AI 中文总结
研究最大长度的有限维完全分次复丝状李代数族的局部和 2 - 局部自同构空间，利用代数过滤等方法，通过分析多个序列和族，明确了不同结构中局部及 2 - 局部自同构的形式特点，划分出非线性与严格线性结构的边界。
AI 中文摘要
本文旨在基于既定分类框架和代数过滤方法，完整描述最大长度的有限维完全分次复丝状李代数族的局部和 2 - 局部自同构空间。系统研究了六个无限序列和五个单参数族。分析利用内部交换边界并在特定参数坐标子空间上构造非线性、非加法变换。证明了对于某些结构局部自同构严格包含自同构群，存在纯局部自同构；而对于另一些结构，局部自同构受刚性幂约束限制为可逆下三角矩阵形式。还表明部分序列有纯非线性 2 - 局部自同构，其余结构严格线性，2 - 局部自同构与真正自同构重合，在最大长度丝状李代数中划分出非线性变换结构与严格线性结构的清晰边界。
英文摘要
This paper aims to provide a complete description of the spaces of local and 2-local automorphisms for the families of finite-dimensional totally graded complex filiform Lie algebras of maximum length, building upon established classification frameworks and algebraic-filtration methods. We systematically investigate six infinite sequences ($\mathfrak{m}_0(n)$, $\mathfrak{m}_2(n)$, $W^+(n)$, $\mathfrak{m}_{0,1}(n)$, $\mathfrak{m}_{0,2}(n)$, $\mathfrak{m}_{0,3}(n)$) and five one-parameter families ($\mathfrak{g}_{k,α}$ for $k=7,\dots,11$). The analysis utilizes internal commutation boundaries and constructs non-linear, non-additive transformations on specialized parametric coordinate subspaces. We prove that for the structures $\mathfrak{m}_0(n)$ and $\mathfrak{m}_{0,1}(n)$, the space of local automorphisms strictly encapsulates the group of automorphisms, confirming the existence of pure local automorphisms. Conversely, for $\mathfrak{m}_2(n)$, $W^+(n)$, $\mathfrak{m}_{0,2}(n)$, $\mathfrak{m}_{0,3}(n)$, and $\mathfrak{g}_{k,α}$, the local automorphisms are restricted to an invertible lower triangular matrix form due to rigid power constraints. Furthermore, the sequences $\mathfrak{m}_{0,1}(n)$, $\mathfrak{m}_{0,2}(n)$, and $\mathfrak{m}_{0,3}(n)$ are shown to possess pure non-linear 2-local automorphismsThe remaining investigated structures adhere strictly to linearity, forcing every 2-local automorphism to coincide with a genuine automorphism. This establishes a clear boundary between structures allowing non-linear transformations and those maintaining strict linearity within filiform Lie algebras of maximum length.
Comments34 pages
