AI 中文总结
研究投射$A$-模两项复形的三角范畴$\Ktwo(\cP)$的平方根归一化霍尔代数，建立 PBW 分解和混合乘法公式，证明关系并确定积，构造霍尔 - 福克模，提供有限域霍尔模型，还表明 BGP 反射有相关性质。
AI 中文摘要
设$Q$是在$k = \mathbb{F}_q$上的一个 Dynkin 箭图，$A = kQ$，$\Ktwo(\cP)$是投射$A$-模的两项复形的三角范畴。我们研究$\Ktwo(\cP)$的平方根归一化霍尔代数。首先建立一个 PBW 型向量空间分解为 Ringel - 霍尔部分和移位投射部分，并导出一个明确的混合乘法公式。对于不可分解投射$P_i$，设$p_i$和$z_i$分别是$(0\to P_i)$和$(P_i\to0)$的归一化霍尔类。证明了$z_ip_i = q^{-1}p_iz_i + 1$，并根据映射$P_i\to P_j$的核与余核确定了所有非对角积$z_ip_j$。每对$(p_i,z_i)$生成一个秩为一的量子外尔代数，每个两两 Hom - 正交的投射族生成一个更高秩的量子外尔子代数。为这些子代数构造了明确的霍尔 - 福克模，$p_i$作为产生算子作用，$z_i$作为$q$ - 湮灭算子作用。这提供了一个与唐纳森 - 托马斯理论中的范畴霍尔型外尔作用平行的有限域霍尔模型。最后，证明了 BGP 反射传输相应的反射子代数并保持混合霍尔系数。
英文摘要
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.
