I am studying the Heisenberg group with the Lie algebra generators $\{ U,V,W \}$ and the structure $[U,W]=[V,W]=0$ and $[U,V]=W$.
This group has an infinite-dimensional unitary representation on the Hilbert space $L^2(\mathbb{R}\to\mathbb{C})$ of square integrable complex functions. The corresponding generators of the Heisenberg group of this representation are the anti-Hermitian linear operators $\partial_x$, $\mathcal{i}x$ on $f(x) \in L^2$ and the identity operator $1$.
My questions are now the following:
1) How are $\partial_x$ and $\mathcal{i}x$ operators on $L^2$? They are both unbounded and their image should be a superset of $L^2$. How does that work with the choice of the Hilbert space for the representation?
2) Is it correct to say that the universal enveloping algebra generated by $\{\partial_x,\mathcal{i}x\}$ is dense in the set of continuous linear operators on $L^2$? If yes, how can I show this? If not, what is the space of the closure of the enveloping algebra?
3) Can someone recommend a good literature source for the topic? My current knowledge is built from several incomplete puzzle pieces that I have encountered in the mathematical physics literature.
