Generalized derivations of $3$-Lie algebras

TL;DR

This paper introduces and studies generalized derivations, quasiderivations, and quasicentroids of 3-Lie algebras, revealing their structural relations and embedding properties, with a focus on diagonalized tours and module actions.

Contribution

It systematically investigates the structures of quasiderivations and quasicentroids of 3-Lie algebras, establishing their algebraic relations and module properties under diagonalized tours.

Findings

$GDer(A)$ equals $QDer(A) + Q extGamma(A)$ for any 3-Lie algebra $A$

Quasiderivations can be embedded as derivations in larger algebras

$QDer(A)$ normalizes $Q extGamma(A)$, i.e., $[QDer(A), Q extGamma(A)] \\subseteq Q extGamma(A)$

Abstract

Generalized derivations, quasiderivations and quasicentroid of $3$-algebras are introduced, and basic relations between them are studied. Structures of quasiderivations and quasicentroid of $3$-Lie algebras, which contains a maximal diagonalized tours, are systematically investigated. The main results are: for all $3$-Lie algebra $A$, 1) the generalized derivation algebra $GDer(A)$ is the sum of quasiderivation algebra $QDer(A)$ and quasicentroid $Q\Gamma(A)$; 2) quasiderivations of $A$ can be embedded as derivations in a larger algebra; 3) quasiderivation algebra $QDer(A)$ normalizer quasicentroid, that is, $[QDer(A), Q\Gamma(A)]\subseteq Q\Gamma(A)$; 4) if $A$ contains a maximal diagonalized tours $T$, then $QDer(A)$ and $Q\Gamma(A)$ are diagonalized $T$-modules, that is, as $T$-modules, $(T, T)$ semi-simplely acts on $QDer(A)$ and $Q\Gamma(A)$, respectively.