Infinitesimal CR Symmetries of Accidental CR Structures

TL;DR

This paper explicitly lists infinitesimal CR automorphisms for specific CR models with exceptional Lie algebra structures, using computer-aided methods to derive explicit vector field generators in holomorphic coordinates.

Contribution

It provides complete explicit lists of infinitesimal CR automorphisms for certain models with exceptional Lie algebra structures, expanding understanding of their symmetry groups.

Findings

Explicit automorphism generators for ${f E}_{II}, {f E}_{III}, rak{so}( ext{ell}-1, ext{ell}+1), rak{su}(p,q)$ models.

Use of symbolic computation to derive embedded vector fields.

Representation of symmetries in extrinsic holomorphic coordinates.

Abstract

In this companion paper to our article {\em Accidental CR structures} (arxiv.org, January 2023), thought of as an appendix not submitted for publication, we provide complete explicit lists of infinitesimal CR automorphisms for the concerned CR models having respective Lie algebra structures: $${\bf E}_{II}, \qquad\ {\bf E}_{III}, \qquad\ \mathfrak{so}(\ell-1,\ell+1), \qquad\ \mathfrak{su}(p,q).$$ We start from our lists of {\em quadric} CR submanifolds $M^{2n+c} \subset \mathbb{C}^{n+c}$ of codimension $c >1$ which are shown to be {\em accidental}, in the sense that their CR symmetry groups are {\em equal to} (and not smaller than) the symmetry groups of the underlying real distribution structures -- after forgetting the complex structure. Thanks to intensive symbolic computer explorations, we then determine embedded vector field generators of these CR symmetries Lie algebras, and we express them in {\em extrinsic} holomorphic coordinates, because intrinsic formulas would be too extended to be shown.