Canonical Isomorphism of Two Lie Algebras Arising in CR-geometry

A CR-structure on a smooth real manifold M of dimension m is a smooth distribution of subspaces in the tangent spaces T c p (M) ⊂ Tp(M), p ∈ M , with operators of complex structure Jp : T c p (M) → T c p (M), J 2 p ≡ −id, that depend smoothly on p. A manifold M equipped with a CR-structure is called a CR-manifold. It follows that the number CRdimM := dimCT c p (M) does not depend on p; it is called the CR-dimension of M . The number CRcodimM := m − 2CRdimM is called the CR-codimension of M . CRstructures naturally arise on real submanifolds in complex manifolds. Indeed, if, for example, M is a real submanifold of C , then one can define the distribution T c p (M) as follows: T c p (M) := Tp(M) ∩ iTp(M).