Bianchi's classification of 3-dimensional Lie algebras revisited

TL;DR

This paper revisits Bianchi's classification of 3-dimensional Lie algebras, providing a coordinate-free, representation-theoretic perspective, and analyzes automorphism groups, orbit dimensions, and orbit closures within the algebraic variety of Lie brackets.

Contribution

It offers a new, coordinate-free proof of Bianchi's classification, computes automorphism groups, and clarifies the topology of the orbit space of 3D Lie algebras.

Findings

Coordinate-free classification of 3D Lie algebras

Automorphism groups and orbit dimensions computed

Topology of orbit space clarified

Abstract

We present Bianchi's proof on the classification of real (and complex) $3$-dimensional Lie algebras in a coordinate free version from a strictly representation theoretic point of view. Nearby we also compute the automorphism groups and from this the orbit dimensions of the corresponding orbits in the algebraic variety $X\subseteq\Lambda^2V^*\otimes V$ describing all Lie brackets on a fixed vector space $V$ of dimension $3$. Moreover we clarify which orbits lie in the closure of a given orbit and therefore the topology on the orbit space $X/G$ with $G=\mathrm{Aut}(V)$.