2. Representation s of SU(2 )
The group SU(2 ) is the grou p o f 2 x 2 complex matrice s o f determinan t 1, which
leave invarian t th e hermitian for m x lxl + x 2x2\ the y ar e exactly th e matrice s
•-(:"')
where th e comple x numbers a, b are linked b y th e relatio n \a\ 2 4 - \b\ 2 = 1. Thi s group is
isomorphic t o th e grou p S
3
o f quaternion s o f nor m 1, the isomorphis m s \-+ q
s
bein g de-
fined by q
s
= a + bf (for th e usual embedding o f the field C in the field of quaternion s H),
the verificatio n o f th e relatio n q ss, = q sqs, bein g immediate. I f on e remember s that , when
R3 i s identified wit h th e spac e of pure quaternion s (i.e . those suc h that q = -q) b y th e
mapping (x
v
x
2
, x
3
) i- xxi + x
2
f + x
3
k, an y rotation ca n be writte n p(q): x h- qxq~ l,
where q i s a quaternion o f nor m 1, determined u p to it s sign, one obtain s a surjective homo -
morphism s (- p(qs) o f SU(2 ) onto th e grou p SO(3 , R) of rotation s i n R
3;
th e kerne l o f
that homomorphis m bein g the grou p {-1,1} o f 2 elements (whic h is the cente r o f SU(2)) ,
SO(3, R) is isomorphic t o th e quotien t SU(2)/{-I , I} . Thi s isomorphism ca n be made mor e
precise b y usin g the descriptio n o f p(q): i f q = a 4 - t, wher e a E R an d t i s a pure quaterni -
on ¥" 0, p(q) i s the rotatio n whos e axi s is the lin e through t, an d th e angl e 6 is given by
tg(0/2) = \\t\\lx (i n the plan e orthogona l t o t
y
oriente d i n suc h a way that i f (c
lf
c 2) i s a
sequence o f tw o vectors in tha t plan e whic h is direct, the sequenc e (c
lf
c 2, t) i n R
3
i s di-
rect). W e shall in particular nee d th e fact s that , whe n
(13)
p(qs) i s the rotatio n o f angl e 2ip around Ox x, an d whe n
cos 6 i sin 6
(14) s
K
i sin 6 co s 0
p(qs) i s the rotatio n o f angl e -26 aroun d Ox
3
.
The grou p SU(2 ) is clearly a subgroup o f th e grou p SL(2 , C) of al l complex 2 x 2
matrices o f determinan t 1. No w there i s a simple wa y o f definin g representation s o f arbi-
trary degre e TV + 1 of SL(2 , C): consider th e comple x vecto r spac e EN
+ l
of homogeneous
polynomials o f degre e TV in two comple x variables , with comple x coefficient s
4
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