# Four-Dimensional Lie Algebras Revisited

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.

## Verifications of Theorems 1 and 2

We used Macaulay2 (version 1.20) to verify Theorems 1 and 2 from the paper.

The file `Lie 4 Component`

includes the explicit generators of the irreducible components \(C_1, C_2, C_3, C_4\) which are explained in Section 3 of our paper.
We calculate the dimension, degree, Betti numbers and the Hilbert polynomial of each of these component.
We also verify that our idels for \(C_1, C_3, C_4\) are prime.
To show that \(C_1\) and \(C_3\) are prime it is enough to run the isPrime command in Macaulay2.
To show \(C_4\) is prime we run

```
# minimalPrimes C4;
radical C4 == C4
```

Since we get the output 1 and true we see that \(C_4\) is prime. Finally we take the intersection of these components to get the radical ideal of \(\operatorname{Lie}_4\) and calculate its dimension, degree and Betti numbers.

In the file `C2 prime`

we verify that our ideal for \(C_2\) is prime.
We do this by representing the birational parametrization of \(C_2\) mentioned in Section 5 of our paper.

Project page created: 30/08/2022.

Project contributors: Laurent Manivel, Bernd Sturmfels and Svala Sverrisdóttir.

Corresponding author of this page: Svala Sverrisdóttir, svalasverris@berkeley.edu

Software used: Macaulay2 (version 1.20).