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Harmonic analysis is concentrated mostly on studying locally compact groups. I would like to ask people about examples of non-locally compact topological groups that are interesting in connection with some applications in mathematics (or physics). I must confess that I know only two examples:

• topological vector spaces are studied as examples of topological groups (with the additive group operation) to which Pontryagin duality is sometimes transferred from the class of locally compact abelian groups,




• the groups of diffeomorphisms of smooth manifolds are studied in the theory of infinite dimensional manifolds.

Can people enlighten me about other similar subjects? (If possible, with motivations.)

There are two quite popular areas related to "big" or "large" groups. One concerns extreme amenability and fixed point properties (see Pestov), and the other concerns harmonic analysis and the representation theory (see Borodin - Olshanski).

• 
  Infinite-dimensional Lie groups; e.g. locally convex groups, pro-Lie groups, ind-groups.


• Central extensions thereof; e.g. Virasoro group.


• 
  Loop groups, Current groups.


• Central extensions thereof, Kac-Moody groups.

A couple of common classes of examples you may have overlooked:

• The rationals $mathbb{Q}$ with their usual topology. More generally, any non-closed subgroup of a locally compact Hausdorff group will be a non-locally compact group (in its subspace topology). There are lots of situations where it's convenient to prove something about a locally compact group by working on a dense subgroup.




• Infinite products of non-compact groups with their product topology. Some of these are topological vector spaces like $mathbb{R}^omega$, but for instance, $mathbb{Z}^omega$ is another simple example (it's homeomorphic to Baire space, and to the set of irrationals in $mathbb{R}$). Products arise naturally any time you want to say "give me a whole sequence of these". The failure of such groups to have a Haar measure is a frequent sticking point in analysis and probability.

This is far from the use of topological groups in analysis, but:

In homotopy theory a basic background fact that is that a (based connected) space is always the classifying space of a topological group. In a little more detail: the following four categories (with suitable notions of weak homotopy equivalence) are equivalent for purposes of homotopy theory.

1. Based path-connected spaces


2. Based connected simplicial sets


3. Simplicial groups


4. Topological groups

So in some sense arbitrary topology groups play a role in homotopy theory.

In practice homotopy-theorists work with 2 as a substitute for 1, and with 3 as a tool for 2, but 4 is part of the picture. Roughly speaking, the topological group associated with a based space $X$ is the space of based loops in $X$, but in order to represent this homotopy type by an actual topological group one ordinarily goes through simplicial groups.

Quantum mechanics: The unitary group $U$ of a Hilbert space. If you use the norm topology, then it is a Banach Lie group with bounded skew hermitian operators as Lie algebra. But unbounded self-adjoint operators $A$ (Schrödinger operators) lead to 1-parameter semigroups $e^{itA}$ in $U$, where now the strong operator topology plays a role.

The question mentions topological vector spaces, but aren't Banach spaces a special case of sufficient interest to merit explicit mention? Every infinite dimensional Banach space is a non-locally compact topological group under addition and the norm topology.