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.
1
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