Let A be the path algebra of a Dynkin quiver Q over a finite field, and $\mathcal{P}$ be the category of projective A-modules. Denote by $C^1(\mathcal{P})$ the category of 1-cyclic complexes over $\mathcal{P}$, and ${\tilde{\mathfrak{n}}}^{+}$ the vector space spanned by the isomorphism classes of indecomposable and non-acyclic objects in $C^1(\mathcal{P})$. In this paper, we prove the existence of Hall polynomials in $C^1(\mathcal{P})$, and then establish a relationship between the Hall numbers for indecomposable objects therein and those for A-modules. Using Hall polynomials evaluated at 1, we define a Lie bracket in ${\tilde{\mathfrak{n}}}^{+}$ by the commutators of degenerate Hall multiplication. The resulting Hall Lie algebras provide a broad class of nilpotent Lie algebras. For example, if Q is bipartite, ${\tilde{\mathfrak{n}}}^{+}$ is isomorphic to the nilpotent part of the corresponding semisimple Lie algebra; if Q is the linearly oriented quiver of type ${\mathbb{A}}_n$, ${\tilde{\mathfrak{n}}}^{+}$ is isomorphic to the free 2-step nilpotent Lie algebra with n-generators. Furthermore, we give a description of the root systems of different ${\tilde{\mathfrak{n}}}^{+}$. We also characterize the Lie algebras ${\tilde{\mathfrak{n}}}^{+}$ by generators and relations. When Q is of type $\mathbb{A}$, the relations are exactly the defining relations. As a byproduct, we construct an orthogonal exceptional pair satisfying the minimal Horseshoe lemma for each sincere non-projective indecomposable A-module.

令A为 Dynkin quiver Q在有限域上的路径代数，并且$\mathcal{磷}$是射影A 模的范畴。表示为$C^1(\mathcal{磷})$ 1-循环配合物的范畴 $\mathcal{磷}$， 和 ${\stackrel{～}{\mathfrak{n}}}^{+}$ 由不可分解和非循环对象的同构类跨越的向量空间 $C^1(\mathcal{磷})$. 在本文中，我们证明了霍尔多项式在$C^1(\mathcal{磷})$，然后建立其中不可分解物体的霍尔数与A-模数的关系。使用计算为 1 的霍尔多项式，我们在${\stackrel{～}{\mathfrak{n}}}^{+}$由简并霍尔乘法的换向器。由此产生的霍尔李代数提供了广泛的幂零李代数。例如，如果Q是二分的，${\stackrel{～}{\mathfrak{n}}}^{+}$与相应的半单李代数的幂零部分同构；如果Q是类型的线性定向箭袋${\mathbb{一种}}_n$, ${\stackrel{～}{\mathfrak{n}}}^{+}$同构于具有n 个发生器的自由 2 步幂零李代数。此外，我们还描述了不同的根系统${\stackrel{～}{\mathfrak{n}}}^{+}$. 我们还刻画了李代数${\stackrel{～}{\mathfrak{n}}}^{+}$通过生成器和关系。当Q是类型时$\mathbb{一种}$，这些关系正是定义关系。作为副产品，我们为每个真实的非投影不可分解的A 模构造了满足最小马蹄引理的正交异常对。