16
$\begingroup$

I know the definitions of Lie group and topological group are different. Can you give me an example of topological group which is not a Lie group.

$\endgroup$
22
$\begingroup$

Another example is the rationals under addition with the subspace topology induced from inclusion in $\mathbb R$. Since the rationals are countable, they can't be a manifold of dimension exceeding $0$, so the only possibility would be a $0$ manifold. However, the topology on the rationals is not discrete, so they do not form a $0$-manifold either.

$\endgroup$
13
$\begingroup$

The Cantor set is a topological group. It is homeomorphic to $\{0,1\}^{\omega}$ in the product topology, which is a topological vector space over $\mathbb{Z}_2$. As the set is totally disconnected and not discrete it is easy to see that it cannot be a manifold.

$\endgroup$
9
$\begingroup$

The $p$-adic numbers are a topological group, but not a Lie group.

There are many profinite groups which are not Lie groups, for example the profinite group completion of a knot group.

$\endgroup$
  • $\begingroup$ (But of course, the $p$-adic numbers provide an example of a $p$-adic Lie group). $\endgroup$ – Watson Nov 26 '18 at 13:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.