TITLE: Can the Quantum Torus be realized as a Hall Algebra?
QUESTION [11 upvotes]: 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 $\mathbb{C}^\times\times \mathbb{C}^\times$; 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 $M,N\in A$ and $i\in \mathbb{Z}$.  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
$$0\rightarrow N\rightarrow R\rightarrow M\rightarrow 0$$
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?

REPLY [10 votes]: As far as I understand, the Hall algebra of a category (say, with finite length of objects) is graded by the Grothendieck monoid of this category, spanned by simple objects over $\Bbb Z_+$, and it must have the ground field in degree $0$. The quantum torus algebra does not seem to have such a grading (it has a $\Bbb Z^2$-grading, not a $\Bbb Z_+^m$-grading). Maybe one should ask this question for the q-Weyl algebra $xy=qyx$ (not allowing negative powers of x and y)? Note that this algebra appears as a subalgebra of a Hall algebra (the Hall algebra for the quiver $A_2$ is $U_q(n_+)$, where $n_+$ is the nilpotent subalgebra of $sl(3)$; the q-Weyl algebra is generated by $e_{12}$ and $e_{13}$ inside this algebra).