Yes. My original statement was that harmonic analysis is limited to Lie groups. jsteinhardt observed that any locally compact abelian group can have harmonic analysis done on it—some of these (say, p-adic groups) are not Lie groups, since they have no smooth structure, though they are still topological groups.
So I was trying to be less specific by changing my term from Lie group to topological group.
You can do harmonic analysis on any locally compact abelian group, see e.g. Pontryagin duality.
“locally compact” implies you have a topology—maybe I should be saying “topological group” rather than “Lie group,” though.
All Lie groups already have a topology. They’re manifolds, after all.
Yes. My original statement was that harmonic analysis is limited to Lie groups. jsteinhardt observed that any locally compact abelian group can have harmonic analysis done on it—some of these (say, p-adic groups) are not Lie groups, since they have no smooth structure, though they are still topological groups.
So I was trying to be less specific by changing my term from Lie group to topological group.
Oh. That makes more sense.