geometric rep theory - Can the Quantum Torus be realized as a Hall Algebra?

Background

The Quantum Torus

Let $q$ be an arbitrary complex number, and define (the algebra of) the quantum torus to be

$$T_q:=mathbb{C}langle x^{pm 1},y^{pm 1}rangle/xy-qyx$$
For $q=1$, this is the commutative ring of functions on the torus $mathbb{C}^timestimes mathbb{C}^times$; hence, for general $q$, this is regarded as a quantization of the torus.

Hall Algebras

Consider a small abelian category $A$, with the property that $Hom_A(M,N)$ and $Ext^i_A(M,N)$ are always finite sets for any $M,Nin A$ and $iin mathbb{Z}$. Let $overline{A}$ denote the set of isomorphism classes in $A$, and let

$$H(A)=oplus_{[M]in overline{A}}mathbb{C}[M]$$
denote the complex vector space spanned by $overline{A}$. Endow $H(A)$ with a multiplication by the formula
$$[M]cdot [N]=sqrt{langle [M],[N]rangle)}sum_{[R]in overline{A}}frac{a_{MN}^R}{|Aut(M)||Aut(N)|}[R]$$
where $a_{MN}^R$ is the number of short exact sequences
$$0rightarrow Nrightarrow Rrightarrow Mrightarrow 0$$
and
$$langle [M],[N]rangle = sum (-1)^i |Ext^i_A(M,N)|$$
is the Euler form. This multiplication makes $H(A)$ into an associate algebra called the Hall algebra of $A$; the proof can be found e.g. here.

Finite Fields and Quantization

The categories $A$ appearing in the construction of a Hall algebra are usually linear over some finite field $mathbb{F}_q$. Often, it is possible to simultaneously define a category $A_q$ for each finite field $mathbb{F}_q$; usually by considering modules on the $mathbb{F}_q$-points of some scheme over $mathbb{Z}$. The corresponding Hall algebras $H(A_q)$ will then usually be closely related, and can often be defined by relations that are functions in $q$.

The Question

I know that there are cases where an algebra is deformed by a parameter $q$, and then the resulting family of algebras `magically' coincides with a family of Hall algebras $H(A_q)$ in the special cases when $q$ is a prime power. I think this happens in the case of the Hecke algebra (discussed here), and the case of quantum universal enveloping algebras (discussed here). I somewhat understand that this is a symptom of a related convolution algebra on the scheme used to define $A_q$.

Is there a family of categories $A_q$ such that the corresponding Hall algebras $H(A_q)$ are isomorphic to the Quantum Torus $T_q$ for all $q$ a prime power? If so, is there a convolution algebra realization of the Quantum Torus?

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