When doing quantum mechanics, a natural object to consider is $\mathcal{U}( \mathbb{H} )$, the set of unitary operators that act on square integrable wavefunctions from the Hilbert space $\mathbb{H}$.
Suppose $\mathbb{H}$ is $L^2(S^1)$, $L^2(\mathbb{R})$, or $L^2(S^2)$. Then $\mathcal{U}(\mathbb{H})$ is an infinite dimensional Lie group (see https://ncatlab.org/nlab/show/infinite-dimensional+Lie+group and What Lie group structure is used for infinite-dimensional Unitary Groups in Quantum Mechanics?).
I am interested in writing down abelian subgroups of $\mathcal{U}(\mathbb{H)}$ for these cases (and how to proceed for more general $L^2(\mathcal{X})$).