The variety of Lie algebra representations

TL;DR

This paper investigates the geometric structure of the variety of Lie algebra representations, using invariant theory and energy functionals to analyze critical points and prove rigidity and structural theorems for semi-simple Lie algebra homomorphisms.

Contribution

It introduces a geometric invariant theory approach to study Lie algebra representations, proving new results on the structure and rigidity of semi-simple homomorphisms.

Findings

Semi-simple pairs minimize energy globally within their orbits.

Elementary proof of rigidity of semi-simple homomorphisms.

New proof of Mostow's theorem on Cartan involutions.

Abstract

We study the affine variety $L_{n}(\mathfrak{g})$ of Lie algebra representations, the collection of all homomorphisms from an arbitrary $n$-dimensional Lie algebra into a fixed real semi-simple Lie algebra $\mathfrak{g}$. Using techniques from real Geometric Invariant Theory, we equip this variety with a natural moment map and associated energy functional arising from the action of the real reductive group $GL(n,\mathbb{R}) \times \text{Inn}(\mathfrak{g})$. We analyze the critical points of the energy functional and describe their structure. In particular, we prove that every semi-simple pair, that is representations of semi-simple Lie algebras, will globally minimize the energy in its orbit. As consequences, we obtain an elementary proof of the rigidity of semi-simple homomorphisms and derive a new proof of the Mostow theorem on the existence of compatible Cartan involutions for semi-simple subalgebras. Subsequent results concerning the structure of critical points of higher energy are also obtained.