Infinitesimal CR Symmetries of Accidental CR Structures

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: $${\mathbf{E}}_{II},{\mathbf{E}}_{III},\mathfrak{so}(\ell -1,\ell +1),\mathfrak{su}(p,q).$$ 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.