On $\delta$-derivations of Lie algebras and superalgebras

TL;DR

This paper investigates $oldsymbol{ ext{}oldsymbol{ ext{delta}}}$-derivations in Lie algebras and superalgebras, providing classifications, counterexamples, and new insights into gradings and derivation structures.

Contribution

It computes $oldsymbol{ ext{}oldsymbol{ ext{delta}}}$-derivations for specific Lie algebras, answers a question of Filippov, and extends results to Lie superalgebras using Grassmann envelopes.

Findings

Identifies $oldsymbol{ ext{}oldsymbol{ ext{delta}}}$-derivations with zero divisors.

Provides examples of non-semigroup gradings of Lie algebras.

Shows prime Lie superalgebras lack nontrivial $oldsymbol{ ext{}oldsymbol{ ext{delta}}}$-derivations.

Abstract

We study $\delta$-derivations -- a construction simultaneously generalizing derivations and centroid. First, we compute $\delta$-derivations of current Lie algebras and of modular Zassenhaus algebra. This enables us to provide examples of Lie algebras having 1/2-derivations which are divisors of zero, thus answering negatively a question of Filippov. Second, we note that $\delta$-derivations allow, in some circumstances, to construct examples of non-semigroup gradings of Lie algebras, in addition to the recent ones discovered by Elduque. Third, we note that utilizing the construction of the Grassmann envelope allows to obtain results about $\delta$-(super)derivations of Lie superalgebras from the corresponding results about Lie algebras. In this way, we prove that prime Lie superalgebras do not possess nontrivial $\delta$-(super)derivations, generalizing the recent result of Kaygorodov.