Overview
This research investigates the relationship between local derivations and local automorphisms within the context of five-dimensional naturally graded nilpotent associative algebras. Specifically, the study focuses on algebras $\pi_2$ and $\pi_3$. The primary aim is to characterize the general form of matrices corresponding to local derivations and local automorphisms for these algebras. A key finding involves the distinction between the general form of matrices for local transformations and their global counterparts (automorphisms and derivations). Furthermore, the study explores the algebraic structure of the sets of local derivations.
Research Context
The study operates within the field of algebra, concentrating on specific types of associative algebras: five-dimensional naturally graded nilpotent associative algebras. These algebraic structures provide a framework for examining the properties of derivations and automorphisms. The work directly addresses certain problems posed by Ayupov, Eldique, and Kudaybergenov concerning local derivations, which serve as a foundational aspect of this research.
Approach
The methodology employed involves a detailed analysis of local derivations and local automorphisms for the specified algebras, $\pi_2$ and $\pi_3$. This includes clarifying the general form of the matrices associated with these transformations. The study then compares these forms to the general forms of matrices for standard automorphisms and derivations on the same algebras. An exponential expression is utilized to establish a relationship between local automorphisms and local derivations. Finally, the research investigates whether the sets of local derivations for $\pi_2$ and $\pi_3$ satisfy the properties of Lie algebras under Lie brackets.
Findings
- The general form of the matrices for local derivations and local automorphisms of algebras $\pi_2$ and $\pi_3$ has been clarified.
- The general form of the matrix of an automorphism on these algebras does not coincide with the local automorphism's matrix's general form.
- The general form of the matrix of a derivation on these algebras does not coincide with the local derivation's matrix's general form.
- These associative algebras ($\pi_2$ and $\pi_3$) possess local automorphisms that are not automorphisms.
- These associative algebras ($\pi_2$ and $\pi_3$) possess local derivations that are not derivations.
- A relationship between local automorphisms and local derivations has been established via an exponential expression.
- The sets of local derivations of algebras $\pi_2$ and $\pi_3$ form Lie algebras with respect to the Lie brackets.
- 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 $\pi_2$ and $\pi_3$.