2026/07/25 by Farkhodzhon Arzikulov, Mirzobek Shodiev
#math.RA #math.AG
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 (\mathfrakm0(n), \mathfrakm2(n), W+(n), \mathfrakm0,1(n), \mathfrakm0,2(n), \mathfrakm0,3(n)) and five one-parameter families (\mathfrakgk,α for k=7,…,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 \mathfrakm0(n) and \mathfrakm0,1(n), the space of local automorphisms strictly encapsulates the group of automorphisms, confirming the existence of pure local automorphisms. Conversely, for \mathfrakm2(n), W+(n), \mathfrakm0,2(n), \mathfrakm0,3(n), and \mathfrakgk,α, the local automorphisms are restricted to an invertible lower triangular matrix form due to rigid power constraints. Furthermore, the sequences \mathfrakm0,1(n), \mathfrakm0,2(n), and \mathfrakm0,3(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.