12 1. Introduction
One can also define algebraic groups such as GLn over the p-adic ratio-
nals Qp; thus for instance GLn(Qp) is the group of invertible n × n matrices
with entries in the p-adics. This is still a locally compact group, and is
certainly not Lie.
Exercise 1.1.3 (Solenoid). Let p be a prime. Let G be the solenoid group
G
:= (Zp×R)/ZΔ,
where
:=
{(n, n) : n Z} is the diagonally embedded
copy of the integers in Zp × R. (Topologically, G can be viewed as the set
Zp ×[0, 1] after identifying (x+1, 1) with (x, 0) for all x Zp.) Show that G
is a compact additive group that is connected but not locally connected (and
thus not a Lie group). Thus one cannot eliminate p-adic type behaviour from
locally compact groups simply by restricting attention to the connected case
(although we will see later that one can do so by restricting to the locally
connected case).
We have now seen several examples of locally compact groups that are
not Lie groups. However, all of these examples are “almost” Lie groups in
that they can be turned into Lie groups by quotienting out a small compact
normal subgroup. (It is easy to see that the quotient of a locally compact
group by a compact normal subgroup is again a locally compact group.) For
instance, a group with the trivial topology becomes Lie after quotienting out
the entire group (which is “small” in the sense that it is contained in every
open neighbourhood of the origin). The infinite-dimensional torus
(R/Z)N
can be quotiented into a finite-dimensional torus
(R/Z)d
(which is of course
a Lie group) by quotienting out the compact subgroup
{0}d ×(R/Z)N;
note
from the definition of the product topology that these compact subgroups
shrink to zero in the sense that every neighbourhood of the group identity
contains at least one (and in fact all but finitely many) of these subgroups.
Similarly, with the p-adic group Zp, one can quotient out by the compact
(and open) subgroups
pjZp
(which also shrink to zero, as discussed above) to
obtain the cyclic groups
Z/pjZ,
which are discrete and thus Lie. Quotienting
out Qp by the same compact open subgroups
pjZp
also leads to discrete
(hence Lie) quotients; similarly for algebraic groups defined over Qp, such
as GLn(Qp). Finally, with the solenoid group G := (Zp ×
R)/ZΔ,
one can
quotient out the copy of
pjZp
× {0} in G for j = 0, 1, 2,... (which are
another sequence of compact subgroups shrinking to zero) to obtain the
quotient group
(Z/pjZ
×
R)/ZΔ,
which is isomorphic to a (highly twisted)
circle R/Z and is thus Lie.
Inspired by these examples, we might be led to the following conjecture:
if G is a locally compact group, and U is a neighbourhood of the identity,
then there exists a compact normal subgroup K of G contained in U such
that G/K is a Lie group. In the event that G is Hausdorff, this is equivalent
to asserting that G is the projective limit (or inverse limit) of Lie groups.
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