The motivic Hall algebra of a finite quiver

Let $Q=(Q_0,Q_1,s,t)$ be a finite quiver over a field $k$. A representation of $Q$ assigns a finite-dimensional $k$-vector space $V_i$ to each vertex $i\in Q_0$ and a linear map $V_{s(a)}\to V_{t(a)}$ to each arrow $a\in Q_1$. Its dimension vector is $\mathbf{d}=(\dim V_i{)}_{i\in Q_0}\in {\mathbb{N}}^{Q_0}$.

For $\mathbf{d}\in {\mathbb{N}}^{Q_0}$ put ${\mathrm{Rep}}_{\mathbf{d}}(Q)={\prod }_{a\in Q_1}{\mathrm{Hom}}_k(k^{d_{s(a)}},k^{d_{t(a)}}),G_{\mathbf{d}}={\prod }_{i\in Q_0}{\mathrm{GL}}_{d_i}(k),$and let ${\mathfrak{M}}_{\mathbf{d}}(Q)=[{\mathrm{Rep}}_{\mathbf{d}}(Q)\mathrm{/}{G}_{\mathbf{d}}]$ be the quotient Artin stack. It is finite type and has affine stabilizers: the representation variety and the acting group are finite type and affine, and each stabilizer is a closed subgroup of $G_{\mathbf{d}}$. For dimension vectors $\mathbf{d},\mathbf{e}$, let ${\mathfrak{E}}_{\mathbf{d},\mathbf{e}}(Q)$ be the stack of short exact sequences of representations$0⟶B_{\mathbf{e}}⟶E_{\mathbf{d}+\mathbf{e}}⟶A_{\mathbf{d}}⟶0.$The endpoints are part of the data, and isomorphisms are isomorphisms of the whole exact sequence. Equivalently, after choosing the standard $\mathbf{e}$-dimensional subspaces at each vertex, this is the quotient by the product parabolic preserving those subspaces of the affine space of block upper-triangular arrow maps. Thus ${\mathfrak{E}}_{\mathbf{d},\mathbf{e}}(Q)$ is a finite-type Artin stack with affine stabilizers. More explicitly, on a local splitting $E_i=B_i\oplus A_i$, the arrow $a:i\to j$ has block matrix$\left(\begin{array}{cc}{B}_a& {\xi }_a\\ 0& A_a\end{array}\right),{\xi }_a:A_i\to B_j\mathrm{.}$Changing the splitting by $h_i:A_i\to B_i$ changes ${\xi }_a$ to ${\xi }_a+B_a{h}_i-h_j{A}_a$. This is the translation action of the vector bundle ${⨁}_i\mathrm{Hom}(A_i,B_i)$ on ${⨁}_{a:i\to j}\mathrm{Hom}(A_i,B_j)$ used in the extension stack calculation. Let $q:{\mathfrak{E}}_{\mathbf{d},\mathbf{e}}\to {\mathfrak{M}}_{\mathbf{d}}$ send a sequence to its quotient, $s:{\mathfrak{E}}_{\mathbf{d},\mathbf{e}}\to {\mathfrak{M}}_{\mathbf{e}}$ send it to its subrepresentation, and $m:{\mathfrak{E}}_{\mathbf{d},\mathbf{e}}\to {\mathfrak{M}}_{\mathbf{d}+\mathbf{e}}$ send it to its middle representation. The Hall correspondence is

& \mathfrak E_{\mathbf d,\mathbf e}(Q) \ar[dl,"(q,s)"'] \ar[dr,"m"] & \\ \mathfrak M_{\mathbf d}(Q)\times\mathfrak M_{\mathbf e}(Q) && \mathfrak M_{\mathbf d+\mathbf e}(Q)

Its arrows record the ordered quotient, subrepresentation, and middle object of the displayed short exact sequence; the diagram fixes the convolution convention. Define$H_Q:={⨁}_{\mathbf{d}\in {\mathbb{N}}^{Q_0}}\mathcal{K}({\mathfrak{M}}_{\mathbf{d}}(Q))\mathrm{.}$For $\alpha \in \mathcal{K}({\mathfrak{M}}_{\mathbf{d}})$ and $\beta \in \mathcal{K}({\mathfrak{M}}_{\mathbf{e}})$, define$\alpha \ast \beta :=m_{!}((q,s{)}^{\ast }(\alpha ⊠\beta ))\in \mathcal{K}({\mathfrak{M}}_{\mathbf{d}+\mathbf{e}})\mathrm{.}$ Thus the first factor is the quotient and the second factor is the subrepresentation, so the sequence is $0\to B\to E\to A\to 0$. Each homogeneous product is defined by one finite-type stack correspondence; no completion or finiteness of a sum over isomorphism classes is required. This is the stack-function product without a potential or a vanishing-cycle weight.