A generalization on derivations of Lie algebras

TL;DR

This paper introduces a broad new class of generalized derivations for finite-dimensional Lie algebras, unifying existing concepts and analyzing their properties, including computational methods for specific cases.

Contribution

It proposes a novel generalization of derivations in Lie algebras, connecting it to associative ring theory and providing computational techniques for explicit examples.

Findings

Unified many known generalized derivations under a new framework

Analyzed properties and rationality of Hilbert series of these derivations

Developed computational methods for the three-dimensional special linear Lie algebra

Abstract

We initiate a study on a range of new generalized derivations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero. This new generalization of derivations has an analogue in the theory of associative prime rings and unites many well-known generalized derivations that have already appeared extensively in the study of Lie algebras and other nonassociative algebras. After exploiting fundamental properties, we introduce and analyze their interiors, especially focusing on the rationality of the corresponding Hilbert series. Applying techniques in computational ideal theory we develop an approach to explicitly compute these new generalized derivations for the three-dimensional special linear Lie algebra over the complex field.