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

Farkhodzhon Arzikulov, Utkir Khakimov, Abduqaxxor Qurbonov

TL;DR

The paper analyzes local automorphisms and local derivations of two five‑dimensional naturally graded nilpotent associative algebras π₂ and π₃ with C(A)=(3,2). It provides explicit matrix characterizations for local maps, proves that local automorphisms and local derivations can exist beyond ordinary automorphisms and derivations, and establishes an exponential correspondence between local derivations and local automorphisms for these algebras. It further shows that LocDer(π₂) and LocDer(π₃) are Lie algebras, yielding a positive solution to the Lie algebra aspect of the Ayupov–Eldique–Kudaybergenov problems, while LocAut(π₂) forms a Lie group but LocAut(π₃) does not. These results illuminate the interplay between local and global structure in finite‑dimensional algebras and highlight distinct group/ manifold properties between the two algebras.

Abstract

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 other problems from the Ayupov-Eldique-Kudaybergenov problems have a positive (negative) solution for algebra $π_2$ (resp. algebra $π_3$)

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

TL;DR

The paper analyzes local automorphisms and local derivations of two five‑dimensional naturally graded nilpotent associative algebras π₂ and π₃ with C(A)=(3,2). It provides explicit matrix characterizations for local maps, proves that local automorphisms and local derivations can exist beyond ordinary automorphisms and derivations, and establishes an exponential correspondence between local derivations and local automorphisms for these algebras. It further shows that LocDer(π₂) and LocDer(π₃) are Lie algebras, yielding a positive solution to the Lie algebra aspect of the Ayupov–Eldique–Kudaybergenov problems, while LocAut(π₂) forms a Lie group but LocAut(π₃) does not. These results illuminate the interplay between local and global structure in finite‑dimensional algebras and highlight distinct group/ manifold properties between the two algebras.

Abstract

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 and 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 and 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 other problems from the Ayupov-Eldique-Kudaybergenov problems have a positive (negative) solution for algebra (resp. algebra )

Paper Structure

This paper contains 9 sections, 16 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local automorphisms of five-dimensional naturally graded associative algebra with characteristic sequence $C(A)=(3,2)$
  4. Principles for derivations and local derivations of finite dimensional algebras
  5. Local derivations of five-dimensional naturally graded associative algebra with characteristic sequence $C(A)=(3,2)$
  6. The local derivations of $\pi_2$ and $\pi_2$ form a Lie algebra
  7. The relationship between local derivations and local automorphisms of $\pi_2$ and $\pi_2$
  8. Conclusion
  9. Statements and Declarations

Key Result

Theorem 1

1.1 A linear operator on associative algebra $\pi_2$ is an automorphism if and only if the matrix of this linear operator has the following matrix form

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Theorem 2
  • Definition 4
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • ...and 17 more