Algèbres de Lie et Représentations, 1ère partie (Cours

sl2
g
α
∆(g,h)
K→Q(→R)
g

K
sl2glnsontnun
g
g⇔g∗sln
sl2
sl2
r f(vi) = vi+1 e(vi) = µivi−1
sl2
sl2
Ker(e)⊂Pi>0Vi(h)Vi(h)h
i
dim g<∞K K =K
<∞
[g,gi]⊂gi+1
z(g)6={0}
g gl(V)
Vg:= {v∈V|g·v={0}}
(V, σ)gσ(x)x∈g
(Vi)ig·Vi⊂Vi−1
g⇔adx∀x∈g

g
g⇔a g/a
Vg a g a g
Vg(Vi)ig·Vi⊂Vi
g⇔D(g) := [g,g]
(V, σ)gβσ:g×g→K
(x, y)7→ tr(xy)g
σ(g)⇔βσ(D(g),g) = {0} ⇔ βσ(D(g), D(g)) = {0}
Lg:g×g→Ka g La= (Lg)|a×a
g⇔βσ(D(g),g) = {0} ⇔ βσ(D(g), D(g)) = {0}
rad g a g rad(g/a) = {0}
rad gD(g)L
a g rad a=a∩rad g
dim g>1
rad g={0}
slnsontnsl
g⇔L⇔g
{0}
⇒
g g
g=D(g)
g
Homg(V, W ) = {V→W}Endg(V) := Homg(V, V ) HomK(V, W ) :=
{KV→W}glK(V) := HomK(V, V )
Pgλ∈ P Vλ
multV(Vλ) = dimKHomg(Vλ, V ) = dimKHomg(V, Vλ)
gHomK(V, W )

g→VgVder(g, V )
der(g)ider
g
der(g, V ) = ider(g, V )
V
g⇔g
x=xs+xn
µ∈Kx∈gl(V)V(µ)(x)Vµ(x)
x V µ Vµ(a)a
gl(V)gVgµ∈a∗g(µ)(h)gµ(h)
[g(λ),g(mu)]⊂g(λ+µ)[g(λ),g(mu)]⊂g(λ+µ)
g∈g(adg(g))s= adg(gs) (adg(g))n= adg(gn)
g⊂gl(V)
h⊂g h adg
h⇒h
h g =Pα∈h∗gα(h)
L|gα×gβ={0}α6=−β
L|gα×g−α
h⇒cg(g0(h)) := {x∈g|[x, g0(h)] = 0}
h⊂g⇔h h =g0(h)
g
L|h×h
∆(g,h)
g=h⊕Lα∈∆∆ = −∆
hαα(·) = L(hα,·)h∗×h∗
<·,·>
[gα,g−α] = Khαα(hα)6= 0
Hα=2
L(hα,hα)hαsl2(eα, Hα, fα)
dim gα= 1 sα:= gα⊕[gα,g−α]⊕g−αsl2
L(h, h0) = Pα∈∆α(h)α(h0)< β, γ >=Pα∈∆< α, β >< α, γ >

α
aβα := β(Hα)=2<β,α>
<α,α>
∀α, β ∈∆, aβα ∈Z{t∈Z|β+tα ∈∆∪ {0}} = [[−p, q]] p, q ∈Np−q=aβα
β−aβαα∈∆
aβα
α∈∆Kα∩∆ = {±α}α, β, β +α∈∆ [gα,gβ] = gα+β
1n−1−1
1
∃a∈V, ∃!a∨∈V∗a∨(x)=2 s=sa,a∨:x7→ x−a∨(x)a
V=h∗sα,Hαα∈∆
sα,Hα(∆) = ∆
charK= 0 K=K K V
∆⊂V
∆V0/∈∆
∀α∈∆∃α∨∈V∗α∨(α)=2 sα,α∨(∆) = ∆
∀α, β ∈∆, α∨(β)∈Z
Kα∩∆ = {±α}
α∨
∆(g,h)h∗
V
(·|·)
α sα,α∨sα
α−α
α∨(β)(β|α)
(α|α)
⇒
2 3
2G2
∆(g,h)
exp(adx)x∈g
exp(adx)g
Aute(g)g
Aute(g)