2026/07/24 by Qinghua Chen, Yonggang Hu
#math.RT
Let Q be a Dynkin quiver over k=\mathbb Fq, let A=kQ, and let \Ktwo(\cP) be the extriangulated category of two-term complexes of projective A-modules. We study the square-root normalized Hall algebra of \Ktwo(\cP). We first establish a PBW-type vector-space factorization into the Ringel--Hall part and the shifted-projective part, and derive an explicit mixed multiplication formula. For the indecomposable projectives Pi, let pi and zi be the normalized Hall classes of (0→ Pi) and (Pi→0), respectively. We prove zipi=q-1pizi+1, and determine all off-diagonal products zipj in terms of kernels and cokernels of maps Pi→ Pj. Hence each pair (pi,zi) generates a rank-one quantum Weyl algebra, while every pairwise Hom-orthogonal family of projectives generates a higher-rank quantum Weyl subalgebra. For these subalgebras we construct explicit Hall--Fock modules, on which the pi act as creation operators and the zi act as q-annihilation operators. This provides a finite-field Hall model parallel to categorical Hall-type Weyl actions in Donaldson--Thomas theory. Finally, we show that BGP reflection transports the corresponding reflection subalgebras and preserves the mixed Hall coefficients.