1. Introductio n

The most importan t functio n i n Analysis certainly i s the exponentia l functio n x i- e x,

and it s most fundamenta l propert y (fro m whic h al l others may b e derived ) is the functiona l

equation

(1) e x+y = e xey.

It mus t b e interpreted i n term s of group theory : x H * e* is a homomorphism o f th e

additive grou p R (or C ) into th e multiplicativ e grou p R * (o r C*) . I f we restrict th e expo -

nential t o purel y imaginar y value s id o f x, th e comple x numbe r e lQ ma y be identified wit h

the rotatio n

(x, y)

H

(x co s 0 - y si n 0, x si n 0 + y co s 0)

of angl e 0 in R

2,

an d th e equatio n e

l{Q +e'*

= e

i6eld'may

b e writte n

/cos(0 + 0' ) sin( 0 + 0') \ _ / co s 0 si n »\ /co s 0' si n 0' \

\-sin(0 + 0' ) cos( 0 + e')J y-si n 0 co s 0/\-si n 0 ' co s 0')'

One is thus led t o generaliz e thes e propertie s b y considerin g a Lie group G (which in

case (2) will be the uni t circl e U ) and a mapping

(3) s h + U(s) = (U if(s))

where, for eac h sEG, U(s ) is an n x n invertibl e matri x wit h complex element s dependin g

on 5, and fo r an y tw o element s s, t i n G, we have

(4) U(st) = U(s)U(f )

or equivalentl y

(5) U

if

(st) = £ U

ik

{s)Ukfit) fo r 1 / , / n.

k=l

Such a mapping is called a /wazr representation o f G of dimension (o r degree) n. I t i s a

group homomorphism o f G into th e grou p GL(n , C ) of invertibl e n x n comple x matrices ,

and fro m genera l group theor y on e has

(6) u(e ) = l

n

(e neutral elemen t o f G , l

n

uni t matrix) ,

(7) U( s

-1

) = (U(s))

_1fo

r al l s G G.

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