Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local and 2-local $\frac{1}{2}$-derivations on finite-dimensional Lie algebras

Abror Khudoyberdiyev, Bakhtiyor Yusupov

Abstract

In this work, we introduce the notion of local and $2$-local $δ$-derivations and describe local and $2$-local $\frac{1}{2}$-derivation of finite-dimensional solvable Lie algebras with filiform, Heisenberg, and abelian nilradicals. Moreover, we describe the local $\frac{1}{2}$-derivation of oscillator Lie algebras, Schr{ö}dinger algebras, and Lie algebra with a three-dimensional simple part, whose radical is an irreducible module. We prove that an algebra with only trivial $\frac{1}{2}$-derivation does not admit local and $2$-local $\frac{1}{2}$-derivation, which is not $\frac{1}{2}$-derivation.

Local and 2-local $\frac{1}{2}$-derivations on finite-dimensional Lie algebras

Abstract

In this work, we introduce the notion of local and -local -derivations and describe local and -local -derivation of finite-dimensional solvable Lie algebras with filiform, Heisenberg, and abelian nilradicals. Moreover, we describe the local -derivation of oscillator Lie algebras, Schr{ö}dinger algebras, and Lie algebra with a three-dimensional simple part, whose radical is an irreducible module. We prove that an algebra with only trivial -derivation does not admit local and -local -derivation, which is not -derivation.
Paper Structure (10 sections, 18 theorems, 59 equations)

This paper contains 10 sections, 18 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local and 2-local $\frac{1}{2}$-derivation on finite-dimensional solvable Lie algebras
  4. Local and 2-local $\frac{1}{2}$-derivation on solvable Lie algebras with naturally graded filiform nilradical
  5. Local and 2-local $\frac{1}{2}$-derivations on solvable Lie algebras with Heisenberg nilradical
  6. Local and 2-local $\frac{1}{2}$-derivation on solvable Lie algebras with abelian nilradical
  7. Local and 2-local $\frac{1}{2}$-derivations on oscillator Lie algebras
  8. Local and 2-local $\frac{1}{2}$-derivations of some non-solvable Lie algebras
  9. Local and 2-local $\frac{1}{2}$-derivation on semidirect products of ${\mathfrak {sl}}_2$ and irreducible modules.
  10. Local and 2-local $\frac{1}{2}$-derivation on Schrödinger Lie algebras.

Key Result

Theorem 4

Let $\mathfrak{L}$ be an algebra, whose all $\frac{1}{2}$-derivation are trivial. Then any local and $2$-local $\frac{1}{2}$-derivation of $\mathfrak{L}$ is a trivial $\frac{1}{2}$-derivation.

Theorems & Definitions (37)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 4
  • proof
  • Definition 5
  • Corollary 6
  • Definition 7
  • Theorem 8
  • proof
  • ...and 27 more