Given an infinite-dimensional Hilbert space $H$, the set $U(H)$ of all unitary operators on $H$ forms a group, known as the unitary group. Now several subgroups of this group play an important role in quantum mechanics, for instance the group of time-translation operators and the group of spatial translation operators. And these subgroups are treated as Lie groups, which I assume means that $U(H)$ is also a Lie group. So my question is, what is the differentiable manifold structure on $U(H)$ used in quantum mechanics?
And what is the topology used in quantum mechanics for this group? Is it the uniform (norm) topology, or the strong operator topology, or what?
EDIT: Assuming the strong operator topology is the right one for quantum mechanics, the second page of this paper mentions "the Frechet Lie group $U(H)$ consisting of all unitary operators on H, equipped with the strong operator topology". That means $U(H)$ has a Frechet manifold structure, i.e. it’s locally isomorphic to an infinite-dimensional Frechet space. (As opposed to an ordinary manifold which is locally isomorphic to a finite-dimensional Euclidean space.) But what is that Frechet manifold structure?