Representation of Weyl-Heisenberg Lie Algebra

Question

I'm reading the book: Coherent State in Quantum Physics, by Jean-Pierre Gazeau. In the page no. 35 of this book, the Weyl-Heisenberg Lie Algebra $\mathfrak{w}_m$ has been given that $$\mathfrak{w}_m=linear\,span\{iQ,iP,iI_d\};$$ where $Q=\frac{a+a^\dagger}{\sqrt{2}}$, $P=\frac{a-a^\dagger}{i\sqrt{2}}$ and $I_d$ is the identity operator with $a$ is annihilation operator and $a^\dagger$ is creation operator. One can see that a generic element of $\mathfrak{w}_m$ is written as $$\mathfrak{w}_m\ni X=isI_d+i(pQ-qP)=isI_d+(za^\dagger-\overline{z}a);$$ where $s\in\mathbb{R}$ and $z=\frac{q+ip}{\sqrt{2}}$. Here $X$ is the anti-self-adjoint and the infinitesimal generator of the unitary operator: $$e^X = e^{is}e^{za^\dagger-\overline{z}a}:=e^{is}D(z).$$ After this some discussion on the action of the D-Function. My question is, in the page no. 37, it is said that the map $X\longmapsto e^X=e^{is}D(z)$ is irreducible unitary representation of Weyl-Heisenberg Algebra, but first of all it, to say this map $X\longmapsto e^X=e^{is}D(z)$ is representation of Weyl-Heisenberg Algebra, I think it should satisfy the relation $$e^{[X,Y]}=[e^X,e^Y],$$ for all $X,Y \in \mathfrak{w}_m$, since it is Lie algebra. But I'm unable to obtain this relation. Could anyone please help me...!!! I'm looking for a explanation. Thank you in advance for any help.

Answer 1

M. Tharan

First I am grateful to you for reading my 2009 book and to help me to clarify some points in its content. Actually, what is written in page 37 is exactly the following

The operators X, through the map $X \mapsto e^X = e^{is}D(z)$, thus define an irreducible unitary representation of the Weyl–Heisenberg algebra $\mathfrak{w}$.

I did not write that the map itself is a representation of the algebra.

I wished to mean that the representation of the algebra itself comes from infinitesimal generators of the representation of the group. In an updated version I will improve that sentence in order to make the point clearer. Thank you again!

More informations are given in the recent arXiv:1703.08443v1 [quant-ph]

WEYL-HEISENBERG INTEGRAL QUANTIZATION(S): A COMPENDIUM