On transitive contact and $CR$ algebras

TL;DR

This paper proves that under certain contact structure conditions, the automorphisms of locally homogeneous CR manifolds form a finite dimensional Lie group.

Contribution

It establishes a new condition based solely on contact structure that guarantees finite dimensionality of CR automorphism groups.

Findings

CR automorphism groups are finite dimensional under the new contact condition

The condition depends only on the underlying contact structure

Results apply to locally homogeneous CR manifolds

Abstract

We consider locally homogeneous $CR$ manifolds and show that, under a condition only depending on their underlying contact structure, their $CR$ automorphisms form a finite dimensional Lie group.

Equations

$${\widetilde{\text{mathpzc}L}}_0=\{\mathrm{Re}(Z)\mid Z\in \text{mathpzc}L\},{\breve{\text{mathpzc}L}}_0=\text{mathpzc}L\cap \overline{\text{mathpzc}L}\cap {\mathfrak{g}}_0.$$

$$\cdots \supseteq \text{mathpzc}{G}_{-h}\supseteq \text{mathpzc}{G}_{1-h}\supset \cdots \text{mathpzc}{G}_{-1}\supseteq \text{mathpzc}{G}_0\supseteq \text{mathpzc}{G}_1\supseteq \cdots \text{mathpzc}{G}_h\supseteq \text{mathpzc}{G}_{h+1}\supseteq \cdots$$

$$\{\begin{array}{ll}\text{mathpzc}{G}_{-1}=\text{mathpzc}{L}_0,& \\ \text{mathpzc}{G}_h=\text{mathpzc}{G}_{h+1}+[\text{mathpzc}{G}_{h+1},\text{mathpzc}{G}_{-1}],& \text{if h<−1,}\\ \text{mathpzc}{G}_h=\{X\in \text{mathpzc}{G}_{-1}\mid [X,\text{mathpzc}{G}_{-1}]\subseteq \text{mathpzc}{G}_{h-1}\}& \text{if h≥0.}\end{array}$$

$$\{0\}=(\text{mathpzc}{G}_{-1}/\text{mathpzc}{G}_{-1})\subseteq (\text{mathpzc}{G}_{-2}/\text{mathpzc}{G}_{-1})\subseteq \cdots (\text{mathpzc}{G}_{-h}/\text{mathpzc}{G}_{-1})\subseteq (\text{mathpzc}{G}_{-h-1}/\text{mathpzc}{G}_{-1})\subseteq \cdots$$

$$c_0={\bigcap }_{h\in Z}\text{mathpzc}{G}_h.$$

$$[\text{mathpzc}{G}_0,\text{mathpzc}{G}_h]$$

$$\subseteq [\text{mathpzc}{G}_0,\text{mathpzc}{G}_{h+1}]+[[\text{mathpzc}{G}_0,\text{mathpzc}{G}_{h+1}],\text{mathpzc}{G}_{-1}]+[\text{mathpzc}{G}_{h+1},[\text{mathpzc}{G}_0,\text{mathpzc}{G}_{-1}]]$$

$$\subseteq \text{mathpzc}{G}_{h+1}+[\text{mathpzc}{G}_{h+1},\text{mathpzc}{G}_{-1}]=\text{mathpzc}{G}_h.$$

$$[[\text{mathpzc}{G}_0,\text{mathpzc}{G}_h],\text{mathpzc}{G}_{-1}]$$

$$\subseteq [\text{mathpzc}{G}_h,\text{mathpzc}{G}_{-1}]+[\text{mathpzc}{G}_0,\text{mathpzc}{G}_{h-1}]\subseteq \text{mathpzc}{G}_{h-1}$$

$$[[\text{mathpzc}{G}_{h_1},\text{mathpzc}{G}_{h_2}],\text{mathpzc}{G}_{-1}]$$

$$\subseteq [\text{mathpzc}{G}_{h_1-1},\text{mathpzc}{G}_{h_2}]][\text{mathpzc}{G}_{h_1},\text{mathpzc}{G}_{h_2-1}]\subseteq \text{mathpzc}{G}_{h_1+h_2-1}$$

$$\text{vspace}-21pt[{\mathfrak{h}}_0,\text{mathpzc}{G}_h]\subseteq \text{mathpzc}{G}_h,\forall h\in Z.$$

$$\{\begin{array}{l}[X_1]=X_1,\\ [X_1,X_2]=\text{Lie product of X1 and X2 in g},\\ [X_1,\dots ,X_k,X_{k+1}]=[[X_1,\dots ,X_k],X_{k+1}],\text{for k≥2},\\ \text{if}{X}_1,\dots ,X_{k+1}\in \mathfrak{g},\end{array}$$

$$\forall X\in \text{mathpzc}{L}_0\setminus {\mathfrak{h}}_0,\exists X_0,\dots ,X_k\in \text{mathpzc}{L}_0\text{such that}[X,X_0,\dots ,X_k]\notin \text{mathpzc}{L}_0.$$

$$\text{vspace}-20pt\forall Z\in \mathfrak{q},\exists Z_0,\dots ,Z_k\in \mathfrak{q}\text{such that}[\bar{Z},Z_0,\dots ,Z_k]\notin (\mathfrak{q}+\bar{\mathfrak{q}}).$$

$${\bar{\mathfrak{q}}}^{(0)}\supseteq {\bar{\mathfrak{q}}}^{(1)}\supseteq \cdots \supseteq {\bar{\mathfrak{q}}}^{(h)}\supseteq {\bar{\mathfrak{q}}}^{h+1}\supseteq \cdots$$

$${\tilde{\mathfrak{q}}}^{(0)}\supseteq {\tilde{\mathfrak{q}}}^{(1)}\supseteq \cdots \supseteq {\tilde{\mathfrak{q}}}^{(h)}\supseteq {\tilde{\mathfrak{q}}}^{h+1}\supseteq \cdots$$

$$\{\begin{array}{l}\begin{array}{rl}& {\bar{\mathfrak{q}}}^{(0)}=\bar{\mathfrak{q}},\\ & {\tilde{\mathfrak{q}}}^{(0)}=\mathfrak{q}+\bar{\mathfrak{q}},\end{array}\text{for h=0,}\\ \begin{array}{rl}& {\bar{\mathfrak{q}}}^{(h)}=\{Z\in {\bar{\mathfrak{q}}}^{(h-1)}\mid [Z,\mathfrak{q}]\subseteq {\tilde{\mathfrak{q}}}^{(h-1)}\},\\ & {\tilde{\mathfrak{q}}}^{(h)}=\mathfrak{q}+{\bar{\mathfrak{q}}}^{(h)},\end{array}\text{for h≥1.}\end{array}$$

$${\bigcap }_{h\ge 0}{\bar{\mathfrak{q}}}^{(h)}={\bigcap }_{h\ge 1}\{Z\in \bar{\mathfrak{q}}\mid [Z,W_1,\dots ,W_h]\in \tilde{\mathfrak{q}},\forall W_1,\dots ,W_h\in \mathfrak{q}\}.$$

$${\mathfrak{q}}^{\prime }=\mathfrak{q}+{\bigcap }_{h\ge 0}{\bar{\mathfrak{g}}}^{(h)}={\bigcap }_{h\ge 0}{\tilde{\mathfrak{q}}}^{(h)}$$

$$\mathfrak{q}\subseteq {\mathfrak{q}}^{\prime }\subseteq \mathfrak{q}+\bar{\mathfrak{q}}.$$

$$[[Z_1,Z_2],\mathfrak{q}]$$

$$\subseteq [Z_1,{\bar{\mathfrak{q}}}^{(h)}]+[Z_1,\mathfrak{q}]+[Z_2,{\bar{\mathfrak{q}}}^{(h)}]+[Z_2,\mathfrak{q}]\subseteq {\bar{\mathfrak{q}}}^{(h)}+\mathfrak{q},$$

$$V_0\supseteq V_1\supseteq \cdots \supseteq V_h\supseteq V_{h+1}\supseteq \cdots$$

$${\mathfrak{g}}_0$$

$$\mathfrak{q}$$

$$\text{piup}:G_0\ni x\to x\cdot {\mathrm{p}}_0\in M_0$$

$$x\cdot {\mathcal{L}}^{\ast }={\mathcal{L}}^{\ast },\forall x\in G_0.$$

$$L_{p_0}=\{{\Theta }_{p_0}\mid \Theta \in {\mathcal{L}}^{\ast }\}\subseteq T_{{\mathrm{p}}_0}{M}_0$$

$$\mathrm{Ad}(x)(\text{mathpzc}{L}_0)=\text{mathpzc}{L}_0,\forall x\in {\mathbf{H}}_0,$$

$$\{X_{p_0}^{\ast }\in T_{{\mathrm{p}}_0}{M}_0\mid X^{\ast }\in {\mathfrak{g}}_0^{\ast }\}=T_{p_0}{M}_0.$$

$$[X,Y{]}^{\ast }=-[X^{\ast },Y^{\ast }],\forall X,Y\in {\mathfrak{g}}_0.$$

Full text

The authors were partially supported by the Project MIUR “Real and Complex Manifolds: Geometry, Topology and Harmonic Analysis” and by GNSAGA of INdAM.

