Groupes locaux analytiques et abstraits. (Analytical and abstract local groups).

*(French)*Zbl 0694.22001The first half of this paper is an interesting account of approaches to the integrability of (finite-dimensional, real) Lie algebras. The author has written profoundly on this subject in the past [for example, Nederl. Akad. Wet., Proc., Ser. A 65, 391-408, 409-425 (1962; Zbl 0105.024, Zbl 0109.020)], and the present account is an overview of new work contained in two more detailed papers [the author, Sémin. Sud-Rhodan. Géom. VIII; 1986, Trav. Cours 27, 83-96 (1988; Zbl 0652.17002); the author and M. A. M. van der Lee, ibid., 97-127 (1988; Zbl 0657.22007)] and considers also, in its second half, the use of these methods in more general integrability problems.

Since the introduction of rigorous global methods into Lie group theory by E. Cartan and H. Weyl, there have been essentially two methods of proving the integrability of Lie algebras. Firstly, one may use the Levi- Malcev decomposition to reduce the problem to the two separate cases of solvable and semi-simple Lie algebras, in each of which cases integrability is easily proved. Secondly, one may integrate a general Lie algebra directly to a local Lie group and then seek to embed this into a global Lie group. In both approaches a crucial role is played by the vanishing of a second order cohomology or homotopy group: in the first case, the Levi-Malcev decomposition depends upon the Whitehead lemmas; in the second, globalizability depends upon the vanishing of the second homotopy group of a semi-simple Lie group.

The general problem of globalizing an abstractly given local (Lie) group or group extension was studied by A. Malcev [“Sur les groupes locaux et complets”, Dokl. Akad. Nauk SSSR 32, 606-608 (1941)] and P. A. Smith [Ann. Math., II. Ser. 54, 371-424 (1951; Zbl 0044.198)]. The paper under review gives a careful modern treatment of this work, interpreting the results in cohomological terms, and showing that the results of Malcev and Smith, which appear to differ in significant respects, are in fact dual in a precise sense.

In the second half of the paper, the author considers generalizations of this work to Lie groupoids, and, more briefly, infinite dimensional Lie groups. He considers a simplicial version of the following problem, which is closely related to the integrability problem for transitive Lie algebroids: Let F be a bundle of Lie groups (“pinceau de groupes”) on base E; under what conditions is it the bundle of vertex (or isotropy, or gauge) groups of a locally trivial Lie groupoid on E? For this problem to be well-posed, it is necessary to stipulate the curvature of possible connections in the required groupoid, and the author shows how some steps of the problem may be reduced to the globalizability criteria of Malcev and Smith.

(Reviewer’s remark: A full account of this last-mentioned problem, in the \(C^{\infty}\) category, is given in the reviewer’s paper [J. Pure Appl. Algebra 58, 181-208 (1989; Zbl 0673.55015)].)

Since the introduction of rigorous global methods into Lie group theory by E. Cartan and H. Weyl, there have been essentially two methods of proving the integrability of Lie algebras. Firstly, one may use the Levi- Malcev decomposition to reduce the problem to the two separate cases of solvable and semi-simple Lie algebras, in each of which cases integrability is easily proved. Secondly, one may integrate a general Lie algebra directly to a local Lie group and then seek to embed this into a global Lie group. In both approaches a crucial role is played by the vanishing of a second order cohomology or homotopy group: in the first case, the Levi-Malcev decomposition depends upon the Whitehead lemmas; in the second, globalizability depends upon the vanishing of the second homotopy group of a semi-simple Lie group.

The general problem of globalizing an abstractly given local (Lie) group or group extension was studied by A. Malcev [“Sur les groupes locaux et complets”, Dokl. Akad. Nauk SSSR 32, 606-608 (1941)] and P. A. Smith [Ann. Math., II. Ser. 54, 371-424 (1951; Zbl 0044.198)]. The paper under review gives a careful modern treatment of this work, interpreting the results in cohomological terms, and showing that the results of Malcev and Smith, which appear to differ in significant respects, are in fact dual in a precise sense.

In the second half of the paper, the author considers generalizations of this work to Lie groupoids, and, more briefly, infinite dimensional Lie groups. He considers a simplicial version of the following problem, which is closely related to the integrability problem for transitive Lie algebroids: Let F be a bundle of Lie groups (“pinceau de groupes”) on base E; under what conditions is it the bundle of vertex (or isotropy, or gauge) groups of a locally trivial Lie groupoid on E? For this problem to be well-posed, it is necessary to stipulate the curvature of possible connections in the required groupoid, and the author shows how some steps of the problem may be reduced to the globalizability criteria of Malcev and Smith.

(Reviewer’s remark: A full account of this last-mentioned problem, in the \(C^{\infty}\) category, is given in the reviewer’s paper [J. Pure Appl. Algebra 58, 181-208 (1989; Zbl 0673.55015)].)

Reviewer: K.Mackenzie

##### MSC:

22E05 | Local Lie groups |

58H05 | Pseudogroups and differentiable groupoids |

17B05 | Structure theory for Lie algebras and superalgebras |

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

20L05 | Groupoids (i.e. small categories in which all morphisms are isomorphisms) |

22E60 | Lie algebras of Lie groups |

55R10 | Fiber bundles in algebraic topology |