Four-Dimensional Lie Algebras Revisited

TL;DR

This paper analyzes the structure of 4-dimensional Lie algebras by computing prime ideals of their classification components, correcting previous results and answering a longstanding question in the field.

Contribution

It provides explicit computations of prime ideals, degrees, and Hilbert polynomials for the irreducible components of 4-dimensional Lie algebra structures, correcting earlier work.

Findings

Identified four irreducible components of the variety of 4D Lie algebras

Computed prime ideals, degrees, and Hilbert polynomials for each component

Resolved a question posed by Kirillov and Neretin in 1987

Abstract

The projective variety of Lie algebra structures on a 4-dimensional vector space has four irreducible components of dimension 11. We compute their prime ideals in the polynomial ring in 24 variables. By listing their degrees and Hilbert polynomials, we correct an earlier publication and we answer a 1987 question by Kirillov and Neretin.