发表机构
V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences; Andijan State University; Bukhara State University（乌兹别克斯坦科学院V.I.罗曼诺夫斯基数学研究所; 安集延国立大学; 布哈拉国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文完整刻画了最大长度完全分次纤维型李代数上的$\frac{1}{2}$-导子，证明了局部$\frac{1}{2}$-导子的线性刚性，并利用边界行参数构造了非线性的2-局部$\frac{1}{2}$-导子，确定了刚性失效的边界。
AI 中文摘要
本文对最大长度的$n$维完全分次复纤维型李代数上的$\frac{1}{2}$-导子、局部$\frac{1}{2}$-导子和2-局部$\frac{1}{2}$-导子给出了完整的代数描述。基于Janez Bernik（2020）建立的基础分类框架，我们系统地确定了六个无限结构序列（$m_0(n)$、$m_2(n)$、$W^+(n)$、$m_{0,1}(n)$、$m_{0,2}(n)$、$m_{0,3}(n)$）和五个例外单参数族（$g_{7,\alpha}$至$g_{11,\alpha}$）的$\frac{1}{2}$-导子向量空间。通过参数矩阵系统分析逐点局部求值方程，我们建立了局部$\frac{1}{2}$-导子的结构线性和刚性。相反，我们证明了位于$\frac{1}{2}$-导子矩阵边界行中的独立参数提供了足够的自由度来绕过线性约束。利用这些边界构型，我们借助一次齐次函数$f(z_1, z_2) = z_1^3 / (z_1^2 + z_2^2)$显式构造了纯非线性和非加性的2-局部$\frac{1}{2}$-导子，从而定义了局部刚性失效的精确边界。
英文摘要
This article provides a complete algebraic description of $\frac{1}{2}$-derivations, local $\frac{1}{2}$-derivations, and 2-local $\frac{1}{2}$-derivations on $n$-dimensional totally graded complex filiform Lie algebras of maximum length. Based on the foundational classification framework established by Janez Bernik (2020), we systematically determine the vector spaces of $\frac{1}{2}$-derivations for the six infinite structural sequences ($m_0(n)$, $m_2(n)$, $W^+(n)$, $m_{0,1}(n)$, $m_{0,2}(n)$, $m_{0,3}(n)$) and the five exceptional one-parameter families ($g_{7,α}$ through $g_{11,α}$). By analyzing the pointwise local evaluation equations via parametric matrix systems, we establish the structural linearity and rigidity of local $\frac{1}{2}$-derivations. In contrast, we demonstrate that the independent parameters residing in the boundary rows of the $\frac{1}{2}$-derivation matrices provide sufficient degrees of freedom to bypass linearity constraints. Exploiting these boundary configurations, we explicitly construct pure non-linear and non-additive 2-local $\frac{1}{2}$-derivations leveraging the homogeneous function of degree one, $f(z_1, z_2) = z_1^3 / (z_1^2 + z_2^2)$, thereby defining the exact boundary where local rigidity fails.
Comments25 pages
