Mirzobek Shodiev | Searcharxiv

arXiv2026

Local and 2 local $\frac{1}{2}$-derivation of $n$-dimensional totally graded filiform Lie algebras

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.

math.RA↗

arXiv2026

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

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.

math.RA↗