Farkhodzhon Arzikulov

arXiv2026

The relationship between local derivations and local automorphisms of some associative algebras

In the present paper, local derivations and local automorphisms of five-dimensional naturally graded nilpotent associative algebras are studied. Namely, a general form of the matrices of local derivations and local automorphisms of algebras $π_2$ and $π_3$ is clarified. It turns out that the general form of the matrix of an automorphism (derivation) on these algebras does not coincide with the local automorphism's (resp. local derivation's) matrix's general form on these algebras. Therefore, these associative algebras have local automorphisms (resp. local derivations) that are not automorphisms (resp. derivations). We also establish a relationship between local automorphisms and local derivations via an exponential expression. We prove that the sets of local derivations of algebras $π_2$ and $π_3$ form Lie algebras with respect to the Lie brackets. Thus, we show that the Lie algebra problem from the Ayupov-Eldique-Kudaybergenov problems for local derivations of the algebras under consideration has a positive solution. The remaining problems from the Ayupov-Eldique-Kudaybergenov problems also have a positive solution for algebras $π_2$ and $π_3$.

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↗