Generalized conformal derivations of Lie conformal algebras

TL;DR

This paper explores various types of conformal derivations in Lie conformal algebras, introduces a hierarchy of derivation algebras, and characterizes all conformal $( ext{α,β,γ})$-derivations for finite simple cases.

Contribution

It introduces the generalized conformal derivation algebra and studies their connections, extending the understanding of derivations in Lie conformal algebras.

Findings

Established the hierarchy of derivation algebras: $CDer(R) \\subseteq QDer(R) \\subseteq GDer(R) \\subseteq gc(R)$

Characterized all conformal $( ext{α,β,γ})$-derivations for finite simple Lie conformal algebras

Connected various generalized conformal derivations with conformal $( ext{α,β,γ})$-derivations.

Abstract

Let $R$ be a Lie conformal algebra. The purpose of this paper is to investigate the conformal derivation algebra $CDer(R)$, the conformal quasiderivation algebra $QDer(R)$ and the generalized conformal derivation algebra $GDer(R)$. The generalized conformal derivation algebra is a natural generalization of the conformal derivation algebra. Obviously, we have the following tower $CDer(R)\subseteq QDer(R)\subseteq GDer(R)\subseteq gc(R)$, where $gc(R)$ is the general Lie conformal algebra. Furthermore, we mainly research the connection of these generalized conformal derivations. Finally, the conformal $(\alpha,\beta,\gamma)$-derivations of Lie conformal algebras are studied. Moreover, we obtain some connections between several specific generalized conformal derivations and the conformal $(\alpha,\beta,\gamma)$-derivations. In addition, all conformal $(\alpha,\beta,\gamma)$-derivations of finite simple Lie conformal algebras are characterized.