Local and 2-Local automorphisms of n-dimensional totally graded filiform Lie algebras

Abstract

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 (${\mathfrak{g}}_{k,\alpha }$ 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 ${\mathfrak{g}}_{k,\alpha }$, 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.

34 pages