1. 1 Definitions and preliminaries
2. 2 Descending chain of a contact pair
3. 3 Descending chain of a $CR$ algebra
4. 4 Homogeneous contact structures
5. 5 $\operatorname{\mathbb{Z}}$-graded Lie algebras and a Tanaka’s theorem
6. 6 A finitness criterion for $CR$ algebras
7. 7 Transitive pairs and generalised contact distributions
8. 8 Extensions

Introduction

In the past years some of the Authors introduced and investigated the notion of $CR$ algebra (see [2, 14]) to describe the local structure of homogeneous $CR$ manifolds. The understanding of local models is important e.g. for applying the method of E. Cartan to describe the differential invariants of $CR$ structures. A key point is to find under which conditions the infinitesimal automorphisms of the structure form a finite dimensional Lie algebra. A $CR$ structure can be defined by the datum of a smooth involutive complex distribution. The real parts of its vectors define a real distribution. A strong version of the condition that the $CR$ manifold is not foliated by $CR$ submanifolds of lower dimension is that this real distribution is a (generalised) contact distribution, i.e. that its iterated commutators span the full tangent space. The strong interplay between $CR$ and underlying contact structures was clearly exploited in the work of N. Tanaka (see [23, 24]). He considered, at each point, the nilpotent $\operatorname{\mathbb{Z}}$-graded real Lie algebra $\textswab{m}$ canonically associated to a contact structure. To describe the infinitesimal automorphisms, one needs to consider extensions, or prolongations $\mathfrak{G}$ of $\textswab{m}.$ In this setting, they can be described recursively in terms of derivations of $\textswab{m}.$ When a $CR$ structure is imposed on the contact distribuiton, finite dimensionality of the maximal prolongation is, for this $\operatorname{\mathbb{Z}}$-graded model, equivalent to the fact that the vector valued Levi form has trivial kernel. Thus Cartan’s method applies, via Tanaka’s theory, to the case where the associated $\operatorname{\mathbb{Z}}$-graded Levi-Tanaka algebras (cf. [13]) are isomorphic at all points and the Levi form is nondegenerate.

The idea of introducing $CR$ algebras in [14] originated from the observation that many interesting homogeneous examples of $CR$ manifolds lead to infinite dimensional Levi-Tanaka algebras, their Levi forms having nontrivial kernels. An obvious generalization of the nondegeneracy condition is to require that the iterated Levi forms have a trivial kernel. This condition, that was called weak nondegeneracy in [14] and is equivalent to the notion of (Levi) $k$-nondegeneracy used by other authors (see e.g. [5]), is indeed equivalent to the fact that the corresponding $CR$ manifold is not the total space of a $CR$ fibration with complex fibres. The differential invariants for $CR$ manifolds of hypersurface type in real dimension $5$ satisfying the notion of $k$-nondegenericity have been so far studied by several authors with different techniques (see e.g. [9, 15, 12, 18, 16]). Further developments in higher dimensions appeared in [21, 19]. A theory of invariants for 2-nondegenerate $CR$ hypersurfaces in arbitrary dimension, modeled on Tanaka’s approach, has been recently developed by Porter and Zelenko in [20].

In this paper we address the question on finite dimensionality of the full group of the automorphisms of $CR$ manifolds of arbitrary $CR$ dimension and $CR$ codimension, whose $CR$ structure is locally homogeneous. Weak nondegeneracy is a much more restrictive condition that the one we found to guarantee the finite dimensionality of the Lie algebra of infinitesimal $CR$ automorphisms. In fact, our criterion only involves the underlying contact structure. We found this fact very interesting. Indeed, it is preliminary to an approach where this (generalized) contact structure is a priori given as a characteristic of the manifold on which the addition of a $CR$ structure is meant to modelling different geometrical or physical situations. Our condition was called ideal nondegeneracy in [14], where the fact that it was a sufficient criterion for the finite dimensionality of the maximal extension was correctly stated; however, in the proof given there there was a gap that we fill here in §6.

Our proof of the existence of maximal extensions of $CR$ algebras relies on a review of the classical work on transitive geometry (see e.g. [6, 7, 10, 11]), allowing us to substitute formal power series to the canonical construction of Tanaka in the $\operatorname{\mathbb{Z}}$-graded case. Our discussion is restrained mostly at a purely algebraic level. Thus, for a better understanding of the geometrical significance of our results, we refer the reader to [4] for a thorough introduction to $CR$ and homogeneous $CR$ manifolds.

Let us briefly describe the contents of the paper.

§1 collects some general notions we thought relevant for the exposition. Contact and transitive pairs and triples and $CR$ algebras are defined, not restraining to finite dimensionality. We explicitly required that the (possibly infinite dimensional) Lie algebras involved have a topological structure, although this structure is implicitly defined by the requirement that the isotropy subalgebra is closed and has finite codimension. We also list various nondegeneracy conditions that will be investigated in the later sections.

In §2 we construct a canonical descending chain of subspaces which is associated to a contact pair to explain contact nondegeneracy.

An analogous construction in §3, characteristic of $CR$ algebras, describes weak $CR$-nondegeneracy. We show by an example that it is in fact a much more restrictive condition than nondegeneracy of the underlying contact structure.

We found convenient to explain in §4 the way the abstract contact triples of §1 relate to actual homogeneous contact manifolds, to motivate our later use of transitive contact geometry.

In §5 we introduce graded Lie algebras and the finiteness criterion of Noburu Tanaka.

By using Tanaka’s criterion, we prove in §6 our main result, which states that $CR$ algebras whose corresponding contact triple is nondegenerate are finite dimensional.

In §7 we deal with the general construction of the representation of transitive contact pairs by structures involving vector fields with formal power series coefficients. This is the main tool in the transitive geometry of [11]: in this purely algebraic setting a germ of homogeneous space is substituted by a topological Lie algebra in which the isotropy subalgebra is closed and has finite codimension. This describes a situation in which the values of the infinitesimal automorphisms of the structure span the full tangent space at a point.

In the final §8 we utilize transitive geometry to construct maximal extensions of $CR$ algebras. Then the result of §6 yields the theorem that locally homogeneous $CR$ manifolds with a nondegenerate underlying contact structure have a finite dimensional Lie algebra of infinitesimal $CR$ automorphisms and hence, in particular, their $CR$ automorphisms make a finite dimensional Lie group.

1. Definitions and preliminaries

In this section we introduce some notions which are relevant for an infinitesimal description of homogeneous (generalised) contact manifolds and of various geometrical structures that can be defined on them. We are particularly interested in partial complex structures, and $CR$ algebras (see [14]) fall in this realm.

A topological Lie algebra is a Lie algebra $\mathfrak{g}_{0}$ over a topological field $\Bbbk$ with a fixed structure of topological Hausdorff vector space for which the Lie product is continuous. We say that $\mathfrak{g}_{0}$ is linearly compact if the intersection of any family of affine subspaces of $\mathfrak{g}_{0}$ having the finite intersection property has a nonempty intersection. (For details, see e.g. [7]).

In the following we assume that $\mathfrak{g}_{0}$ is real and denote by $\mathfrak{g}$ its complexification. Conjugation in $\mathfrak{g}$ is always understood with respect to the real form $\mathfrak{g}_{0}.$ For a $\operatorname{\mathbb{C}}$-linear subspace $\mathpzc{L}$ of $\mathfrak{g}$ we set

[TABLE]

Definition 1.1**.**

• •

A contact pair is the pair $(\mathfrak{g}_{0},\mathpzc{L}_{0})$ consisting of a linearly compact topological real Lie algebra $\mathfrak{g}_{0}$ and a closed linear subspace $\mathpzc{L}_{0}$ of $\mathfrak{g}_{0}$ having a finite dimensional complement in $\mathfrak{g}_{0}$ and spanning $\mathfrak{g}_{0}$ as a Lie algebra.

• •

A contact $\operatorname{\mathbb{C}}$-pair is the pair $(\mathfrak{g}_{0},\mathpzc{L})$ consisting of a linearly compact topological real Lie algebra $\mathfrak{g}_{0}$ and a closed complex linear subspace $\mathpzc{L}$ of $\mathfrak{g}$ such that $(\mathfrak{g}_{0},\tilde{\mathpzc{L}}_{0})$ is a contact pair.

• •

A transitive pair $(\mathfrak{g}_{0},{\mathfrak{h}}_{0})$ consists of a linearly compact topological Lie algebra $\mathfrak{g}_{0}$ and a closed subalgebra ${\mathfrak{h}}_{0}$ having finite codimension in $\mathfrak{g}_{0}$ and not containing nontrivial ideals of $\mathfrak{g}_{0}.$

• •

A contact triple is a triple $(\mathfrak{g}_{0},{\mathfrak{h}}_{0},\mathpzc{L}_{0})$ such that $(\mathfrak{g}_{0},\mathpzc{L}_{0})$ is a contact pair, $(\mathfrak{g}_{0},{\mathfrak{h}}_{0})$ a transitive pair, ${\mathfrak{h}}_{0}\subseteq\mathpzc{L}_{0}$ and $[{\mathfrak{h}}_{0},\mathpzc{L}_{0}]\subseteq\mathpzc{L}_{0}.$

• •

A contact $\operatorname{\mathbb{C}}$-triple is a triple $(\mathfrak{g}_{0},{\mathfrak{h}}_{0},\mathpzc{L}),$ such that $(\mathfrak{g}_{0},{\mathfrak{h}}_{0})$ is transitive, $(\mathfrak{g}_{0},\mathpzc{L})$ is a contact $\operatorname{\mathbb{C}}$-pair, ${\mathfrak{h}}_{0}\subset\mathpzc{L}$ and $[{\mathfrak{h}}_{0},\mathpzc{L}]\subseteq\mathpzc{L}.$

• •

A fundamental almost $CR$ pair is a contact $\operatorname{\mathbb{C}}$-pair $(\mathfrak{g}_{0},\mathpzc{L})$ for which $\breve{\mathpzc{L}}_{0}$ is a Lie subalgebra of $\mathfrak{g}_{0}$ and $[\breve{\mathpzc{L}}_{0},\mathpzc{L}]\subseteq\mathpzc{L}.$ The contact triple $(\mathfrak{g}_{0},\breve{\mathpzc{L}}_{0},\tilde{\mathpzc{L}}_{0})$ is said to be associated to $(\mathfrak{g}_{0},\mathpzc{L}).$

• •

A fundamental $CR$ algebra is an almost $CR$ pair $(\mathfrak{g}_{0},\mathfrak{q})$ such that $\mathfrak{q}$ is a complex Lie subalgebra of $\mathfrak{g}.$

Remark 1.1**.**

In the definition of a $CR$ algebra $(\mathfrak{g}_{0},\mathfrak{q})$ of [14] it was not required that $(\mathfrak{q}+\bar{\mathfrak{q}})\cap\mathfrak{g}_{0}$ generates $\mathfrak{g}_{0}$ as a Lie algebra. When this is not the case and $(\mathfrak{g}_{0},\mathfrak{q}_{0})$ is a Lie algebra associated to a homogeneous $CR$ manifold $M_{0}$, then the manifold $M_{0}$ can be described, at least locally, as the product $M^{\prime}_{0}\times{N}_{0}$ of a $CR$ manifolds $M_{0}^{\prime}$ having the same $CR$ dimension of $M_{0}$ and a totally real $N_{0}.$ For many purposes we could reduce to $M_{0}^{\prime}$, which is a homogeneous $CR$ manifold whose $CR$ algebra at a point has the same $\mathfrak{q},$ while $\mathfrak{g}_{0}$ is substituted by the span of $\tilde{\mathfrak{q}}_{0}.$

We will use the following notions.

