Pictures of compact Lie groups (after Serre)

TL;DR

This paper details a method for visualizing compact real Lie groups, specifically producing pictures for certain rank 2 groups, including the complex and less understood G2 group, linking them to algebraic varieties.

Contribution

It provides a detailed visualization procedure for compact Lie groups, including the complex G2 group, expanding on Serre's outline and connecting to algebraic geometry.

Findings

Generated pictures for three rank 2 Lie groups

Connected the G2 group picture to algebraic varieties

Extended previous visualizations to less studied groups

Abstract

We fill in the details in a procedure outlined by Serre for drawing pictures of compact real Lie groups. In the case of Sp($2n$), the picture generated by the method is connected with abelian varieties over a number field or a finite field. We follow the procedure to produce pictures for the three simply connected simple groups of rank 2. The pictures for two of these have previously been discussed in the literature in a different setting. The remaining one, type $G_2$, has the most complicated picture.