Subobject-first Hall completion and quantum tori for finite quivers

Fix $Q$, $k={\mathbb{F}}_q$, and $Z$ as in Finite-field stability data for an acyclic quiver. For a representation $M$, write $a_M=\mathrm{\mid }{\mathrm{Aut}}_{{\mathcal{A}}_Q}(M)\mathrm{\mid }$ and let ${\delta }_M$ be the basis function of its isomorphism class. Let ${\mathcal{S}}_Q$ be the small finitary abelian skeleton constructed in Finite acyclic quiver representations: heredity, Euler form, and finite-field finitarity. Define $H_Q:=H_{{\mathcal{S}}_Q,\mathbb{Q}}$, identifying its basis with the isomorphism classes of ${\mathcal{A}}_Q$. Fullness and transport of subobjects and cokernels along isomorphisms preserve the Hall coefficients. Thus $H_Q$ is the direct sum over those classes, with rational coefficients and the quotient-first untwisted Hall product of The untwisted Hall algebra of a finitary abelian category. In that convention $[A]{\ast }_{qf}[B]$ counts sequences $0\to B\to L\to A\to 0$, and identify the basis symbol $[M]$ with the characteristic function ${\delta }_M$. Define the subobject-first product by $f\star g:=g{\ast }_{qf}f$. Then $(f\star g)(L)={\sum }_{U\subseteq L}f(U)g(L\mathrm{/}U),$ where the sum is over subobject classes of $L$: monomorphisms into $L$ modulo isomorphisms of their domains commuting with the monomorphism. Distinct embedded subobjects are counted separately even when their domains are isomorphic. Put ${\kappa }_M=a_M{\delta }_M$.

For $\mathbf{d}\in {\mathbb{N}}^{Q_0}$, let $H_{Q,\mathbf{d}}$ be the span of classes of dimension vector $\mathbf{d}$. It is finite-dimensional: the representation set on the standard spaces $k^{d_i}$ is finite. Define the dimension completion ${\widehat{H}}_Q={\prod }_{\mathbf{d}\in {\mathbb{N}}^{Q_0}}{H}_{Q,\mathbf{d}}\mathrm{.}$ The product is extended degreewise; each coefficient uses finitely many decompositions $\mathbf{d}=\mathbf{a}+\mathbf{b}$. There is no analytic convergence assertion. The all-object and phase factors are $A_{\mathrm{a}\mathrm{l}\mathrm{l}}={\sum }_{[M]}{\delta }_M,S_Z(\varphi )={\delta }_0+{\sum }_{\begin{array}{c}[M]\\ M\mathrm{\ne }0,{\varphi }_Z(M)=\varphi ,MZ\text{-semistable}\end{array}}{\delta }_M\mathrm{.}$ Each is an element of ${\widehat{H}}_Q$ because each homogeneous component has finite support.

Let $t=\sqrt{q}>0$ and $K_q=\mathbb{Q}(\sqrt{q})$. Take the completed Euler torus and skew torus over $K_q$, with bases $X^{\mathbf{d}}$ and $x^{\mathbf{d}}$, respectively, and products $X^{\mathbf{d}}{X}^{\mathbf{e}}=q^{-{\chi }_Q(\mathbf{e},\mathbf{d})}{X}^{\mathbf{d}+\mathbf{e}},x^{\mathbf{d}}\ast x^{\mathbf{e}}=t^{-⟨\mathbf{e},\mathbf{d}⟩}{x}^{\mathbf{d}+\mathbf{e}}\mathrm{.}$ Their completions use the same dimension product. The next theorem proves associativity and gives the continuous Euler and skew integration maps.