Definition 1.2** (Nondegenracy conditions).**

We say that a contact triple $(\mathfrak{g}_{0},{\mathfrak{h}}_{0},\mathpzc{L}_{0})$ is

• •

strictly nondegenerate if $\{X\in\mathpzc{L}_{0}\mid[X,\mathpzc{L}_{0}]\subseteq\mathpzc{L}_{0}\}={\mathfrak{h}}_{0}.$

• •

nondegenerate if any ideal of $\mathfrak{g}_{0}$ which is contained in $\mathpzc{L}_{0}$ is already contained in ${\mathfrak{h}}_{0}.$

A fundamental almost $CR$-pair $(\mathfrak{g}_{0},\mathpzc{L})$ is

• •

strictly nondegenerate if $\{Z\in\bar{\mathpzc{L}}\mid[Z,\mathpzc{L}]\subset\mathpzc{L}+\bar{\mathpzc{L}}\}=\mathpzc{L}\cap\bar{\mathpzc{L}}.$

• •

weakly non-degenerate if there is no almost $CR$ pair $(\mathfrak{g}_{0},\mathpzc{L}^{\prime})$ with $\mathpzc{L}\subsetneqq\mathpzc{L}^{\prime}\subseteq\mathpzc{L}+\bar{\mathpzc{L}}.$

• •

contact nondegenerate if the associated contact triple $(\mathfrak{g}_{0},\breve{\mathpzc{L}}_{0},\tilde{\mathpzc{L}}_{0})$ is nondegenerate.

Remark 1.2**.**

For a fundamental almost $CR$ pair, strict nondegeneracy is equivalent to strict nondegeneracy of the associated contact triple and implies weak nondegeneracy, which in turn implies contact nondegeneracy.

2. Descending chain of a contact pair

Given a contact pair $(\mathfrak{g}_{0},\mathpzc{L}_{0})$, we construct a descending chain of $\operatorname{{\mathbb{R}}}$-linear subspaces of $\mathfrak{g}_{0}$

[TABLE]

defining by recurrence

[TABLE]

Since, by assumption, $\mathpzc{L}_{0}$ is a subspace of finite codimension that generates $\mathfrak{g}_{0}$ as a Lie algebra, there is a nonnegative integer $\muup$ such that $\mathpzc{G}_{-\muup}=\mathfrak{g}_{0}.$ Indeed the ascending chain of subspaces

[TABLE]

of the finite dimensional vector space $(\mathfrak{g}_{0}/\mathpzc{G}_{-1})$ stabilizes and, by their definition, if $(\mathpzc{G}_{-h}/\mathpzc{G}_{-1})=(\mathpzc{G}_{-h-1}/\mathpzc{G}_{-1}),$ then $\mathpzc{G}_{-r}=\mathpzc{G}_{-h}$ for all $r>h.$ For a contact pair ${\bigcup}_{h}\mathpzc{G}_{-h}=\mathfrak{g}_{0},$ and hence $\mathpzc{G}_{-h}=\mathpzc{G}_{-h-1}=\mathfrak{g}_{0}$ for some $h>0.$

Definition 2.1**.**

The smallest nonnegative integer $\muup$ for which $\mathpzc{G}_{-\muup}=\mathfrak{g}_{0}$ is called the depth, or kind, of the contact pair $(\mathfrak{g}_{0},\mathpzc{L}_{0}).$

Proposition 2.1**.**

Let $(\mathfrak{g}_{0},\mathpzc{L}_{0})$ be a contact pair, (2.1) the associated descending chain. Set

[TABLE]

Then:

1. (1)

All $\mathpzc{G}_{h}$ are closed subspaces of $\mathfrak{g}_{0}.$ 2. (2)

$\operatorname{\mathrm{c}}_{0}$* is the largest ideal of $\mathfrak{g}_{0}$ contained in $\mathpzc{L}_{0}.$* 3. (3)

(2.1) is a filtration of $\mathfrak{g}_{0}.$ 4. (4)

For all $h\geq{0},$ $\mathpzc{G}_{h}$ is a Lie subalgebra of $\mathfrak{g}_{0}.$

Proof.

(1) The closedness of $\mathpzc{G}_{-1}$ was assumed in the definition of a contact pair. For $h\geq{0}$ the statement follows by recurrence, because each $\mathpzc{G}_{h}$ is an intersection of the inverse images of the closed subspace $\mathpzc{G}_{h-1}$ by the continuous linear maps $\mathpzc{G}_{-1}\ni{X}\to[X,Y]\in\mathfrak{g}_{0},$ for $Y$ varying in $\mathpzc{G}_{-1}.$

For $h<-1$, the statement is true because all $\operatorname{{\mathbb{R}}}$-linear subspaces $V$ with $\mathpzc{L}_{0}\subseteq{V}\subseteq\mathfrak{g}_{0}$ are closed in $\mathfrak{g}_{0},$ since $\mathpzc{L}_{0}$ is closed and has finite codimension in $\mathfrak{g}_{0}.$ This completes the proof of (1).

(2) and the fact that $\mathpzc{G}_{0}$ is a Lie subalgebra of $\mathfrak{g}_{0}$ are streighforward consequences of the defintions.

To complete the proof, it suffices to check that (2.1) if a filtration. We begin by checking the commutators of elements belonging to subspaces with negative indices. If $h<-1,$ and we assume that $[\mathpzc{G}_{0},\mathpzc{G}_{h+1}]\subseteq\mathpzc{G}_{h+1}$, then

[TABLE]

This implies by recurrence that $[\mathpzc{G}_{0},\mathpzc{G}_{h}]\subseteq\mathpzc{G}_{h}$ for all $h\leq{0}.$

Let now $h>0$ and assume that $[\mathpzc{G}_{0},\mathpzc{G}_{h-1}]\subset\mathpzc{G}_{h-1}.$ By (2.2) we already have $[\mathpzc{G}_{h},\mathpzc{G}_{-1}]\subseteq\mathpzc{G}_{h-1}.$ Then

[TABLE]

shows, by recurrence, that $[\mathpzc{G}_{0},\mathpzc{G}_{h}]\subseteq\mathpzc{G}_{h}$ for all $h\geq{0.}$

By (2.2) we have $[\mathpzc{G}_{h},\mathpzc{G}_{-1}]\subseteq\mathpzc{G}_{h-1}$ for all integers $h$ and this implies that $[\mathpzc{G}_{h_{1}},\mathpzc{G}_{h_{2}}]\subseteq\mathpzc{G}_{h_{1}+h_{2}}$ when either $h_{1}\leq{0}$ or $h_{2}\leq{0}.$ When both $h_{1},h_{2}>0$, we can argue by recurrence on $h_{1}+h_{2}.$ In fact

[TABLE]

if we assumed that $[\mathpzc{G}_{h^{\prime}},\mathpzc{G}_{h^{\prime\prime}}]\subseteq\mathpzc{G}_{h^{\prime}+h^{\prime\prime}}$ when $h^{\prime}+h^{\prime\prime}<h_{1}+h_{2}.$ This completes the proof of the fact that (2.1) is a filtration and hence of the Proposition. ∎

Lemma 2.2**.**

If ${\mathfrak{h}}_{0}$ is any Lie subalgebra of $\mathfrak{g}_{0}$ such that $[{\mathfrak{h}}_{0},\mathpzc{L}_{0}]\subseteq\mathpzc{L}_{0}$, then

[TABLE]

∎

In particular, if $(\mathfrak{g}_{0},{\mathfrak{h}}_{0},\mathpzc{L}_{0})$ is a contact triple, then all subspaces $\mathpzc{G}_{h}$ of the canonical filtration (2.1) are ${\mathfrak{h}}_{0}$-modules.

Lemma 2.3**.**

Let (2.1) be the canonical filtration of the contact pair of a contact triple $(\mathfrak{g}_{0},{\mathfrak{h}}_{0},\mathpzc{L}_{0}).$ Then,

1. (1)

$(\mathfrak{g}_{0},{\mathfrak{h}}_{0},\mathpzc{L}_{0})$* is strictly nondegenerate if and only if ${\mathfrak{h}}_{0}=\mathpzc{G}_{0}.$* 2. (2)

$(\mathfrak{g}_{0},{\mathfrak{h}}_{0},\mathpzc{L}_{0})$* is nondegenerate, if and only if there is a positive integer $k$ such that $\mathpzc{G}_{k}\subseteq{\mathfrak{h}}_{0}.$*

Proof.

Statement $(1)$ follows immediately from the definitions.

