# Unitary representation of the Heisenberg group and the universal enveloping algebra

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.

• Where I can find references of "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 $\partial_x,\, ix$ on $f(x)\in L^2(\mathbb{R})$ and the identity operator 1"? Dec 7, 2023 at 2:37