Quantum Weyl Relations arising from Two-Term Complexes

Let $Q$ be a Dynkin quiver over $k=\mathbb F_q$, 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 $P_i$, let $p_i$ and $z_i$ be the normalized Hall classes of $(0\to P_i)$ and $(P_i\to0)$, respectively. We prove $$z_ip_i=q^{-1}p_iz_i+1,$$ and determine all off-diagonal products $z_ip_j$ in terms of kernels and cokernels of maps $P_i\to P_j$. Hence each pair $(p_i,z_i)$ 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 $p_i$ act as creation operators and the $z_i$ 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.