To prove $(2),$ we note that, with $\operatorname{\mathrm{c}}_{0}={\bigcap}_{h\in\operatorname{\mathbb{Z}}}\mathpzc{G}_{h}$ as in (2.3), the nondegeneracy condition can be restated by saying that $\operatorname{\mathrm{c}}_{0}\subseteq{\mathfrak{h}}_{0}.$ This is equivalent to the fact that the intersection of all subspaces $(\mathpzc{G}_{h}+{\mathfrak{h}}_{0})/{\mathfrak{h}}_{0}$ in $\mathfrak{g}_{0}/{\mathfrak{h}}_{0}$ is $\{0\}.$ The statement follows because these subspaces form a descending chain of vector subspaces of the finite dimensional vector space $\mathfrak{g}_{0}/{\mathfrak{h}}_{0}.$ ∎

Definition 2.2**.**

The order of degeneracy of a contact triple $(\mathfrak{g}_{0},{\mathfrak{h}}_{0},\mathpzc{L}_{0})$ is the smallest positive integer $k$ for which $\mathpzc{G}_{k}\subseteq{\mathfrak{h}}_{0}.$

We observe that [math] is strict nondegeneracy, and degenerate corresponds to $\infty$-degenerate.

Example 2.4**.**

To a contact pair $(\mathfrak{g}_{0},\mathpzc{L}_{0})$ we can always associate the triples $(\mathfrak{g}_{0},\{0\},\mathpzc{L}_{0})$ and $(\mathfrak{g}_{0}/\operatorname{\mathfrak{a}}_{0},\mathpzc{G}_{0}/\operatorname{\mathfrak{a}}_{0},\mathpzc{L}_{0}/\operatorname{\mathfrak{a}}_{0}),$ where $\operatorname{\mathfrak{a}}_{0}$ is the largest ideal of $\mathfrak{g}_{0}$ which is contained in $\mathpzc{G}_{0}.$ The first one is a contact triple iff $\mathfrak{g}_{0}$ is finite dimensional. The second one is a contact triple provided $\mathfrak{g}_{0}/\mathpzc{G}_{0}$ is finite dimensional.

Remark 2.5**.**

In §4 we will explain how a contact triple is canonically associated to a homogeneous contact manifold, providing in this way a geometrical motivation for Definition 1.1.

It is useful to reformulate the nondegeneracy conditions of §1 in terms of iterated Lie brackets. We define by recurrence

[TABLE]

Proposition 2.6**.**

A necessary and sufficient condition in order that a contact triple $(\mathfrak{g}_{0},{\mathfrak{h}}_{0},\mathpzc{L}_{0})$ be $k$-nondegenerate is that

[TABLE]

A necessary and sufficient condition in order that a $CR$ algebra $(\mathfrak{g}_{0},\mathfrak{q})$ be weakly nondegenerate is that

[TABLE]

∎

3. Descending chain of a $CR$ algebra

Let $(\mathfrak{g}_{0},\mathfrak{q})$ be a $CR$ algebra. We already noted that strict nondegeneracy implies weak nondegeneracy. It is well known that the two conditions are not equivalent. Set $\tilde{\mathfrak{q}}=(\mathfrak{q}+\bar{\mathfrak{q}})$ and $\tilde{\mathfrak{q}}_{0}=\tilde{\mathfrak{q}}\cap\mathfrak{g}_{0}.$ These are not, in general, subalgebras, but only linear subspaces of $\mathfrak{g}$ and $\mathfrak{g}_{0},$ respectively. To better understand weak $CR$ nondegeneracy, we construct recursively descending chains of complex Lie subalgebras and of complex vector subspaces of $\mathfrak{g}:$

[TABLE]

by setting

[TABLE]

Lemma 3.1**.**

We have

[TABLE]

Proof.

Let us denote by $\mathpzc{A},$ $\mathpzc{B}$ the left and right hand side of (3.4), respectively. Since $[\bar{\mathfrak{q}}^{(h)},\mathfrak{q}]\subset\mathfrak{q}+\bar{\mathfrak{q}}^{(h-1)}$ for all $h>0,$ we obtain that $\mathpzc{A}\subseteq{\mathpzc{B}}.$

Vice versa, if $Z\in\bar{\mathfrak{q}}$ does not belong to $\mathpzc{A},$ then there is a positive integer $h$ with $Z\notin\bar{\mathfrak{q}}^{(h)}.$ This means that there is $W_{1}\in\mathfrak{q}$ such that $[Z,W_{1}]\notin\mathfrak{q}+\bar{\mathfrak{q}}^{(h-1)}.$ If $h=1,$ this suffices to show that $Z\notin{\mathpzc{B}}.$ If $h>1,$ write $[Z,W_{1}]=Z_{1}+W^{\prime}_{1}$ with $Z_{1}\in\bar{\mathfrak{q}}$ and $W_{1}\in\mathfrak{q}.$ Since, by assumption, $Z_{1}\notin\bar{\mathfrak{q}}^{(h-1)}$ we can find $W_{2}\in\mathfrak{q}$ such that $[Z_{1},W_{2}],$ and hence also $[Z,W_{1},W_{2}],$ does not belong to $\mathfrak{q}+\bar{\mathfrak{q}}^{(h-2)}.$ Iterating this argument, we show that there are $W_{1},\ldots,W_{h}\in\mathfrak{q}$ such that $[Z,W_{1},\ldots,W_{h}]\notin\mathfrak{q}+\bar{\mathfrak{q}}$ and therefore $Z$ does not belong to $\mathpzc{B}.$ This completes the proof. ∎

Lemma 3.2**.**

All $\bar{\mathfrak{q}}^{(h)}$ are complex Lie algebras and

[TABLE]

is the largest complex Lie subalgebra satisfying

[TABLE]

Proof.

Let us show first that the $\bar{\mathfrak{q}}^{(h)}$’s are Lie subalgebras. This holds true for $h=0,$ because the conjugate $\bar{\mathfrak{q}}$ of $\mathfrak{q}$ with respect to the real form $\mathfrak{g}_{0}$ is a complex Lie subalgebra of $\mathfrak{g}.$ Assume that we already know that $\bar{\mathfrak{q}}^{(h)}$ is a Lie algebra for some $h\geq{0}.$ If $Z_{1},Z_{2}\in\bar{\mathfrak{q}}^{(h+1)},$ we have $[Z_{1},Z_{2}]\in\bar{\mathfrak{q}}^{(h)}$ because $\bar{\mathfrak{q}}^{(h+1)}\subseteq\bar{\mathfrak{q}}^{(h)}$ and, by our inductive assumption, $\bar{\mathfrak{q}}^{(h)}$ is a Lie algebra. Moreover,

[TABLE]

shows that $[Z_{1},Z_{2}]\in\bar{\mathfrak{q}}^{(h+1)}.$ Clearly the right hand side of (3.5) is a Lie subalgebra of $\mathfrak{g}$ with $\mathfrak{q}\subseteq\mathfrak{q}^{\prime}\subseteq\mathfrak{q}+\bar{\mathfrak{q}}.$ By Lemma 3.1 it is the maximal complex Lie subalgebra of $\mathfrak{g}$ containing $\mathfrak{q}$ and contained in $\mathfrak{q}+\bar{\mathfrak{q}}.$ In fact it contains $\mathfrak{l}\cap\bar{\mathfrak{q}}$ for all complex Lie subalgebras with $\mathfrak{q}\subseteq\mathfrak{l}\subseteq\mathfrak{q}+\bar{\mathfrak{q}}.$ ∎

Proposition 3.3**.**

The following are equivalent

1. (1)

$(\mathfrak{g}_{0},\mathfrak{q})$* is weakly $CR$ nondegenerate;* 2. (2)

${\bigcap}_{h\geq{0}}\bar{\mathfrak{q}}^{(h)}=\mathfrak{q}\cap\bar{\mathfrak{q}};$** 3. (3)

$\bigcap_{h\geq{0}}\tilde{\mathfrak{q}}^{(h)}={\mathfrak{q}}.$* ∎*

Sequences (3.1) and (3.2) can be used to measure weak nondegeneracy. Let the lenght of a descending chain of vector spaces

[TABLE]

be the smallest integer $\nuup$ such that $V_{h}=V_{\nuup}$ for all $h\geq\nuup.$ By Lemma 3.2 we have the statement:

Proposition 3.4**.**

The sequences (3.1) and (3.2) have the same lenght $\nuup$ and all their terms with indices smaller than $\nuup$ are different. ∎

Definition 3.1**.**

We say that $(\mathfrak{g}_{0},\mathfrak{q})$ is $(\nuup-1)$-nondegenerate if the descending chains (3.1) and (3.2) have finite lenght $\nuup.$

Remark 3.5**.**

Strict nondegeneracy is [math]-nondegeneracy, while weak nondegeneracy is $\nuup$-nondegeneracy for some $\nuup<+\infty.$

Example 3.6**.**

For $CR$ algebras, contact is a weaker notion than weak nondegeneracy. A score of examples can be obtained by considering real orbits in complex flag manifolds ( see [1, 3]) whose $CR$ algebras are fundamental, but not weakly nondegenerate. We give here a simple example, consisting of the minimal orbit $M_{0}$ of $\mathbf{SU}(1,5)$ in the complex flag manifold $M$ consisting of triples $(\ell_{2},\ell_{3},\ell_{4})$ of complex $2,3,4$-planes in $\operatorname{\mathbb{C}}^{6}$ with $\ell_{2}\subset\ell_{3}\subset\ell_{4}.$ A point of $M_{0}$ is a flag in the $\mathbf{SU}(1,5)$-orthogonal of an isotropic line. Let us give the explicit matrix representation. We define $\mathfrak{g}_{0}\simeq\mathfrak{su}(1,5)$ and $\mathfrak{q},\mathfrak{q}^{\prime}\subset\mathfrak{sl}_{6}(\operatorname{\mathbb{C}})$ by

[TABLE]

Then $(\mathfrak{g}_{0},\mathfrak{q})$ is not weakly nondegenerate, because $\mathfrak{q}\subsetneqq\mathfrak{q}^{\prime}\subset\mathfrak{q}+\bar{\mathfrak{q}},$ while $(\mathfrak{g}_{0},\breve{\mathfrak{q}}_{0},\tilde{\mathfrak{q}}_{0})$ is a contact triple which is nondegenerate because $\mathfrak{su}(1,5)$ is simple and therefore does not contain nontrivial ideals.

