$\delta$-superderivations of simple finite-dimensional Jordan and Lie superalgebras

TL;DR

This paper introduces and studies the concept of $\, ext{delta}$-superderivations in superalgebras, providing classifications and proving their triviality in certain classes of simple finite-dimensional Jordan and Lie superalgebras over algebraically closed fields of characteristic zero.

Contribution

It offers a complete description of $\, ext{delta}$-superderivations for Cartan-type Lie superalgebras and proves their triviality in simple finite-dimensional Jordan superalgebras, advancing the understanding of superderivations.

Findings

$\, ext{delta}$-superderivations are trivial on simple finite-dimensional Jordan superalgebras of degree ≥ 2.

Complete classification of $\, ext{delta}$-derivations for Cartan-type Lie superalgebras.

Nontrivial $\, ext{delta}$-superderivations do not exist in the studied classes.

Abstract

We introduce the concept of a $\delta$-superderivation of a superalgebra. $\delta$-Derivations of Cartan-type Lie superalgebras are treated, as well as $\delta$-superderivations of simple finite-dimensional Lie superalgebras and Jordan superalgebras over an algebraically closed field of characteristic 0. We give a complete description of $\{1}{2}$-derivations for Cartan-type Lie superalgebras. It is proved that nontrivial $\delta$-(super)derivations are missing on the given classes of superalgebras, and as a consequence, $\delta$-superderivations are shown to be trivial on simple finite-dimensional noncommutative Jordan superalgebras of degree at least 2 over an algebraically closed field of characteristic 0. Also we consider $\delta$-derivations of unital flexible and semisimple finite-dimensional Jordan algebras over a field of characteristic not 2.