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

Abstract

We show that the maximal prolongation of a certain algebra associated with a non-degenerate Hermitian form on $C^n\times C^n$ 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 $\text{CR}$-manifolds.