4. Homogeneous contact structures

Let $\operatorname{\mathbf{G}}_{0}$ be a Lie group, acting transitively on a smooth manifold $M_{0}.$ Fix a point ${\mathrm{p}}_{0}$ of $M_{0}.$ The injective quotient of

[TABLE]

yields the idenfication $M_{0}\thickapprox\operatorname{\mathbf{G}}_{0}/\mathbf{H}_{0}$ of $M_{0}$ with the quotient of $\operatorname{\mathbf{G}}_{0}$ by the stabilizer $\mathbf{H}_{0}$ of ${\mathrm{p}}_{0}.$ A $\operatorname{\mathbf{G}}_{0}$-equivariant contact structure on $M_{0}$ is the datum of a constant rank distribution $\mathscr{L}^{*}$ on $M_{0},$ which is invariant for the left translations by elements of $\operatorname{\mathbf{G}}_{0}:$

[TABLE]

The pullback ${\mathscr{L}}=\piup^{*}(\mathscr{L}^{*})$ is a left-invariant distribution on $\operatorname{\mathbf{G}}_{0}$ generated by a subspace $\mathpzc{L}_{0}$ of left-invariant vector fields on $\operatorname{\mathbf{G}}_{0},$ containing the Lie algebra ${\mathfrak{h}}_{0}$ of $\mathbf{H}_{0}.$ Moreover, the vector subspace

[TABLE]

must be invariant for the differential at $p_{0}$ of the translations by elements of $\mathbf{H}_{0}.$ This yields

[TABLE]

which also implies that $[{\mathfrak{h}}_{0},\mathpzc{L}_{0}]\subseteq\mathpzc{L}_{0}$ for the Lie algebra ${\mathfrak{h}}_{0}$ of $\mathbf{H}_{0}.$

Vice versa, il $\mathpzc{L}_{0}$ is an $\operatorname{\mathrm{Ad}}(\mathbf{H}_{0})$-invariant linear subspace of the Lie algebra $\mathfrak{g}_{0}$ of $\operatorname{\mathbf{G}}_{0},$ then the push-forward on $M_{0}$ of the distribution on $\operatorname{\mathbf{G}}_{0}$ generated by the left-invariant vector fields corresponding to the elements of $\mathpzc{L}_{0}$ is a smooth distribution $\mathscr{L}^{*}=\mathpzc{L}_{0}^{*}$ on $M_{0},$ which is invariant by the $\operatorname{\mathbf{G}}_{0}$-translations on $M_{0}.$

Assume now that we do not know a priori that $M_{0}$ is a homogeneous space, but we are given a constant rank distribution $\mathscr{L}^{*}$ on $M_{0}$ and a Lie algebra ${\mathfrak{g}}_{0}^{*}$ of smooth vector fields on $M_{0}$ which leave $\mathscr{L}^{*}$ invariant: this means that $[\mathfrak{g}_{0}^{*},\mathscr{L}^{*}]\subseteq\mathscr{L}^{*}.$

We say that ${\mathfrak{g}}_{0}^{*}$ is transitive at ${\mathrm{p}}_{0}$ if

[TABLE]

Let $\mathfrak{g}_{0}={{\mathfrak{g}}_{0}^{*}}^{\mathrm{opp}},$ where the superscript  “opp”  means that, if we denote by $X$ the element of $\mathfrak{g}_{0}$ corresponding to the vector field $X^{*}$ of $\mathfrak{g}_{0}^{*},$ then

[TABLE]

With $L_{{\mathrm{p}}_{0}}=\{\Theta_{{\mathrm{p}}_{0}}\mid\Theta\in\mathscr{L}^{*}\},$ let us set

[TABLE]

Proposition 4.1**.**

If ${\mathfrak{g}}_{0}^{*}$ is transitive, then $(\mathfrak{g}_{0},{\mathfrak{h}}_{0},\mathpzc{L}_{0})$ is a transitive contact triple.

Proof.

The quotient $\mathfrak{g}_{0}/{\mathfrak{h}}_{0}$ maps injectively into $T_{{\mathrm{p}}_{0}}M_{0}$ and therefore is finite dimensional. Let $X\in\mathpzc{L}_{0}$ and $Y\in{\mathfrak{h}}_{0}.$ Then we can find a vector field $\Theta,$ vanishing at ${\mathrm{p}}_{0},$ such that $X^{*}+\Theta\in\mathpzc{L}^{*}_{0}.$ Then

[TABLE]

Since $[Y^{*},\Theta]$ vanishes at ${\mathrm{p}}_{0},$ this means that $[X,Y]^{*}_{{\mathrm{p}}_{0}}\in{L}_{p_{0}},$ showing that $[Y,X]\in\mathpzc{L}_{0}.$ ∎

5. $\operatorname{\mathbb{Z}}$-graded Lie algebras and a Tanaka’s

theorem

We will use the following criterion ([24, §11]):

Proposition 5.1** (N.Tanaka).**

Let

[TABLE]

be a $\operatorname{\mathbb{Z}}$-graded real Lie algebra, with $\dim_{\operatorname{{\mathbb{R}}}}(\mathfrak{G}_{h})<+\infty$ for all $h\in\operatorname{\mathbb{Z}},$ having finitely many summands with negative index. Assume that $\mathfrak{G}$ is transitive: this means that

[TABLE]

Then a necessary and sufficient condition for $\mathfrak{G}$ to be finite dimensional is that

[TABLE]

be finite dimensional.∎

Let us comment on this criterion. In the following, we assume that $\mathfrak{G}$ is transitive.

Clearly, $\mathfrak{G}^{\prime}$ is a $\operatorname{\mathbb{Z}}$-graded Lie subalgebra of $\mathfrak{G}$ and

[TABLE]

Given real vector spaces $V,W,$ let $\mathpzc{M}^{h}(V,W)$ denote the space of $W$-valued $h$-multilinear forms on $V$ and $\mathpzc{Symm}^{h}(V,W)$ the subspace consisting of those which are symmetric.

We define a map $\etaup\to\etaup_{h}$ of $\mathfrak{G}$ into $\mathpzc{M}^{h}(\mathfrak{G}_{-1},\mathfrak{G})$ by associating to each $\etaup\in\mathfrak{G}$ the multilinear form

[TABLE]

We also consider the alternate $\mathfrak{G}_{-2}$-valued bilinear form on $\mathfrak{G}_{-1}:$

[TABLE]

and set

[TABLE]

Lemma 5.2**.**

For each $h\geq{0},$ the maps

[TABLE]

are injective.

[TABLE]

Proof.

The fact that the maps in (5.6) are injective is a straightforward consequence of transitivity.

If $\etaup\in\mathfrak{G}_{0}$ and $[\etaup,\mathfrak{G}_{-2}]=0,$ then

[TABLE]

shows that $\etaup_{1}\in\mathfrak{sp}(\omegaup).$ In the same way, if $\etaup\in\mathfrak{G}$ and $[\etaup,\mathfrak{G}_{-2}]=0,$ we obtain that

[TABLE]

This gives at once that $[\etaup,\xiup_{1},\xiup_{2},\xiup_{3}]=[\etaup,\xiup_{2},\xiup_{1},\xiup_{3}]$ for $\etaup\in\mathfrak{G}^{\prime}$ and $\xiup_{1},\xiup_{2},\xiup_{3}\in\mathfrak{G}_{-1},$ while

[TABLE]

shows that also $[\etaup,\xiup_{1},\xiup_{2},\xiup_{3}]=[\etaup,\xiup_{1},\xiup_{3},\xiup_{2}]$ for $\etaup\in\mathfrak{G}^{\prime}$ and $\xiup_{1},\xiup_{2},\xiup_{3}\in\mathfrak{G}_{-1}.$ This yields symmetry on the triples. Arguing recursively on $k,$ we obtain, for all ${k}>3,$

[TABLE]

This shows that $[\etaup,\xiup_{1},\ldots,\xiup_{k-1},\xiup_{k}]=[\etaup,\xiup_{1},\ldots,\xiup_{k},\xiup_{k-1}].$ Thus $\etaup_{k}$ stays invariant under the transposition $(k-1,k).$ By the recursive assumption, it is also invariant under the transpositions $(j-1,j)$ for $2\leq{j}<k$ and thus is invariant under the full permutation group of $\{1,\ldots,k\}.$ ∎

Example 5.3**.**

Let $V$ be a real vector space of finite dimension $n,$ viewed as a degree $(-1)$-homogeneous Abelian real Lie algebra. Then its maximal Levi-Tanaka extension is isomorphic to the $\operatorname{\mathbb{Z}}$-graded Lie algebra $\mathfrak{P}$ of vector fields with polynomial coefficients in $\operatorname{{\mathbb{R}}}^{n},$ the grading being defined by

[TABLE]

Here $\operatorname{{\mathbb{R}}}_{h+1}[x_{1},\ldots,x_{n}]$ denotes the vector space of homogeneous polynomials of degree $(h+1)$ in the $x_{1},\ldots,x_{n}$ variables.

If $n=2m$ and $V$ has a complex structure, then the Levi-Tanaka extension of $V\oplus\mathfrak{gl}_{\operatorname{\mathbb{C}}}(V)$ is isomorphic to the $\operatorname{\mathbb{Z}}$-graded complex Lie algebra $\mathfrak{P}^{\operatorname{\mathbb{C}}},$ of homolorphic complex vector fields with holomorphic polynomial coefficients, with the gradation defined by

[TABLE]

Here $\operatorname{\mathbb{C}}_{h+1}[z_{1},\ldots,z_{n}]$ denotes the vector space of homogeneous holomorphic polynomials of degree $(h+1)$ in the $z_{1},\ldots,z_{n}$ variables.

