A characterization of CR quadrics with a symmetry property

TL;DR

This paper characterizes CR quadrics with a weaker symmetry property $( ilde S)$ in terms of their Levi-Tanaka algebras, explores when this implies the stronger $(S)$ property, and provides a new example of a quadric with an unusually large automorphism algebra.

Contribution

It introduces a characterization of CR quadrics satisfying the $( ilde S)$ property via Levi-Tanaka algebras and examines the relationship between $( ilde S)$ and $(S)$ properties, including a novel example.

Findings

Characterization of $( ilde S)$-satisfying quadrics via Levi-Tanaka algebras

$( ilde S)$ implies $(S)$ for many cases, especially compact quadrics

New example of a quadric with a larger automorphism algebra than its dimension

Abstract

We study CR quadrics satisfying a symmetry property $(\tilde S)$ which is slightly weaker than the symmetry property $(S)$, recently introduced by W. Kaup, which requires the existence of an automorphism reversing the gradation of the Lie algebra of infinitesimal automorphisms of the quadric. We characterize quadrics satisfying the $(\tilde S)$ property in terms of their Levi-Tanaka algebras. In many cases the $(\tilde S)$ property implies the $(S)$ property; this holds in particular for compact quadrics. We also give a new example of a quadric such that the dimension of the algebra of positive-degree infinitesimal automorphisms is larger than the dimension of the quadric.