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Local derivations and local automorphisms on the super Virasoro algebras

Qingyan Wu, Shoulan Gao, Dong Liu, Chang Ye

TL;DR

The work addresses the local-to-global behavior of derivations and automorphisms on the super-Virasoro algebras SVir[ε]. By first analyzing the Virasoro subalgebra and then extending to SVir[0] and SVir[1/2], the authors prove that all local derivations are derivations and all local or 2-local automorphisms are automorphisms, highlighting a rigidity phenomenon in these Lie superalgebras. The results rely on decomposing local maps via inner derivations and exploiting the known automorphism structure from Zhao's classification. The findings contribute to a deeper understanding of the local structure determining global symmetries in infinite-dimensional Lie superalgebras, with potential implications for representation theory and mathematical physics. The methods illustrate how local-to-global arguments can enforce strong constraints on algebra homomorphisms in the super setting.

Abstract

This paper aims to study the local derivations, 2-local automorphisms and local automorphisms on the super-Virasoro algebras. The primary focus is to establish that every local derivation of the super-Virasoro algebras is indeed a derivation, and to demonstrate that every local or 2-local automorphism of the super-Virasoro algebras is an automorphism.

Local derivations and local automorphisms on the super Virasoro algebras

TL;DR

The work addresses the local-to-global behavior of derivations and automorphisms on the super-Virasoro algebras SVir[ε]. By first analyzing the Virasoro subalgebra and then extending to SVir[0] and SVir[1/2], the authors prove that all local derivations are derivations and all local or 2-local automorphisms are automorphisms, highlighting a rigidity phenomenon in these Lie superalgebras. The results rely on decomposing local maps via inner derivations and exploiting the known automorphism structure from Zhao's classification. The findings contribute to a deeper understanding of the local structure determining global symmetries in infinite-dimensional Lie superalgebras, with potential implications for representation theory and mathematical physics. The methods illustrate how local-to-global arguments can enforce strong constraints on algebra homomorphisms in the super setting.

Abstract

This paper aims to study the local derivations, 2-local automorphisms and local automorphisms on the super-Virasoro algebras. The primary focus is to establish that every local derivation of the super-Virasoro algebras is indeed a derivation, and to demonstrate that every local or 2-local automorphism of the super-Virasoro algebras is an automorphism.
Paper Structure (5 sections, 15 theorems, 46 equations)

This paper contains 5 sections, 15 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local derivations on the Virasoro subalgebra
  4. Local derivation on the super Virasoro algebras
  5. Local and 2-Local Automorphisms on the super Virasoro algebras

Key Result

Lemma 2.6

Suppose that $\sigma\in {\rm Aut\,(SVir}[\epsilon])$. Then where $\varepsilon=\pm 1,a\in \mathbb{C}^{*},m\in\mathbb{Z},r\in\mathbb{Z}+\epsilon$.

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 23 more