On the variety of four dimensional lie algebras

TL;DR

This paper investigates the structure of four-dimensional Lie algebras by analyzing the geometric distribution of their structure constants within a projective space, revealing the exact count and types of such algebras over complex numbers.

Contribution

It provides a detailed enumeration and classification of four-dimensional Lie algebras by geometric and algebraic methods, answering a question posed by Kirillov and Neretin.

Findings

Exactly 1033 points correspond to four-dimensional Lie algebras in a generic subspace.

Among these, 660 are isomorphic to gl_2.

121 have a three-dimensional abelian derived algebra.

Abstract

Lie algebras of dimension $n$ are defined by their structure constants , which can be seen as sets of $N = n^2 (n -- 1)/2$ scalars (if we take into account the skew-symmetry condition) to which the Jacobi identity imposes certain quadratic conditions. Up to rescaling, we can consider such a set as a point in the projective space $P^{N--1}$. Suppose $n =4$, hence $N = 24$. Take a random subspace of dimension $12$ in $P^{23}$ , over the complex numbers. We prove that this subspace will contain exactly $1033$ points giving the structure constants of some four dimensional Lie algebras. Among those, $660$ will be isomorphic to $gl\_2$ , $195$ will be the sum of two copies of the Lie algebra of one dimensional affine transformations, $121$ will have an abelian, three-dimensional derived algebra, and $57$ will have for derived algebra the three dimensional Heisenberg algebra. This answers a question of Kirillov and Neretin.