The Euler-twisted Hall algebra

Let $\mathcal{A}$ be a finite-field hereditary Hall category over ${\mathbb{F}}_q$, and choose the positive square root $\nu =\sqrt{q}$. For objects $M,N$, put ${\chi }_0(M,N)={\dim }_{{\mathbb{F}}_q}\mathrm{Hom}(M,N)-{\dim }_{{\mathbb{F}}_q}{\mathrm{YExt}}^1(M,N)$. This integer depends only on the isomorphism classes of $M,N$. At this point ${\chi }_0$ is only an object-level exponent; additivity on $K_0$ and the resulting bicharacter are separate proof obligations. The coefficient field is $\mathbb{Q}(\nu )$. The Euler-twisted Hall algebra is the graded vector space $H_{\mathcal{A},\nu }={⨁}_{[M]}\mathbb{Q}(\nu )[M],\mathrm{deg}[M]=[M]\in K_0(\mathcal{A}),$ with bilinear operation $[M]\star [N]={\nu }^{{\chi }_0(M,N)}{\sum }_{[L]}{g}_{M,N}^L[L],$ where The untwisted Hall algebra of a finitary abelian category fixes $0\to N\to L\to M\to 0$. This operation is distinct from the untwisted product: its field, Euler exponent and square-root parameter are additional data. Finite support and associativity are verified in the following items.