440 X A Character Study
(iii) thegeneral linear group G L(n)of all real nonsingularn×nmatrices under matrix
multiplication;
(iv) theorthogonal group O(n)of all matricesX∈GL(n)such thatXtX=In;
(v) theunitary group U(n)of all complexn×nmatricesXsuch thatX∗X=In,
whereX∗is the conjugate transpose ofX;(U(n)may be viewed as a subgroup of
GL( 2 n))
(vi) theunitary symplectic group S p(n)of all quaternionn×nmatricesXsuch that
X∗X=In,whereX∗is the conjugate transpose ofX.(Sp(n)maybeviewedas
a subgroup ofGL( 4 n))
The definition implies that any Lie group is a locally compact topological group.
The fifth Paris problem of Hilbert (1900) asks for a characterization of Lie groups
among all topological groups. A complete solution was finally given by Gleason,
Montgomery and Zippin (1953): a topological group can be given the structure of a
Lie group if and only if it islocally Euclidean, i.e. there is a neighbourhood of the
identity which is homeomorphic toRnfor somen.
The advantage of Lie groups over arbitrary topological groups is that, by replacing
them by their Lie algebras, they can be studied by the methods oflinearanalysis.
A real (resp. complex)Lie algebrais a finite-dimensional real (resp. complex) vec-
tor spaceLwith a map(u,v)→[u,v]ofL×LintoL, which is linear inuand inv
and has the properties
(i) [v,v]=0foreveryv∈L,
(ii) [u,[v,w]]+[v,[w,u]]+[w,[u,v]]=0forallu,v,w∈L. (Jacobi identity)
It follows from (i) and the linearity of the bracket product that[u,v]+[v,u]=0forallu,v∈L.An example of a real (resp. complex) Lie algebra is the vector spacegl(n,R)(resp.
gl(n,C))ofalln×nreal (resp. complex) matricesXwith [X,Y]=XY−YX.Other
examples are easily constructed as subalgebras.
ALie subalgebraof a Lie algebraLis a vector subspaceMofLsuch thatu∈M
andv∈Mimply [u,v]∈M. Some Lie subalgebras ofgl(n,C)are
(i) the setAnof allX∈gl(n+ 1 ,C)with trX=0,
(ii) the setBnof allX∈gl( 2 n+ 1 ,C)such thatXt+X=0,
(iii) the setCnof allX∈gl( 2 n,C)such thatXtJ+JX=0, where
J=
(
0 In
−In 0)
,
(iv) the setDnof allX∈gl( 2 n,C)such thatXt+X=0.
The manifold structure of a Lie groupGimplies that with eachs∈Gthere is asso-
ciated a real vector space, thetangent spaceats. The group structure of the Lie group
Gimplies that the tangent space at the identityeofGis a real Lie algebra, which will
be denoted byL(G). For example, ifG=GL(n)thenL(G)=gl(n,R). The proper-
ties of Lie groups are mirrored by those of their Lie algebras in the following way.