Canonical Isomorphism of Two Lie Algebras Arising in CR-geometry

Abstract

We show that the maximal prolongation of a certain algebra associated with a non-degenerate Hermitian form on \mathbb{C}ⁿ×\mathbb{C}ⁿ with values in \mathbb{R}^k is canonically isomorphic to the Lie algebra of infinitesimal holomorphic automorphisms of the corresponding quadric in \mathbb{C}^n+k. This fact creates a link between different approaches to the equivalence problem for Levi-nondegenerate strongly uniform CR-manifolds.