Connected subgroups of SO(2,n) acting irreducibly on $\R^{2,n}$

TL;DR

This paper classifies all connected subgroups of SO(2,n) that act irreducibly on 4,n4, revealing specific subgroups beyond SO_0(2,n) and exploring implications for Lorentzian conformal structures.

Contribution

It provides a complete classification of connected irreducible subgroups of SO(2,n), extending understanding of their structure and applications in conformal geometry.

Findings

Identifies all connected irreducible subgroups of SO(2,n).

Links subgroup classification to holonomy groups of Lorentzian conformal structures.

Uses geometric and Lie theoretical tools like the Karpelevich Theorem.

Abstract

We classify all connected subgroups of SO(2,n) that act irreducibly on $\R^{2,n}$. Apart from $SO_0(2,n)$ itself these are $U(1,n/2)$, $SU(1,n/2)$, if $n$ even, $S^1\cdot SO(1,n/2)$ if $n$ even and $n\ge 2$, and $SO_0(1,2)$ for $n=3$. Our proof is based on the Karpelevich Theorem and uses the classification of totally geodesic submanifolds of complex hyperbolic space and of the Lie ball. As an application we obtain a list of possible irreducible holonomy groups of Lorentzian conformal structures, namely $SO_0(2,n)$, SU(1,n), and $SO_0(1,2)$.