The variety of Lie algebra representations

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.