Maximally degenerate Weyl tensors in Riemannian and Lorentzian signatures

TL;DR

This paper determines the maximum possible symmetry dimensions for Riemannian and Lorentzian conformal structures by classifying subalgebras of orthogonal Lie algebras and analyzing their action on Weyl tensors.

Contribution

It provides the submaximal symmetry dimension for these conformal structures and introduces a method using Lie algebra classification and representation theory.

Findings

Established submaximal symmetry dimensions for Riemannian and Lorentzian conformal structures.

Classified subalgebras of orthogonal Lie algebras relevant to Weyl tensor stabilization.

Applied Dynkin's classification, Mostow's theorem, and Kostant's Bott-Borel-Weil theorem in the analysis.

Abstract

We establish the submaximal symmetry dimension for Riemannian and Lorentzian conformal structures. The proof is based on enumerating all subalgebras of orthogonal Lie algebras of sufficiently large dimension and verifying if they stabilize a non-zero Weyl tensor up to scale. Our main technical tools include Dynkin's classification of maximal subalgebras in complex simple Lie algebras, a theorem of Mostow, and Kostant's Bott-Borel-Weil theorem.