La sphère S3 et les quaternions

S3
S3S1S2
S1
S3S2R3
R2
2×2
R4
2×2
M2(R) 2 ×2
a b
c d 
a, b, c, d
M2(R)
M2(R)
M2(R)
M2(R)
4 4
(x, y)R2
z=x y
−y x 

2
z1+z2=x1+x2y1+y2
−(y1+y2)x1+x2,
λ z
λz =λx λy
−λy λx ,
2
z1z2=x1x2−y1y2y1x2+x1y2
−(y1x2+x1y2)x1x2−y1y2.
M2(R)C
CM2(R) 2
z1z2=z2z1.
z∈C
det(z) = 
x y
−y x 
=x2+y2,
zC
z−1=1
det(z)x−y
y x 
C C
(1,0)
1=1 0
0 1 
(0,1) i
i=0 1
−1 0 

i2=−1
z=x1+yi
x y
zC
z
¯z=x−y
y x 
1
x1=x0
0x
C
z= ¯z
z
z=0y
−y0
2×2
R2
z1(x1, y1)z2
(x2, y2)
(x1, y1)·(x2, y2) = 1
2tr(z1¯z2)
(x, y)z
k(x, y)k2=x2+y2= det(z) = 1
2tr(z¯z)
z
|z| |z1z2|=|z1| |z2|

C U(1)
zC
|z|2=z¯z=x2+y2= 1.
1U(1)
U(1) U(1)
U(1)
U(1) U(1)
S1
CS11
U(1)
tr(z) = 0.
R4
M2(C) 2 ×2
(a, b)
q=a b
−¯ b¯a
2
q1+q2=a1+a2b1+b2
−(b1+b2) ¯a1+ ¯a2,
λ q
λq =λa λb
−λ¯ b λ¯a,
2
q1q2=a1a2−b1¯
b2a1b2+b1¯a2
−a1¯ b2−¯ b2¯a1¯a1¯a2−¯ b1¯ b2.
M2(C)K
C
i0
0−i 0i i0=0−1 1 0 

0i
i0 i0 0−i=0 1 −1 0 
q∈K
det(q) = 
a b
−¯ b¯a
=|a|2+|b|2,
qK
q−1=1
det(q)¯a−b
¯
b a 
K K
4
q
q=t1+xI+yJ+zK
1=1 0
0 1 I=i0 0−iJ=0 1 −1 0 K=0i i0
I2=J2=K2=−1,
I J =−J I =K,J K =−K J =I,K I =−I K =J.
qK
q
¯q=¯a−b
¯
b a 
q
q=t0
0t
tR K
q= ¯q