Download One-parameter subgroups and Hilbert`s fifth problem

ONE-PARAMETER SUBGROUPS AND HILBERT'S F I F T H PROBLEM
A. M,
GLEASON
, ,
The affirmative solution of Hilbert's fifth problem requires that we bridge* thé
gap between topologico-algebraic structure and analytic structure. In building
this bridge we quite naturally seek an intermediate island on which to rest the
piers. Such an island is provided by the one-parameter subgroups. One-parameter
subgroups are themselves a topologico-algebraic concept and their existence 'can
be demonstrated, in some cases at least, by the methods of topological algebra,
On the other hand the one-parameter subgroups are perhaps the most striking
feature of a Lie group and it is known that the analytic structure can be recovered from them.
A one-parameter subgroup of a group G is a subgroup which is a (continuous)
homomorphic image of the additive group of real numbers R. We do not require
that the subgroup be closed. The structure of such a subgroup can be quite
complicated, even if the group G is locally compact, a condition which we shall
assume throughout. If we consider only a part of the subgroup corresponding
to a small segment of the reals including 0, the complications vanish and all
such local one-parameter subgroups look the same. In a Lie group there is a
neighborhood U of the identity e such that every element x oî U is on a oneparameter subgroup. Furthermore this one-parameter subgroup is unique if we
require that it go directly from e to re without leaving U. We shall say that a
group has a canonical family of one-parameter subgroups if there exists a neighborhood U with these properties.
Our intermediate goal is to prove that every locally Euclidean group has a
canonical family of one-parameter subgroups. Quite recently great strides have
been made toward this objective.
An important class of groups is the class of those which contain no small subgroups; that is, those which have a neighborhood of the identity containing no
entire subgroup except (e). It has long been known that a group with no small
subgroups contains a one-parameter subgroup. An extension of this result was
made by Chevalley and the author independently: A locally connected group
of finite dimension which contains no small subgroups has a canonical family of
one-parameter subgroups. Unfortunately little is known about the existence or
nonexistence of small subgroups in locally connected groups of finite dimension,
However, under the stronger hypothesis that the group be locally Euclidean,
Newman has shown that there is a neighborhood of the identity containing no
finite subgroup. Smith has extended his investigation and shown that if there
are arbitrarily small subgroups H in a locally Euclidean group, then some of
them must satisfy the implausible relation dim G/H > dim G.
Without the hypothesis concerning small subgroups we can say less, but
still a great deal. The author proved that every n-dimensional group G (n > 0)
451
452
A. M. GLEASON
contains a one-parameter subgroup. Montgomery and Zippin have shown that,
provided G is not compact, this subgoup can be chosen isomorphic to R. The
method of proof, in both cases, is to prove the existence of subgroups of lower
dimension, eventually winding up with a subgroup of dimension one. This
method does not seem applicable to proving that there is a canonical family of
one-parameter subgroups.
On the other side of our island the situation is not so bright. This is to be
expected; it is here that we must make the transition from topological algebra
to analysis. On the analytical side of the channel, the stringency of the conditions leading to analytic structure have gradually been relaxed from requiring
three times differentiable coordinates to certain rather strong Lipschitz conditions; but all conditions have been truly analytic in character, and it seems safe
to say that the first purely analytic result derived by the methods of topological
algebra will prove decisive.
Consider the class $ of homomorphisms of R into G. (There will of course be
many distinct homomorphisms onto each one-parameter subgroup.) If G is a
Lie group, every homomorphism of R into G has the form t —> exp tX where X
is an element of the Lie algebra fl of G ; hence, we may identify $ with ß. Evidently
we should attempt to introduce the structure of a Lie algebra into $.
To introduce the additive structure into $, we again turn to the theory of Lie
groups for our cue. We have the basic formula
(1)
exp (X + Y) = lim (exp X/n exp
Y/n)\
»-»00
To carry this over to $ we must prove that, for any two homomorphisms <pi
and <P2 of R into a locally Euclidean group, lim„-»Qo(ç?i(l/ri)^(l/w))n exists. If
this is true, we can define addition in $ quite simply by
(2)
(fit + <p2)(t) = lim (<pi(t/n)<p2(t/n))n.
n-*oo
The commutativity of this operation is easily proved, but the associativity is
in doubt. Scalar multiplication is defined, of course, by (a<p)(t) = <p(at), and it
satisfies the distributive law with respect to addition.
If these ideas can be carried out, making $ a linear vector space, it will follow
that G is a Lie group. For if G is the center of G, then G/C will be represented
faithfully by the linear transformations induced on <ï> by the inner automorphisms
of G. Hence G is a generalized Lie group, and, being locally Euclidean, it is a
Lie group.
It may be noted that the program here outlined does not actually require
that we have a canonical family of one-parameter subgroups. It is sufficient
that the set of one-parameter subgroups generate G, and this can be shown
to be no essential restriction.
HARVARD UNIVERSITY,
CAMBRIDGE, MASS., U. S.
A.