442 X A Character Study
(i) a compact Lie group, and even any compact topological group, has only finitely
many connected components;
(ii) a connected compact Lie group is abelian if and only if it is ann-torusTnfor
somen;(iii) a semisimple connected compact Lie groupGhas a finite centre. Moreover the
simply-connected Lie groupG ̃such thatL(G ̃)=L(G)is not only semisimple
but also compact;
(iv) an arbitrary connected compact Lie groupGhas the formG=ZH,whereZ,H
are connected compact Lie subgroups,His semisimple andZis the component
of the centre ofGwhich contains the identitye.
These results essentially reduce the classification of arbitrary compact Lie groups
to the classification of those which are semisimple and simply-connected. It may be
shown that the latter are in one-to-one correspondence with the semisimplecom-
plexLie algebras. Since a semisimple Lie algebra is a direct sum of finitely many
simple Lie algebras, we are thus reduced to the classification of the simple com-
plex Lie algebras. The miracle is that these can be completely enumerated: the
non-isomorphic simple complex Lie algebras consist of the four infinite families
An(n≥ 1 ),Bn(n≥ 2 ),Cn(n≥ 3 ),Dn(n≥ 4 ), of dimensionsn(n+ 2 ),n( 2 n+ 1 ),
n( 2 n+ 1 ),n( 2 n− 1 )respectively, and fiveexceptionalLie algebrasG 2 ,F 4 ,E 6 ,E 7 ,E 8
of dimensions 14, 52, 78, 133, 248 respectively.
To the simple complex Lie algebra An corresponds the compact Lie group
SU(n+ 1 )of all matrices inU(n+ 1 )with determinant 1; toBncorresponds the
compact Lie groupSO( 2 n+ 1 )of all matrices inO( 2 n+ 1 )with determinant 1; to
Cncorresponds the compact Lie groupSp(n)(whose matrices all have determinant 1),
and toDncorresponds the compact Lie groupSO( 2 n)of all matrices inO( 2 n)with
determinant 1. The groupsSU(n)andSp(n)are simply-connected ifn≥2, whereas
SO(n)is connected but has index 2 in its simply-connected covering groupSpin(n)if
n≥5. The compact Lie groups corresponding to the five exceptional simple complex
Lie algebras are all related to the algebra ofoctonionsor Cayley numbers.
Space does not permit consideration here of the methods by which this classifi-
cation has been obtained, although the methods are just as significant as the result.
Indeed they provide a uniform approach to many problems involving the classical
groups, giving explicit formulas for the invariant mean and for the characters of all
irreducible representations. There is also a notable connection withgroups generated
by reflections.
The classification of arbitrary semisimple Lie groups reduces similarly to the clas-
sification of simplerealLie algebras, which have also been completely enumerated.
The irreducible unitary representations of non-compact semisimple Lie groups have
been extensively studied, notably by Harish-Chandra. However, the non-compact case
is essentially more difficult than the compact, since any nontrivial representation is
infinite-dimensional, and the results are still incomplete. Much of the motivation for
this work has come from elementary particle physics where, in the original formula-
tion of Wigner (1939), a particle (specified by its mass and spin) corresponds to an
irreducible unitary representation of the inhomogeneous Lorentz group.