Groupes et Géométrie

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001
100
010
?
H1, H2R3σ1, σ2
σ2σ1H1H2
H1, H2Rns1, s2H1, H2
φ∈SO(n)φ(H1) = H2
H1⊕H⊥
1=Rn=H2⊕H⊥ 2
φs1φ−1=s2
G[a, b] = aba−1b−1a, b ∈G
[G, G]G[O(n), O(n)] ⊆SO(n)
s1s2O(n) [O(n), O(n)] = SO(n)
Gln(R)Rn
G Gln(R)Sn−1
H G (1,0,...,0) ∈RnG/H ∼
=Sn−1
SO(n)G H ∩SO(n) = SO(n−1)
SO(n)/SO(n−1) ∼
=Sn−1
PR3
ˇ
P={x∈R3|<x, y >≤1∀y∈P}.
ˇ
PR3
Pˇ PR3ˇ
ˇ
P=P.
Pˇ P
P
Pˇ P
Pˇ P
Pˇ P
TR3IT
T
ITTITS4
IT∼
=S4.
S4∼
=Idet
−→ Z/2ZT
CR3IC
ICO(3) I+
C:= IC∩SO(3)
IC4C φ :IC→S4
Kerφ={±id}φ
(φ, det) : IC→S4×Z/2Z
I+
CS4
T1T2C
CIT1IT2IT1=IT2
ICS4I+
C