Canonical Isomorphism of Two Lie Algebras Arising in $\textit{CR}$-geometry

Abstract

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