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**Sur les pseudogroupes de Lie.**
*(French)*
Zbl 0068.02101

Colloque de Topologie de Strasbourg, Années 1954-1955, 20 p. (1955).

In this exposition, a rigorous foundation is laid for the theory of infinite (Lie-) groups in the sense of É. Cartan. The principal tool is the theory of jets as developed by Ch. Ehresmann. As these groups are defined as transformation pseudogroups given by partial differential equations, one starts with the variety \(J^q(V_n,V_n)\), of all jets of order \(q\) whose source and goal are on an analytic variety \(V_n\). An elementary analytic completely regular subvariety is a set in a coordinate neighbourhood of \(J^q\), given by a set of partial differential equations of order \(q\) in a canonical form, the Jacobian having constant rank. A pseudogroup \(\Gamma\) of analytical transformations of \(V_n\) is said to be complete of order \(q\), if \(q\) is the least order for which there exists jet-groupoids (sub-groupoids of \(J^q( V_n, V_n)\) determining \(\Gamma\). Finally, \(\Gamma\) is a Lie-pseudogroup if \(\Gamma\) is complete of order \(q\) and if the associated jet-groupoid of order \(q\) is an analytic completely regular (a. c. r.) subvariety. \(\Gamma\) is of finite type, of degree \(r\), if the restriction to an elementary variety of the projection of \(J^r(\Gamma)\) (jets of order \(r\) defined by the mappings in \(\Gamma)\) onto \(J^{r-1}(\Gamma)\) is an isomorphism. It is shown that with \(J^q(\Gamma)\) all \(J^s(\Gamma)\), \(s\le q\), are a. c. r., therefore their solutions also form Lie pseudogroups. The degree of a group is never less than its order. The prolongation of a pseudogroup is defined. It is shown that on any elementary a. c..r. subvariety, the action of \(J^q(\Gamma)\) may be described by linear independent Pfaffian forms \(\omega^j\), with

\[ d\omega^j = c_{ik}^j\omega^i \wedge \omega^k + a_{i\alpha}^j\omega^i\wedge \pi_\alpha, \]

where the \(a_{i\alpha}^j\) and \(c_{ik}^j\) are invariant under the operation of a neighbourhood of the image of the identity in \(J^q\). In this way, the whole of É. Cartan’s theory of infinite Lie “groups” is based on a sound base, topics treated are: infinitesimal isotropy groups, local equivalence of pseudogroups, sub-pseudogroups, the fundamental theorem of É. Cartan [Ann. Sci. Éc. Norm. Supér. (3) 25, 57–194 (1908; JFM 39.0206.04)] (this theorem now appears as a theorem on the “feuilletage” defined by a prolongation of order \(h\) on one of order \(h+1\), and seems to be quite stronger than É. Cartan’s original theorem).

Imprimitive and simple pseudogroups are treated next, as are systatic varieties. In the whole, it turns out that for concrete computations, one may safely use É. Cartan’s methods.

\[ d\omega^j = c_{ik}^j\omega^i \wedge \omega^k + a_{i\alpha}^j\omega^i\wedge \pi_\alpha, \]

where the \(a_{i\alpha}^j\) and \(c_{ik}^j\) are invariant under the operation of a neighbourhood of the image of the identity in \(J^q\). In this way, the whole of É. Cartan’s theory of infinite Lie “groups” is based on a sound base, topics treated are: infinitesimal isotropy groups, local equivalence of pseudogroups, sub-pseudogroups, the fundamental theorem of É. Cartan [Ann. Sci. Éc. Norm. Supér. (3) 25, 57–194 (1908; JFM 39.0206.04)] (this theorem now appears as a theorem on the “feuilletage” defined by a prolongation of order \(h\) on one of order \(h+1\), and seems to be quite stronger than É. Cartan’s original theorem).

Imprimitive and simple pseudogroups are treated next, as are systatic varieties. In the whole, it turns out that for concrete computations, one may safely use É. Cartan’s methods.

Reviewer: H. Guggertheimer

### MSC:

58H05 | Pseudogroups and differentiable groupoids |

22A22 | Topological groupoids (including differentiable and Lie groupoids) |

22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |