∩ H=h(U ∩ H)hU U
H
G
H H G
H G
ϕ:H−→ G
ϕ
Γϕ:H→H×G
h7→ (h, ϕ(h)) .
HΓϕ(H)
pr1|Γϕ(H)H×Gpr1 −→ HΓϕ(H) =
{(h, g)∈H×G, ϕ(h) = g}Γϕ(H)
H×G pr1pr1|Γϕ(H)
Γϕ
pr2ϕ=pr2◦Γϕ
deϕ:TeH−→ TeG
ϕ e deϕ
ϕ
e ϕ deϕ
e ϕ ϕ
ϕ

KR C V
K n
(V)−→ 1(K)
(V)
V V
n(K)
n(K) 1
µn(K)n K∗n K =C
K=R
SymK(V)K K
Φ : V×V−→ K
(V) SymK(V)gΦ(v, w) := Φ(g−1v, g−1w) Φ
GL(V)
O(Φ) := {g∈(V),Φ(gv, gw) = Φ(v, w),∀v, w ∈V}.
Φ Φ
(V)
(V)K=C
O(Φ)
O(n, C) = {M∈Mn(C),tMM =In}.
K=Rp6n
e1,· · · , en
Φ n
X
i=1
xiei,
n
X
i=1
yiei!=
p
X
i=1
xiyi−
n
X
i=p+1
xiyi.
(p, q := n−p) Φ O(Φ)
O(p, q) := M∈Mp+q(R),tM.Dp,q.M =Dp,q 
Dp,q =D(1,· · · ,1,−1,· · · ,−1) 1 p−1
q p =n p = 0 O(n, 0) = O(n)
O(p, q) = O(q, p) O(p, q)q6p
xx0+yy0+zz0−tt0R4
O(3,1)