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2-Local derivations on a Block-type Lie algebra

Qiufan Chen, Xiaohan Guo

TL;DR

This work addresses the problem of characterizing 2-local derivations on the Block-type Lie algebra $\mathcal{B}$, which has an infinite-dimensional structure and an outer derivation. By decomposing 2-local derivations into sums of inner derivations and a multiple of the outer derivation and performing a detailed basis-wise analysis, the authors show that any 2-local derivation must coincide with a global derivation. The proof relies on a sequence of lemmas that constrain the action of $\Delta$ on key basis elements and ultimately reduce to the standard derivation framework. The result extends local-to-global rigidity phenomena from finite-dimensional Lie algebras to this infinite-dimensional Block-type setting, contributing to the understanding of 2-local derivations in Lie theory.

Abstract

The present paper is devoted to study 2-local derivations on the Block-type Lie algebra which is an infinite-dimensional Lie algebra with some outer derivations. We prove that every 2-local derivation on the Block-type Lie algebra is a derivation.

2-Local derivations on a Block-type Lie algebra

TL;DR

This work addresses the problem of characterizing 2-local derivations on the Block-type Lie algebra , which has an infinite-dimensional structure and an outer derivation. By decomposing 2-local derivations into sums of inner derivations and a multiple of the outer derivation and performing a detailed basis-wise analysis, the authors show that any 2-local derivation must coincide with a global derivation. The proof relies on a sequence of lemmas that constrain the action of on key basis elements and ultimately reduce to the standard derivation framework. The result extends local-to-global rigidity phenomena from finite-dimensional Lie algebras to this infinite-dimensional Block-type setting, contributing to the understanding of 2-local derivations in Lie theory.

Abstract

The present paper is devoted to study 2-local derivations on the Block-type Lie algebra which is an infinite-dimensional Lie algebra with some outer derivations. We prove that every 2-local derivation on the Block-type Lie algebra is a derivation.
Paper Structure (3 sections, 8 theorems, 31 equations)

This paper contains 3 sections, 8 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. 2-local derivations on $\mathcal{B}$

Key Result

Lemma 2.1

(cf. XW) $\mathrm{Der}\mathcal{B}=\mathrm{ad}\mathcal{B}\oplus\mathbb{C} d$, where $d$ is an outer derivation defined by $d(L_{\beta, j})=\beta L_{\beta, j}$ for all $\beta\in\mathbb{Z},j\in\mathbb{Z}_+$.

Theorems & Definitions (14)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 4 more