Transitive pseudogroups of local diffeomorphisms preserving geometric structures on manifolds have been studied by many authors; the origins of this subject are classical, and may be said to lie in the works of Sophus Lie and Élie Cartan. The structure of such a pseudogroup Γ acting on a manifold X is reflected in the structure of the Lie algebra of formal infinitesimal transformations of Γ, that is to say, those formal vector fields on X which are formal solutions to the linear partial differential equation which defines the infinitesimal transformations of Γ. The Lie algebras of formal vector fields obtained...

Throughout this section, we denote by K an arbitrary field of characteristic zero which is endowed with the discrete topology.

Let V be a Hausdorff topological vector space over K. Then V is said to be linearly compact if:

(i) V is complete; and

(ii) There exists a fundamental system {V_α} of neighborhoods of 0 in V such that each V_αis a vector subspace of finite codimension in V.

An example of such a space is provided by the local algebra\[\text{F}=\text{K}[[{{\text x}_{1}},\ \ldots \ ,{{\text{x}}_{\text n}}]]\]of formal power series over K in n indeterminates x₁, … , x_n. Denote In by F⁰...

Throughout this section, we denote by K a field of characteristic zero.

Let L be a Lie algebra over K, and let A be a subalgebra of L. By a derivation of A into L, we shall mean a K-linear mapping\[\overline{\text{D}}:\text{A}\to \text{L}\ ,\]such that, for all ξ,$\eta \ \epsilon \ \text{A}$\[\overline{\text{D}}([\xi,\eta ])=[\overline{\text{D}}(\xi),\eta ]+[\xi,\overline{\text{D}}(\eta )]\ ;\]from the Jacobi identity in L, we see that if$\zeta\ \epsilon \ \text{L}$, then ad(ζ) is a derivation of A into L. When A = L, such a linear mapping$\overline{\text{D}}$is simply a derivation of L. Recall that, in the last section, we established the notation Der(R) for the Lie algebra...

Let L be a linearly compact Lie algebra over a field K of characteristic zero. Suppose that V is a finite-dimensional vector space over K. We denote by F the linearly compact local algebra\[\text{F}=\text{F}\{{{\text{V}}^{*}}\}\]of formal power series on V, and we set\[{{\text{F}}^{l}}={{\text{F}}^{l}}\{{{\text V}^{*}}\}\ ,\quad \quad \quad \text{for}\ l\ge -1\ .\]Recall that the completed tensor product$\text{L}{{\hat{\otimes }}_{\text K}}\ \text{F}$, defined in Section one, is a linearly compact vector space over K. There is a natural structure of Lie algebra on the dense subspace$\text{L}{{\otimes }_{\text{K}}}\ \text{F}$of$\text{L}{{\hat{\otimes }}_{\text K}}\ \text{F}$, defined by setting\[\caption {(3.1)} [\xi \otimes \text{f},\eta \otimes \text{g}]=[\xi,\eta ]\otimes (\text{fg})\ ,\]for all ξ,$\eta \ \epsilon \ \text{L}$and f,$\text{g}\ \epsilon \ \text{F}$. One verifies easily that there is a...

To increase the intelligibility of this section, we have reproduced, in Lemma 4.1 through Corollary 4.3, certain technical results from the paper ([11]) of Guillemin.

Let L be a linearly compact Lie algebra over a field K of characteristic zero, and let I be a non-zero closed ideal of L. According to (ii) of Lemma 1.6, there exists a closed ideal J of I which is properly contained in I and strictly maximal. Since J is closed in L, the normalizer N_L(J) of J in L is a closed subalgebra of L. The first series of lemmas of this section...

Throughout this section, if V is a linearly compact topological vector space over${\rm I\!R}$, we denote by\[{{\text{V}}_{\mathbb{C}}}=\mathbb{C}{{\otimes }_{\rm{I\!R}}}\text{V}\]the complexification of V; then${{\text{V}}_{\mathbb{C}}}$, is a linearly compact complex topological vector space.

Let L be a linearly compact Lie algebra over${\rm I\!R}$, and let I be a non-abelian minimal closed ideal of L. Recall that, by Corollary 4.3, there exists a unique maximal closed ideal J of I, and the quotient

R = I/J

is a non-abelian simple transitive Lie algebra. According to Proposition 1.4, the commutator ring K_Rof R is a field which is a...