Polynomial invariants for 3-dimensional Leibniz algebras

TL;DR

This paper classifies polynomial invariants and automorphism groups of 3-dimensional non-Lie Leibniz algebras over complex numbers, providing tools to distinguish non-nilpotent cases using traces and automorphism dimensions.

Contribution

It explicitly describes the polynomial invariants and automorphism groups for all such algebras, offering a new method to differentiate them.

Findings

Polynomial invariants are fully described for each algebra.

Automorphism groups are explicitly determined.

Non-nilpotent algebras are distinguishable by traces and automorphism dimensions.

Abstract

For each 3-dimensional non-Lie Leibniz algebra over the complex numbers, we describe the algebra of polynomial invariants and determine its group of automorphisms. As a consequence, we establish that any two non-nilpotent 3-dimensional non-Lie Leibniz algebras can be distinguished by the traces of degrees $\leqslant 2$ and by the dimensions of their automorphism groups.