Number Theory: An Introduction to Mathematics

8 Generalizations 439

intimately related. This was already shown in§6 of Chapter I, but another version of
the proof will now be given.
The setVof all 2×2 matricesvwhich are skew-Hermitian and have zero trace,

v=

(

αβ

−β ̄ α ̄

)

, whereRα= 0 ,

is a three-dimensional real vector space which may be identified withR^3 by
writingα=iξ 3 ,β=ξ 1 +iξ 2 .Anyg∈G=SU( 2 )defines a linear transformation
Tg:v→gvg−^1 ofR^3 .MoreoverTgis an orthogonal transformation, since if

Tgv=v 1 =

(

α 1 β 1

−β ̄ 1 α ̄ 1

)

then, by the product rule for determinants,

|α 1 |^2 +|β 1 |^2 =|α|^2 +|β|^2.

Hence detTg =±1. In fact, sinceTgis a continuous function ofgandSU( 2 )is
connected, we must have detTg=detTe =1foreveryg∈G. ThusTg∈SO( 3 ).
SinceTgh=TgTh,themapg→Tgis a representation ofG.
Every element ofSO( 3 )is represented in this way, since

ifgφ=

(

e−iφ/^20

0 eiφ/^2

)

thenTgφ=Bφ=

⎛

⎝

cosφ sinφ 0

−sinφ cosφ 0

001

⎞

⎠,

ifhθ=

(

cosθ/ 2 −sinθ/ 2

sinθ/2cosθ/ 2

)

thenThθ=Cθ=

⎛

⎝

10 0

0cosθ sinθ

0 −sinθ cosθ

⎞

⎠,

and everyA∈SO( 3 )can be expressed as a productA=BψCθBφ,whereφ,θ,ψare
Euler’s angles.
IfTg=I 3 is the identity matrix, i.e. ifgv=vgfor everyv∈V,theng=±I 2 ,
since any 2×2 matrix which commutes with both the matrices

(

01

− 10

)

,

(

i 0

0 −i

)

must be a scalar multiple of the identity matrix. It follows thatSO( 3 )is isomorphic to
the factor group SU( 2 )/{±I 2 }.
These examples, and higher-dimensional generalizations, can be treated systemat-
ically by the theory of Lie groups. ALie groupis a groupGwith the structure of a
finite-dimensional real analytic manifold such that the map(x,y)→xy−^1 ofG×G
intoGis real analytic.
Some examples of Lie groups are
(i) aEuclidean spaceRnunder vector addition;
(ii) ann-dimensional torus(orn-torus)Tn, i.e. the direct product ofncopies of the
multiplicative groupT^1 of all complex numbers of absolute value 1;