Generalized \delta-Derivations

TL;DR

This paper introduces generalized -derivations in algebraic structures, characterizes their forms in various algebra types, and constructs new non-trivial -derivations of Lie algebras, expanding understanding of derivation generalizations.

Contribution

It provides a comprehensive description of generalized -derivations across multiple algebra classes and introduces new examples of non-trivial -derivations in Lie algebras.

Findings

Generalized -derivations are either generalized derivations or -derivations in studied algebras.

Explicit descriptions of -superderivations for specific superalgebras.

New non-trivial -derivations constructed for Lie algebras.

Abstract

We defined generalized \delta-derivations of algebra A as linear mapping \chi associated with usual \delta-derivation \phi by the rule \chi(xy)=\delta(\chi(x)y+x\phi(y))=\delta(\phi(x)y+x\chi(y)) for any x,y \in A. We described generalized \delta-derivations of prime alternative algebras, prime Lie algebras and superalgebras, unital algebras, and semisimple finite-dimensional Jordan superalgebras. In this cases we proved that generalized \delta-derivation is a generalized derivation or \delta-derivation. After that we described \delta-superderivations of superalgebras <<KKM Double>>, arising from prime alternative algebras, prime Lie algebras and superalgebras, unital algebras, and semisimple finite-dimensional Jordan superalgebras. In the end, we constructed new examples of non-trivial \delta-derivations of Lie algebras.