# 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 $$\mathbb{C}^\times\times \mathbb{C}^\times$$; 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?

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).