Algèbres semi-simples et théorème de Wedderburn

R06=RM R
06=RM M {0}M
RM R N M
RN0N⊕N0=M
Z
06=RN < M M n ∈N\ {0}
Rn ∼
=R/L L =annRn I
L I/L
R/L R/L ∼
=Rn I/L In
Rn M M =In ⊕P
P x ∈Rn x =αn +x0
α∈I, x0∈P x0=x−αn ∈Rn ∩P
Rn =In ⊕(P∩Rn).
In Rn P ∩Rn
N

M=Pi∈IMi
J⊆I M =
⊕j∈JMj
F:= {E⊆I|Pi∈EMiest une somme directe }
i∈I,{i} ∈ F F 6=∅ F
(Iλ)λ∈ΛFJ=∪λ∈ΛIλ
Fj0∈J x ∈Mj0∩Pi∈J\{j0}Mix=
xi1+xi2+... +xil∈Mj0ij∈J\ {j0}xij∈Mijµ∈Λ
{j0,i1, . . . ,il} ⊆ IµPs∈IµMsx= 0
Pj∈JMjJ∈ F
K⊆IPk∈KMk
i∈I, Mi⊆Pk∈KMkMi
Mi∩Pk∈KMkMi
K Mi∩Pk∈KMk=Mi
Mi⊆Pj∈JMjM=Pj∈JMj=Mi⊕j∈JMj
RM R
M
M
M
1) ⇒2) M1M
M=M1⊕M2M26= 0 M2
M1
2) ⇒3)
3) ⇒1) M=⊕i∈IMiMiN < M N 6=M
F:= {J⊆I| ⊕j∈JMj∩N= 0}
RM R EndR(M)
M M
R n ∈NM R

M(n)M⊕M⊕ ··· ⊕ M n M
R M,M1,M2, . . . ,MnR
EndR(M(n))∼
=Mn(EndR(M)) .
i6=j, HomR(Mi,Mj) = 0
EndR(
n
M
i=1
Mi)∼
=
n
Y
i=1
EndR(Mi).
M1M2R
M:= M1=M2EndR(M)
M1M2HomR(M1,M2) = 0
M2
φ
∼
=M1HomR(M1,M2)
f.g := f◦φ◦g
R
RR
R
R R
R∼
=Qs
i=1 Mni(Ki)Ki
k G
G k kG := {f:Glongrightarrowk ||{x∈
G|f(x)6= 0}| <∞} k f ∈kG f
f=Pg∈supp(f)f(g)g kG ={Pfinie αxx}

f=Pαxx g =Pβyy kG
sup(f)∪sup(g)
f.g =X
x∈supp(f),y∈supp(g))
αxβyxy =X
z
(X
(x,y)|xy=z
αxβy)z
kG G
k G
G k kG
chark |G|

V k
G V
ρ:G−→ GL(V).
dimV
BV n
G GLn(k)
ρ:G−→ GL(V)G µ :G−→
GLn(k) : g7→ MB(ρ(g))
ρ G V
G
G V g.v =ρ(g)(v)
V
C
k=C
G
V
ρ:G−→ Gl(V) : g7→ IdV.
reg G C
V=⊕g∈GCeg{eg|g∈G}V