Proof.

The fact that $\mathfrak{P}$ is a maximal transitive extension of $\mathfrak{P}_{-1}\simeq{V}$ is a consequence of the fact that $[\mathfrak{P}_{h},\mathfrak{P}_{-1}]=\mathfrak{P}_{h-1}$ for $h\geq{0}.$

Analogously, when $V$ has a complex structure, $\mathfrak{P}^{\operatorname{\mathbb{C}}}$ is a maximal transitive extensions of $V\oplus\mathfrak{gl}_{\operatorname{\mathbb{C}}}(V)\simeq\mathfrak{P}^{\operatorname{\mathbb{C}}}_{-1}\oplus\mathfrak{P}^{\operatorname{\mathbb{C}}}_{0}$ because $[\mathfrak{P}_{h}^{\operatorname{\mathbb{C}}},\mathfrak{P}_{-1}^{\operatorname{\mathbb{C}}}]=\mathfrak{P}_{h-1}^{\operatorname{\mathbb{C}}}$ for all $h\geq{0}.$ ∎

6. A finitness criterion for $CR$ algebras

We recall that the contact triple associated to a fundamental $CR$ algebra $(\mathfrak{g}_{0},\mathfrak{q})$ is

[TABLE]

We consider the canonical filtration (2.1) of the contact pair $(\mathfrak{g}_{0},\tilde{\mathfrak{q}}_{0})$ and the corresponding $\operatorname{\mathbb{Z}}$-graded Lie algebra

[TABLE]

Denote by $\piup_{h}:\mathpzc{G}_{h}\to\mathfrak{G}_{h}=\mathpzc{G}_{h}/\mathpzc{G}_{h+1}$ the projections onto the quotients.

Lemma 6.1** (Partial complex structure).**

There is a unique complex structure $J$ on $\mathfrak{G}_{-1},$ defined by

[TABLE]

The operator $\mathrm{J}\in\mathfrak{gl}_{\operatorname{{\mathbb{R}}}}(\mathfrak{G}_{-1})$ satisfies

[TABLE]

Proof.

To show that $\mathrm{J}$ is well defined, we need to verify that, if $X,Y\in\mathfrak{g}_{0},$ $X\in\mathpzc{G}_{0}$ and $X+\mathpzc{i}{Y}\in\mathfrak{q},$ then $Y\in\mathpzc{G}_{0}.$ If $X^{\prime}\in\mathpzc{G}_{-1},$ then we can find $Y^{\prime}\in\mathpzc{G}_{-1}$ such that $X^{\prime}+\mathpzc{i}{Y}^{\prime}\in\mathfrak{q}.$ Then

[TABLE]

Since by assumption both $[X,X^{\prime}]$ and $[X,Y^{\prime}]$ belong to $\mathpzc{G}_{-1},$ then also $[Y,X^{\prime}]$ and $[Y,Y^{\prime}]$ belong to $\mathpzc{G}_{-1}.$ This shows that $[Y,\mathpzc{G}_{-1}]\subset\mathpzc{G}_{-1},$ and then $Y\in\mathpzc{G}_{0}.$ Formula (6.4) holds in general for $X+\mathpzc{i}{Y},X^{\prime}+\mathpzc{i}{Y^{\prime}}\in\mathfrak{q},$ yielding (6.3). ∎

Lemma 6.2**.**

The $\mathfrak{G}_{-2}$-valued form (5.4) is nondegenerate, i.e.

[TABLE]

∎

Lemma 6.3**.**

Let $(\mathfrak{g}_{0},\mathfrak{q})$ be a $CR$ algebra and (2.1) the associated $\operatorname{\mathbb{Z}}$-filtration. assume that there is a nonnegative integer $k$ such that $\mathpzc{G}_{k}\subset{\mathfrak{h}}_{0}=\mathfrak{q}\cap\bar{\mathfrak{q}}\cap\mathfrak{g}_{0}.$ Then $\mathfrak{G}^{\prime}_{2k+1}=\{0\}.$

Proof.

By the assumption, any $Y$ of $\mathpzc{G}_{k}$ belongs to $\mathfrak{q}$ and therefore

[TABLE]

In the complexification of $\mathfrak{G},$ this yelds the equation

[TABLE]

as $\piup_{-2}(\mathfrak{q})=\{0\},$ because $\mathfrak{q}$ is contained in the complexification of $\mathpzc{G}_{-1}.$

Let now $\etaup\in\mathfrak{G}_{2k+1}.$ Fix $\xiup\in\mathfrak{G}_{-1}$ and consider, for $0\leq{h}\leq{2k+3},$ the $(2k+4)$ vectors of $\mathfrak{G}_{-2}:$

[TABLE]

For each choice of $\xiup_{1},\ldots,\xiup_{k+1}$ in $\mathfrak{G}_{-1},$ the multi-commutator $[\etaup,\xiup_{1},\ldots,\xiup_{k+1}]$ belongs to $\mathfrak{G}_{k}$. Thus, setting, for each integer $h$ with $0\leq{h}\leq{k+1},$ $\xiup_{i}^{h}=\mathrm{J}(\xiup)$ if $1\leq{i}\leq{h}$ and $\xiup_{i}^{h}=\xiup$ if $h<i\leq{k+1},$ the real and imaginary parts of

[TABLE]

yield $(2k+4)$ linear combinations of $\mathpzc{v}_{0},\ldots,\mathpzc{v}_{2k+3}$ which sum to zero. These can be written in the form

[TABLE]

where $M_{k}$ is a real $(2k+4)\times(2k+4)$ matrix $M_{k}$ whose columns are the coefficients of the real and imaginary parts of the polynomials $t^{h}(t-i)^{k+2},$ for $0\leq{t}\leq({k+1}).$

It is easy to check that these polynomials form a basis of the $(2k+4)$-dimensional $\operatorname{\mathbb{C}}$-vector space $\operatorname{\mathbb{C}}_{2k+3}[t]$ of polynomials of degree less or equal to $(2k+3).$

In fact their $\operatorname{\mathbb{C}}$-linear span contains the $(2k+4)$ polynomials $(t-i)^{k+h+2}$ and $(t+i)^{k+h+2},$ for $0\leq{h}\leq{k+1}.$ These are linearly independent and hence form a basis of $\operatorname{\mathbb{C}}_{2k+3}[t].$ Indeed, let $a_{h},b_{h}$ be complex coefficients for which

[TABLE]

Let $r\leq{k+1}$ and assume that we already know that $a_{h}=0$ and $b_{h}=0$ if $h>r,$ this being obviously the case when $r=(k+1).$ Then

[TABLE]

shows that also $a_{r}=0$ and $b_{r}=0.$ By recurrence, this proves that all coefficients $a_{h},b_{h}$ must be zero and thus the claimed linear independence of the polynomials $(t-i)^{k+h+2},$ $(t+i)^{k+h+2}.$

Hence $M_{k}$ is nondegenerate and (6.5) tells us that all vectors $\mathpzc{v}_{0},\ldots,\mathpzc{v}_{2k+3}$ are zero. In particular $\mathpzc{v}_{0}=0$ and therefore we proved that

[TABLE]

Since, by Lemma 5.2, the multilinear $\mathfrak{G}_{-2}$-valued form $\etaup_{2k+3}$ is symmetric in its arguments, it follows by polarization (see e.g. [25, p.5]) that

$[\etaup,\xiup_{1},\ldots,\xiup_{2k+3}]=0$ for all $\xiup_{1},\ldots,\xiup_{2k+3}\in\mathfrak{G}_{-1}.$

Since $\omegaup$ is nondegenerate, this yields

$[\etaup,\xiup_{1},\ldots,\xiup_{2k+2}]=0$ for all $\xiup_{1},\ldots,\xiup_{2k+2}\in\mathfrak{G}_{-1}$

and hence, by transitivity, $\etaup=0.$ ∎

Thus we obtain, using also [14, Theorem 10.2],

Theorem 6.4**.**

A $CR$ algebra $(\mathfrak{g}_{0},\mathfrak{q})$ for which the associated contact triple $(\mathfrak{g}_{0},\breve{\mathfrak{q}}_{0},\tilde{\mathfrak{q}}_{0})=(\mathfrak{g}_{0},\mathfrak{q}\cap\bar{\mathfrak{q}}\cap\mathfrak{g}_{0},(\mathfrak{q}+\bar{\mathfrak{q}})\cap\mathfrak{g}_{0})$ is nondegenerate is finite dimensional. ∎

We will prove in §8 that, under the assumptions of Theorem 6.4, $(\mathfrak{g}_{0},\mathfrak{q})$ has a maximal extension and that this is finite dimensional. To this aim we will generalise, in §7, the construction of §4, by a procedure similar to that of [7, 11].

