8 Generalizations 441For every real Lie algebraL, there is a simply-connected Lie groupG ̃such that
L(G ̃)=L. Moreover,G ̃is uniquely determined up to isomorphism byL. A connected
Lie groupGhasL(G)=Lif and only ifGis isomorphic to a factor groupG ̃/D,
whereDis a discrete subgroup of the centre ofG ̃.
ALie subgroupof a Lie groupGis a real analytic submanifoldHofGwhich
is also a Lie group under the restriction toHof the group structure onG.Itmaybe
shown that a subgroupHof a Lie groupGis a Lie subgroup if it is a closed subset of
G, and is a connected Lie subgroup if and only if it is path-connected. Thus any closed
subgroup ofGL(n)is a Lie group.
IfHis a Lie subgroup of the Lie groupG,thenL(H)is a Lie subalgebra ofL(G).
Moreover, ifMis a Lie subalgebra ofL(G), there is a unique connected Lie subgroup
HofGsuch thatL(H)=M.
IfG 1 ,G 2 are Lie groups, then a mapf:G 1 →G 2 is aLie group homomorphism
if it is an analytic map, regardingG 1 ,G 2 as manifolds, and a homomorphism, regard-
ingG 1 ,G 2 as groups. It may be shown that any continuous mapf:G 1 →G 2 which
is a group homomorphism is actually a Lie group homomorphism. (It follows that a
locally Euclidean topological group can be given the structure of a Lie group in only
one way.)
IfL 1 ,L 2 are Lie algebras, then a mapT:L 1 →L 2 is aLie algebra homomor-
phismif it is linear andT[u,v]=[Tu,Tv]forallu,v∈L 1 .IfG 1 ,G 2 are Lie groups
and iff :G 1 →G 2 is a Lie group homomorphism, then the derivative offat the
identity,f′(e):L(G 1 )→L(G 2 ), is a Lie algebra homomorphism. Moreover, ifG 1 is
connected then distinct Lie group homomorphisms give rise to distinct Lie algebra ho-
momorphisms, and ifG 1 is simply-connected then every Lie algebra homomorphism
L(G 1 )→L(G 2 )arises from some Lie group homomorphism. (In particular, the rep-
resentations of a connected Lie group are determined by the representations of its Lie
algebra.)
A Lie algebraLisabelianif [u,v]=0forallu,v∈L. A connected Lie group
is abelian if and only if its Lie algebra is abelian. Since the Euclidean spaceRnis
a simply-connected Lie group with ann-dimensional abelian Lie algebra, it follows
that anyn-dimensional connected abelian Lie group is isomorphic to a direct product
Rn−k×Tk(whereTkis ak-torus) for someksuch that 0≤k≤n.
Anidealof a Lie algebraLis a vector subspaceMofLsuch thatu∈Land
v∈Mimply [u,v]∈M. A connected Lie subgroupHof a connected Lie groupGis
a normal subgroup if and only ifL(H)is an ideal ofL(G).
A Lie algebraLissimpleif it has no ideals except{ 0 }andLand is not one-
dimensional, andsemisimpleif it has no abelian ideal except{ 0 }. It may be shown that
a Lie algebra is semisimple if and only if it is the direct sum of finitely many ideals,
each of which is a simple Lie algebra.
A Lie group issemisimpleif it is connected and has no connected abelian normal
Lie subgroup except{e}. It follows that a connected Lie groupGis semisimple if and
only if its Lie algebraL(G)is semisimple.
We turn our attention now to compact Lie groups. It may be shown that a compact
topological group can be given the structure of a Lie group if and only if it is finite-
dimensional and locally connected. Furthermore, a compact Lie group is isomorphic
to a closed subgroup ofGL(n)for somen. Other basic results are: