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The Weyl algebra or the algebra of the Canonical Commutation Relations (CCR) is generated by the $p,q$ generators subject to the relation $$ [q, p] = i \hbar I \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1) $$ This is isomporphic to the algebra of the OP under the "change of base": $$ A=\frac{1}{\sqrt{2}}(q-ip), \ \ \ \ B=\frac{1}{\sqrt{2}}(q+ip) $$

The Heisenberg-Weyl algebra or the Weyl algebra or the algebra of the Canonical Commutation Relations (CCR) is generated by the $p,q$ generators subject to the relation $$ [q, p] = i \hbar I \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1) $$ This is isomporphic to the algebra of the OP under the "change of base": $$ A=\frac{1}{\sqrt{2}}(q-ip), \ \ \ \ B=\frac{1}{\sqrt{2}}(q+ip) $$

A first simple and well known result, in the sense of a non-go theorem, can be derived by direct inspection of the commutation relation:

The CCR admit no finite dimensional representations.

This follows easily by a trace argument on both sides of (1) (or of $[A,B]=1$) as long as Planck's constant $\hbar$ is different than zero. So, from the beginning we have to focus in infinite dimensional representations of the Weyl algebra and since the problem is motivated by physical considerations it is natural to consider its generators acting on Hilbert spaces.

Initiating from the above remark, Stone and von Neumann studied the integrated forms of $p,q$ i.e. the unitary, one-parameter groups
$$ \begin{array}{cccccc} U(s) = exp(-i s p) & & & & & V(t) = exp(-i t q) \end{array} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2) $$ The above expressions are well defined because $p, \ q$ are self-adjoint (see 2,3). $V(t)$ acts on $f(x)$ by multiplication with $exp(-itx)$, while $U(s)$ acts by horizontal translations (shifts) of $f(x)$ i.e. $$ \begin{array}{cccccc} (U(s)f)(x) = f(x-s) & & & & & (V(t)f)(x) = e^{-itx}f(x) \end{array} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3) $$ for any square integrable complex function of a real variable, i.e. for any $f(x) \in L_{2}(- \infty, \infty)$. The operators $U(s)$, $\ \forall s \in \mathbb{R}$ και $V(t)$, $\ \forall t \in \mathbb{R}$, are unitary and thus bounded. Their domain is now the whole of the Hilbert space (and not only a dense subspace as was the case for the $p,q$ generators of the Weyl algebra).
Furthermore, $U(s)$, $V(t)$ satisfy $$ U(s)V(t) = e^{ist} V(t)U(s) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4) $$ for all $s \in \mathbb{R}$ and for all $t \in \mathbb{R}$. (4) is frequently called the integrated form of the Weyl relations or the integrated form of the CCR.

Initiating from the above remark, Stone and von Neumann studied the integrated forms of $p,q$ i.e. the unitary, one-parameter groups
$$ \begin{array}{cccccc} U(s) = exp(-i s p) & & & & & V(t) = exp(-i t q) \end{array} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2) $$ The above expressions are well defined because $p, \ q$ are self-adjoint (see 2,3). $V(t)$ acts on $f(x)$ by multiplication with $exp(-itx)$, while $U(s)$ acts by horizontal translations (shifts) of $f(x)$ i.e. $$ \begin{array}{cccccc} (U(s)f)(x) = f(x-s) & & & & & (V(t)f)(x) = e^{-itx}f(x) \end{array} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3) $$ for any square integrable complex function of a real variable, i.e. for any $f(x) \in L_{2}(- \infty, \infty)$. The operators $U(s)$, $\ \forall s \in \mathbb{R}$ and $V(t)$, $\ \forall t \in \mathbb{R}$, are unitary and thus bounded. Their domain is now the whole of the Hilbert space (and not only a dense subspace as was the case for the $p,q$ generators of the Weyl algebra).
Furthermore, $U(s)$, $V(t)$ satisfy $$ U(s)V(t) = e^{ist} V(t)U(s) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4) $$ for all $s \in \mathbb{R}$ and for all $t \in \mathbb{R}$. (4) is frequently called the integrated form of the Weyl relations or the integrated form of the CCR.

Remark 1: Formally, the transition between the CCR (1) and their integrated form (3), (4) can be understood as follows: $\bullet$ If we start from the CCR (1) together with (2) written in power series: $$ \begin{array}{cccc} U(s) = exp(-i s p) = \sum_{n=0}^{\infty}\frac{(-isp)^{n}}{n!} & & & V(t) = exp(-i t q) = \sum_{n=0}^{\infty}\frac{(-i t q)^{n}}{n!} \\ \\ \end{array} \ \ \ \ \ \ \ (5) $$ then one can deduce the integrated form of the CCR (3), (4) by applying the Baker–Campbell–Hausdorff formula. $\bullet$ For the converse, the CCR (1) follow formally on taking $\frac{\partial^2}{\partial t \partial s}$ of (4) at $s=t=0$.
However, this description is rather an indication than a strict proof of equivalence between the Weyl algebra and its integrated form: The reason is that $p, \ q$ are unbounded operators and hence cannot be represented by power series on the whole of the space. Consequently, in general, the integrated form of the Weyl relations is not equivalent (at least not in some obvious way) with the Weyl relations themselves (i.e. the CCR).

Remark 1: Formally, the transition between the CCR (1) and their integrated form (3), (4) can be understood as follows: $\bullet$ If we start from the CCR (1) together with (2) written in power series: $$ \begin{array}{cccc} U(s) = exp(-i s p) = \sum_{n=0}^{\infty}\frac{(-isp)^{n}}{n!} & & & V(t) = exp(-i t q) = \sum_{n=0}^{\infty}\frac{(-i t q)^{n}}{n!} \\ \\ \end{array} \ \ \ \ \ \ \ (5) $$ then one can deduce the integrated form of the CCR (3), (4) by applying the Baker–Campbell–Hausdorff formula. $\bullet$ For the converse, the CCR (1) follow formally on taking $\frac{\partial^2}{\partial t \partial s}$ of (4) at $s=t=0$.
However, this description is rather an indication than a strict proof of equivalence between the Weyl algebra and its integrated form: The reason is that $p, \ q$ are unbounded operators and hence the power series expressions cannot be valid on the whole of the space. Consequently, in general, the integrated form of the Weyl relations is not equivalent (at least not in some obvious way) with the Weyl relations themselves (i.e. the CCR).