7. Transitive pairs and generalised contact distributions

7.1. Vector fields with formal power series coefficients

Let $V$ be a finite dimensional vector space. The space $\mathpzc{F}$ of formal power series associated to $V$ is the infinite direct sum

[TABLE]

where $\mathpzc{Symm}^{h}(V)=\mathpzc{Symm}^{h}(V,\operatorname{{\mathbb{R}}})$ are the real-valued homogeneous multilinear symmetric forms of degree $h$ (cf. § 5). The coefficient $\alphaup_{0}$ is the value at [math] of ${\sum}_{h=0}^{\infty}\alphaup_{h}.$ With the standard operations, $\mathpzc{F}$ is a local ring, whose maximal ideal $\mathpzc{F}_{0}$ consists of formal power series vanishing at $0.$

Each vector $\mathpzc{v}$ of $V$ defines a derivation $D{v}$ on $\mathpzc{F},$ whose action on each summand $\mathpzc{Symm}^{h}(V)$ is described by

[TABLE]

The set $\mathpzc{Der}(\mathpzc{F})$ of derivations of $\mathpzc{F}$ is the left $\mathpzc{F}$-module $\mathpzc{F}\otimes_{\operatorname{{\mathbb{R}}}}{V}$ generated by $V.$ Thus any derivation $X^{*}$ is a formal series

[TABLE]

Denote by $\mathpzc{Der}_{0}(\mathpzc{F})$ the Lie subalgebra $\mathpzc{F}_{0}\otimes{V}$ of derivations vanishing at $0.$

7.2. The case of Lie groups

Let $\operatorname{\mathbf{G}}_{0}$ be a real Lie group. We recall that the left and right invariant vector fields on $\operatorname{\mathbf{G}}_{0}$ coincide with the infinitesimal generators $L_{X}$ and $R_{X}$ of the one-parameter groups

[TABLE]

of diffeomorphisms of $\operatorname{\mathbf{G}}_{0},$ respectively. We have the commutation rules

[TABLE]

The exponential map is a diffeomorphism of an open neighbourhood of [math] in its Lie algebra $\mathfrak{g}_{0}$ onto an open neighborhood of the identity, defining a local chart. In order to determine the Taylor series expansions of $L_{X},$ $R_{X}$ in in these coordinates, it is convenient to consider the identities

[TABLE]

for $\mathpzc{v},X\in\mathfrak{g}_{0}.$ Since the maps $X\to{L}_{X}$ and $X\to{R}_{X}$ are linear, we obtain

[TABLE]

showing that

[TABLE]

We can obtain the Taylor series expansions of $L_{X}$ and $R_{X}$ in the $\mathfrak{g}_{0}$-coordinates from the Baker-Campbell-Hausdorff formula (see e.g. [8]):

[TABLE]

where the coefficients $b_{h}$ are defined by

[TABLE]

7.3. Homogeneous spaces

Let $M=\operatorname{\mathbf{G}}_{0}/\mathbf{H}_{0}$ be a homogeneous space, with base point ${\mathrm{p}}_{0}=[\mathbf{H}_{0}].$ Fix a linear complement $V$ of ${\mathfrak{h}}_{0}$ in $\mathfrak{g}_{0}.$ The map

[TABLE]

restricts to a diffeomorphism of the product of a neighborhood of $\{0\}\times\mathbf{H}_{0}$ onto an open neighbourhood of the identity in $\operatorname{\mathbf{G}}_{0}.$ We use the projection on $M$ of $\exp(\mathpzc{v})$ to define coordinates near ${\mathrm{p}}_{0}.$ In fact, if $\varpi:\operatorname{\mathbf{G}}_{0}\to{M}$ is the natural projection, with $\varpi(x)=x\cdot{\mathrm{p}}_{0},$ then $\mathpzc{v}\to\varpi(\exp(\mathpzc{v}))$ is a diffeomorphism of an open neighbourhood of [math] in $V$ onto an open neighbourhood of ${\mathrm{p}}_{0}$ in $M.$ Let $\mathpzc{F}$ be as in §7.1.

In analogy with the definitions of $L_{X}$ and $R_{X}$ in §7.2, we may introduce a couple of vector fields $L^{*}_{X}$ and $R_{X}^{*}$ on $M,$ as infinitesimal generators of the one-parameter groups of diffeomorphisms of $M,$ locally defined by

[TABLE]

Let us find their formal power series expansions. Set

[TABLE]

Using (7.5), we obtain

[TABLE]

yielding by recurrence

[TABLE]

Let $\piup:\mathfrak{g}_{0}\to{V}$ be the projection along ${\mathfrak{h}}_{0}.$ Equations (7.9) can be used to obtain explicit formulae for $\mathfrak{x}_{h}$ and $\mathpzc{h}^{\prime}_{h}$:

[TABLE]

Since the projection $\operatorname{\mathbf{G}}_{0}\ni{x}\to{x}\cdot{\mathrm{p}}_{0}\in{M}$ relates $R_{X}^{*}$ with the right invariant vector field on $\operatorname{\mathbf{G}}_{0}$ corresponding to $X,$ we obtain:

Lemma 7.1**.**

We have

[TABLE]

∎

Analogously, let us set

[TABLE]

From the equation

[TABLE]

we obtain

[TABLE]

Equations (7.13) can be used to obtain recursive formulae for $\mathfrak{y}_{h}$ and $\mathpzc{h}_{h}$:

[TABLE]

Note that $L_{Y}^{*}=0$ when $Y\in{\mathfrak{h}}_{0}.$

To compute the commutator of $R_{X}^{*}$ and $L^{*}_{Y}$ for a pair $X,Y\in\mathfrak{g}_{0}$, we use the infinitesimal description of their flows, which can be obtained from (7.7). Set $\Phi_{X}$ for the flow of $R_{X}^{*}$ and $\Psi_{Y}$ for the flow of $L_{Y}^{*}.$ Then

[TABLE]

By lifting the action to $\operatorname{\mathbf{G}}_{0},$ we have

[TABLE]

Then we obtain the composition

[TABLE]

This yields

Lemma 7.2**.**

For $X,Y\in\mathfrak{g}_{0},$ we have

[TABLE]

where $H(\mathpzc{v})$ and $H^{\prime}(\mathpzc{v})$ are described by (7.14) and (7.10), and the right hand side is a composition of formal power series. ∎

This lemma tells us that the infinitesimal translations of $L^{*}_{\mathpzc{w}},$ for a $\mathpzc{w}\in{V},$ can be expressed as formal power series whose coefficients are $L^{*}_{Y}$’s for $Y$ in the ${\mathfrak{h}}_{0}$-module generated by $\mathpzc{w}.$ In a similar way we also obtain

Lemma 7.3**.**

For $Y_{1},Y_{2}\in\mathfrak{g}_{0},$ we have

[TABLE]

where $H_{i}\in\mathpzc{F}\otimes{\mathfrak{h}}_{0}$ is defined by the first line of (7.7), with $X=Y_{i}$, for $i=1,2.$ ∎

7.4. General transitive pairs

Let us fix a transitive pair. We recall from Definition 1.1 that it is a couple $(\mathfrak{g}_{0},{\mathfrak{h}}_{0})$ consisting of a linearly compact topological Lie algebra $\mathfrak{g}_{0}$ and of a finite codimensional closed subalgebra ${\mathfrak{h}}_{0}$ that does not contain any nontrivial ideal of $\mathfrak{g}_{0}.$

Let us fix a finite dimensional complement $V$ of ${\mathfrak{h}}_{0}$ in $\mathfrak{g}_{0}.$ We define $R_{X}^{*}(\mathpzc{v}),\,L_{X}^{*}(\mathpzc{v}),\,H(\mathpzc{v}),\,H^{\prime}(\mathpzc{v})$ by (7.8), (7.9), (7.12), (7.13), after noticing that to write these formulae it is not needed that $\mathfrak{g}_{0}$ be finite dimensional, because the homogeneous summands in the $V$-coordinates of their Taylor series only involve finite powers of $\operatorname{\mathrm{ad}}(\mathpzc{v}),$ finite linear combinations, and the projection $\piup:\mathfrak{g}_{0}\to{V}$ along ${\mathfrak{h}}_{0}.$ Set

[TABLE]

Theorem 7.4**.**

If $(\mathfrak{g}_{0},{\mathfrak{h}}_{0})$ is a transitive pair and $V$ a linear complement of ${\mathfrak{h}}_{0}$ in $\mathfrak{g}_{0}$, then the map

[TABLE]

is injective and defines an anti-isomorphism of Lie algebras between $\mathfrak{g}_{0}$ and $\mathfrak{g}_{0}^{*},$ with

[TABLE]

The correspondence

[TABLE]

is a bijection between the set of ${\mathfrak{h}}_{0}$-submodules of $\mathfrak{g}_{0}$ containing ${\mathfrak{h}}_{0}$ and $\mathfrak{g}_{0}^{*}$-invariant left $\mathpzc{F}$-modules of $\mathpzc{Der}(\mathpzc{F}).$ Its inverse is given by

[TABLE]

Proof.

Let $X\in\mathfrak{g}_{0}$ and assume that $R_{X}^{*}=0.$ We use (7.10). From $\mathfrak{x}_{0}=0$ se obtain that $X\in{\mathfrak{h}}_{0}.$ Since $b_{1}=\tfrac{1}{2}\neq{0},$ the condition that $\mathfrak{x}_{1}=0$ means that $[\mathpzc{v},X]\in{\mathfrak{h}}_{0}$ for all $\mathpzc{v}\in{V},$ and hence $[Y,X]\in{\mathfrak{h}}_{0}$ for all $Y\in\mathfrak{g}_{0}.$ In general, we find that $\mathfrak{h}_{h}(\mathpzc{v}),$ for each $h\geq{0},$ is a multiple of $(\operatorname{\mathrm{ad}}(\mathpzc{v})^{h}(X)$ and, arguing by recurrence we obtain that $(\operatorname{\mathrm{ad}}(Y)^{h}(X)\in{\mathfrak{h}}_{0}$ for all $h\geq{0}$ and $Y\in\mathfrak{g}_{0}.$ This yields that actually $\operatorname{\mathrm{ad}}(Y_{1})\circ\cdots\circ\operatorname{\mathrm{ad}}(Y_{h})(X)\in{\mathfrak{h}}_{0}$ for all $h$ and $Y_{1},\ldots,Y_{h}\in\mathfrak{g}_{0}.$ To show this fact, we argue again by recurrence on $h,$ as the cases of $h=0,1$ are already settled. For $h>1,$ we note that, for $t_{1},\ldots,t_{h}\in\operatorname{{\mathbb{R}}},$

[TABLE]

By the recursive assumption, the coefficient of the monomial $t_{1}\cdots{t}_{h}$ in the right hand side is an element of ${\mathfrak{h}}_{0}$ and hence $\operatorname{\mathrm{ad}}(Y_{1})\circ\cdots\circ\operatorname{\mathrm{ad}}(Y_{h})(X)\in{\mathfrak{h}}_{0}.$ We proved that the kernel of the map $X\to{R}_{X}^{*}$ is an ideal of $\mathfrak{g}_{0}$ contained in ${\mathfrak{h}}_{0}.$ Therefore it is $\{0\}$ if $(\mathfrak{g}_{0},{\mathfrak{h}}_{0})$ is a transitive pair.

