On (alpha,beta,gamma)-derivations of Lie algebras and corresponding invariant functions

TL;DR

This paper introduces a generalized concept of derivations for finite-dimensional complex Lie algebras, leading to new invariant functions that classify Lie algebra structures and have applications in physics.

Contribution

It generalizes Lie derivations with complex parameters and constructs invariant functions that classify Lie algebras, including a complete invariant for three-dimensional cases.

Findings

Derived new classes of invariant functions for Lie algebras

Established a complete invariant for three-dimensional Lie algebras

Applied invariants to physically motivated eight-dimensional examples

Abstract

We consider finite-dimensional complex Lie algebras. We generalize the concept of Lie derivations via certain complex parameters and obtain various Lie and Jordan operator algebras as well as two one-parametric sets of linear operators. Using these parametric sets, we introduce complex functions with fundamental property - invariance under Lie isomorphisms. One of these basis-independent functions represents a complete set of invariant(s) for three-dimensional Lie algebras. We present also its application on physically motivated examples in dimension eight.