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