The conclusion of Lemma 7.1 only depends on the formal definition of $R_{X}^{*}$ in (7.8) and (7.9) and therefore is still valid, yielding (7.18).

Also the validity of Lemma 7.2 depends only on the formal definitions of $R_{X}^{*}$ and $L_{Y}^{*}$ and therefore shows that, when $\mathpzc{L}_{0}$ contains ${\mathfrak{h}}_{0}$ and $[{\mathfrak{h}}_{0},\mathpzc{L}_{0}]\subseteq\mathpzc{L}_{0},$ then the left $\mathpzc{F}$-module $\mathpzc{L}_{0}^{*}$ generated by $\{L^{*}_{\mathpzc{w}}\mid\mathpzc{w}\in{V}\cap\mathpzc{L}_{0}\}$ satisfies $[\mathfrak{g}_{0}^{*},\mathpzc{L}_{0}^{*}]\subseteq\mathpzc{L}_{0}^{*}.$ Vice versa, if $\mathpzc{L}_{0}^{*}$ is a left $\mathpzc{F}$ submodule of $\mathpzc{Der}(\mathpzc{F})$ with $[\mathfrak{g}_{0}^{*},\mathpzc{L}_{0}^{*}]\subseteq\mathpzc{L}_{0}^{*},$ then the set $\mathpzc{L}_{0}$ of $Y\in\mathfrak{g}_{0}$ such that $\piup(Y)$ is the value in [math] of a vector field in $\mathpzc{L}_{0}^{*}$ is a subspace of $\mathfrak{g}_{0}$ containing ${\mathfrak{h}}_{0}$ and satisfying $[{\mathfrak{h}}_{0},\mathpzc{L}_{0}]\subseteq{\mathfrak{h}}_{0}.$ Indeed $[X,\mathpzc{w}]$ is the value at [math] of $[R_{X}^{*},L^{*}_{\mathpzc{w}}].$ This yields the correspondence (7.20), completing the proof of the theorem. ∎

We already noted that the map $Y\to{L}^{*}_{Y}$ is linear. In particular, it can be extended by $\operatorname{\mathbb{C}}$-linearity to the case where $Y$ belongs to the complexification $\mathfrak{g}$ of $\mathfrak{g}_{0}$. Then the second part of the statement of Theorem 7.4 extends to the case of complex vector distributions. We denote by ${\mathfrak{h}}\subseteq\mathfrak{g}$ the complexification of ${\mathfrak{h}}_{0}.$

Theorem 7.5**.**

The correspondence

[TABLE]

is a bijection between the set of ${\mathfrak{h}}$-submodules of $\mathfrak{g}$ containing ${\mathfrak{h}}$ and $\mathfrak{g}_{0}^{*}$-invariant left $\operatorname{\mathbb{C}}\otimes\mathpzc{F}$-modules of $\operatorname{\mathbb{C}}\otimes\mathpzc{Der}(\mathpzc{F}).$ Its inverse is given by

[TABLE]

∎

8. Extensions

Definition 8.1**.**

We say that a contact triple $(\mathfrak{g}_{0}^{\prime},{\mathfrak{h}}_{0}^{\prime},\mathpzc{L}^{\prime}_{0})$ extends the contact triple $(\mathfrak{g}_{0},{\mathfrak{h}}_{0},\mathpzc{L}_{0})$ if there is an injective homomorphism of real Lie algebras $\phiup:\mathfrak{g}_{0}\to\mathfrak{g}_{0}^{\prime}$ such that $\phiup(\mathpzc{L}_{0})\subseteq\mathpzc{L}^{\prime}_{0},$ $\phiup({\mathfrak{h}}_{0})\subseteq{\mathfrak{h}}_{0}^{\prime}$ and the quotient maps $(\mathfrak{g}_{0}/\mathpzc{L}_{0})\to(\mathfrak{g}_{0}^{\prime}/\mathpzc{L}_{0}^{\prime})$ and $(\mathfrak{g}_{0}/{\mathfrak{h}}_{0})\to(\mathfrak{g}_{0}^{\prime}/{\mathfrak{h}}_{0}^{\prime})$ induced by $\phiup$ are linear isomorphisms.

We say that a $CR$ algebra $(\mathfrak{g}_{0}^{\prime},\mathfrak{q}^{\prime})$ extends the $CR$ algebra $(\mathfrak{g}_{0},\mathfrak{q})$ if there is an injective Lie algebras homomorphism $\phiup:\mathfrak{g}_{0}\to\mathfrak{g}_{0}^{\prime},$ whose complexification we still denote by $\phiup,$ such that $\phiup(\mathfrak{q})\subseteq\mathfrak{q}^{\prime}$ and the induced map on the quotients $\mathfrak{g}_{0}/(\mathfrak{q}\cap\bar{\mathfrak{q}}\cap\mathfrak{g}_{0})\to\mathfrak{g}_{0}^{\prime}/(\mathfrak{q}^{\prime}\cap\bar{\mathfrak{q}}^{\prime}\cap\mathfrak{g}_{0}^{\prime})$ and $(\mathfrak{g}/\mathfrak{q})\to(\mathfrak{g}^{\prime}/\mathfrak{q}^{\prime})$ are linear isomorphisms.

To deal with extensions, it is convenient to introduce a common Lie algebra in which we can embed both a given Lie algebra and its extension.

Proposition 8.1**.**

Let $(\mathfrak{g}_{0},{\mathfrak{h}}_{0},\mathpzc{L}_{0})$ be a contact triple. Then there is a maximal contact triple $(\mathfrak{g}_{0}^{\prime},{\mathfrak{h}}_{0}^{\prime},\mathpzc{L}_{0}^{\prime})$ extending $(\mathfrak{g}_{0},{\mathfrak{h}}_{0},\mathpzc{L}_{0}),$ which is unique modulo isomorphisms.

Let $(\mathfrak{g}_{0},\mathfrak{q})$ be a $CR$ algebra. Then there is a maximal $CR$ algebra $(\mathfrak{g}_{0}^{\prime},\mathfrak{q}^{\prime})$ extending $(\mathfrak{g}_{0},\mathfrak{q}),$ which is unique modulo isomorphisms.

Proof.

The statement follows from Theorems 7.4 and 7.5. If $\mathpzc{L}_{0}^{*}$ is the $\mathpzc{F}$-module corresponding to $\mathpzc{L}_{0}$, we define ${\mathfrak{g}_{0}^{\prime}}^{*}$ as the Lie algebra of formal vector fields stabilising $\mathpzc{L}^{*}_{0}$ and $\mathfrak{g}_{0}^{\prime}$ equals to its opposite Lie algebra.

Likewise, in the case of a $CR$ algebra, we take ${\mathfrak{g}_{0}^{\prime}}^{*}$ to be the stabiliser of ${\tilde{\mathfrak{q}}}^{*}$ in $\mathpzc{Der}(\mathpzc{F})$ and define $\mathfrak{g}_{0}^{\prime}$ to be its opposite Lie algebra. ∎

The finiteness result of §6 applies to give informations about the global and local $CR$ automorphisms on homogeneous and locally homogeneous $CR$ manifolds.

The analytic Lie subgroup of a Lie group $\operatorname{\mathbf{G}}_{0}$ generated by a Lie subalgebra ${\mathfrak{h}}_{0}$ of its Lie algebra $\mathfrak{g}_{0}$ may fail to be closed. In this case the pair $(\mathfrak{g}_{0},{\mathfrak{h}}_{0})$ is associated to a locally $\operatorname{\mathbf{G}}_{0}$-homogeneous manifold, i.e. a smooth open manifold $M_{0}$ having the property that the elements of a small open neighborhood of the origin of $\operatorname{\mathbf{G}}_{0}$ act as a transitive group of local diffeomorphisms on an open neighborhood of a base point ${\mathrm{p}}_{0}$ of $M_{0},$ and ${\mathfrak{h}}_{0}$ is the Lie algebra of the stabilizer of ${\mathrm{p}}_{0}$ for this action (see e.g. [17, 22]). We can give in an obvious way a notion of locally $\operatorname{\mathbf{G}}_{0}$-homogeneous $CR$ manifold, that we employ in the formulation of the following result.

Theorem 8.2**.**

Every contact nondegenerate $CR$ algebra $(\mathfrak{g}_{0},\mathfrak{q})$ admits an essentially unique maximal extension $(\mathfrak{g}_{0}^{\prime},\mathfrak{q}^{\prime}),$ which is finite dimensional and is therefore a $CR$ algebra of a locally homogeneous $CR$ manifold whose $CR$ automorphisms form a Lie group of transformations.∎

Theorem 8.3**.**

Let $\operatorname{\mathbf{G}}_{0}$ be a Lie group and $M_{0}$ a locally $\operatorname{\mathbf{G}}_{0}$-homogeneous $CR$ manifold, with associated $CR$ algebra $(\mathfrak{g}_{0},\mathfrak{q}).$ If $(\mathfrak{g}_{0},\mathfrak{q})$ is fundamental and contact nondegenerate, then the local $CR$ automorphisms of $M_{0}$ generate a finite dimensional Lie group. ∎