Lie groupoids and Lie algebroids in differential geometry.

*(English)*Zbl 0683.53029
London Mathematical Society Lecture Note Series, 124. Cambridge etc.: Cambridge University Press. XV, 327 p.; £17.50; $ 34.50 (1987).

Topological and differentiable (or smooth) groupoids (and more generally categories) were introduced by C. Ehresmann in 1958 [Centre Belge Rech. math., Colloque Géom. diff. globale, Bruxelles du 19 au 22 Déc. 1958, 137-150 (1959; Zbl 0205.282)]. The main motivations at that period were the structure of jet manifolds (with the composition of jets) and the gauge groupoid of a principal bundle, which is a transitive and locally trivial groupoid. The holonomy groupoid of a foliation (later named “graph of the foliation”) gives an important motivation for considering groupoids which are not locally transitive and also groupoids endowed with other structures (such as measure, Riemannian metric, symplectic or Poisson structure), and which may be “structured” in a sense different from Ehresmann’s.

In the book under review, the term of “Lie groupoids” (following Ngô Van Quê) means transitive locally trivial groupoids (note that other authors, such as Kumpera and Weinstein, use it for general differentiable groupoids). As indicated by the title and clearly claimed in the preface, the book is mainly centered on that special case of smooth groupoids and on Ehresmann’s approach of the theory of principal bundles via their gauge groupoid (disregarding certain phenonena which are specific to the nontransitive case). The corresponding infinitesimal object is in that case equivalent to the so-called Atiyah sequence of the bundle; its elements were named “infinitesimal displacements of the fibres” by Ehresmann who used them for a geometric formulation of infinitesimal connections. In the general case the term of Lie algebroid was introduced by the reviewer to describe the precise algebraic-differentiable structure of this object (which is more precise than a general Lie pseudo-algebra) and to emphasize that it is the candidate for generalizing the role of the Lie algebra of a Lie group.

In the transitive case, the infinitesimal connections correspond to the splittings of the Atiyah sequence of Lie algebroids, so that this geometric situation may be viewed as a generalization of the purely algebraic one consisting of the splittings of exact sequences of Lie algebras. The book under review is focused on this formal analogy which is exploited with much virtuosity and leads to a very rich and elegant cohomological theory which unifies the equivariant de Rham cohomology for principal bundles and the Hochschild-Serre theory for Lie algebras. This is a very important and illuminating contributon. Note that it relies deeply on the fact that the kernel in the Atiyah sequence is indeed a Lie algebra bundle, which is specific to the transitive case. By a curious irony of fate, that strategy had been developed by the author in order to give a van Est style cohomological proof of Lie’s third integrability theorem for Lie algebroids, a statement announced by the reviewer in 1968. As pointed out much later by P. Molino and R. Almeida in 1985, it turns out that the global form of this statement is erroneous, counterexamples arising from the transverse theory of foliations. Then the author immediately realized that the precise measure of the obstruction lies in a third order cohomology class resulting from his previous constructions, while Molino and Almeida gave an independent direct and “more elementary” description of this obstruction, pointing out its connection with other geometrical problems considered earlier by A. Weil, Aragnol and Kostant (in quantization theory) [R. Almeida and P. Molino, in Sémin. Géom. Différ. Univ. Sci. Tech. Languedoc 1984/1985, 39-59 (1985; Zbl 0596.57017)].

To conclude, this book is very important and stimulating and must be read by geometers and physicists. It draws attention to the gauge groupoid approach for bundles (inaugurated by Ehresmann, but too much disregarded by geometers and rediscovered by physicists) as well as to the infinitesimal corresponding notion of Lie algebroid (now extensively used, in the nontransitive case, in symplectic geometry). It is certainly not a definitive textbook which closes a classical subject [the author himself announces improvements and developments, Cah. Topologie Géom. Différ. Catégoriques 28, 283-302 (1987; Zbl 0646.22002), ibid., No.1, 29-52 (1987; Zbl 0626.57015) and in J. Pure Appl. Algebra 58, No.2, 181- 208 (1989)], but a stimulating guide for new researches.

{Reviewer’s remarks: The reviewer regrets that the Lie functorial correspondence between Lie groupoids and Lie algebroids is developed only with a fixed base: it is certainly illuminating to interpret conditions of Maurer-Cartan type as defining more general morphisms of Lie algebroids and to state Lie’s second theorem in this wider framework. On the other hand, the reader is warned that the assumption of paracompactness for manifolds is not sufficient for certain statements (notably Corollary 1.9, p. 89) and should be replaced by the second countability condition.}

In the book under review, the term of “Lie groupoids” (following Ngô Van Quê) means transitive locally trivial groupoids (note that other authors, such as Kumpera and Weinstein, use it for general differentiable groupoids). As indicated by the title and clearly claimed in the preface, the book is mainly centered on that special case of smooth groupoids and on Ehresmann’s approach of the theory of principal bundles via their gauge groupoid (disregarding certain phenonena which are specific to the nontransitive case). The corresponding infinitesimal object is in that case equivalent to the so-called Atiyah sequence of the bundle; its elements were named “infinitesimal displacements of the fibres” by Ehresmann who used them for a geometric formulation of infinitesimal connections. In the general case the term of Lie algebroid was introduced by the reviewer to describe the precise algebraic-differentiable structure of this object (which is more precise than a general Lie pseudo-algebra) and to emphasize that it is the candidate for generalizing the role of the Lie algebra of a Lie group.

In the transitive case, the infinitesimal connections correspond to the splittings of the Atiyah sequence of Lie algebroids, so that this geometric situation may be viewed as a generalization of the purely algebraic one consisting of the splittings of exact sequences of Lie algebras. The book under review is focused on this formal analogy which is exploited with much virtuosity and leads to a very rich and elegant cohomological theory which unifies the equivariant de Rham cohomology for principal bundles and the Hochschild-Serre theory for Lie algebras. This is a very important and illuminating contributon. Note that it relies deeply on the fact that the kernel in the Atiyah sequence is indeed a Lie algebra bundle, which is specific to the transitive case. By a curious irony of fate, that strategy had been developed by the author in order to give a van Est style cohomological proof of Lie’s third integrability theorem for Lie algebroids, a statement announced by the reviewer in 1968. As pointed out much later by P. Molino and R. Almeida in 1985, it turns out that the global form of this statement is erroneous, counterexamples arising from the transverse theory of foliations. Then the author immediately realized that the precise measure of the obstruction lies in a third order cohomology class resulting from his previous constructions, while Molino and Almeida gave an independent direct and “more elementary” description of this obstruction, pointing out its connection with other geometrical problems considered earlier by A. Weil, Aragnol and Kostant (in quantization theory) [R. Almeida and P. Molino, in Sémin. Géom. Différ. Univ. Sci. Tech. Languedoc 1984/1985, 39-59 (1985; Zbl 0596.57017)].

To conclude, this book is very important and stimulating and must be read by geometers and physicists. It draws attention to the gauge groupoid approach for bundles (inaugurated by Ehresmann, but too much disregarded by geometers and rediscovered by physicists) as well as to the infinitesimal corresponding notion of Lie algebroid (now extensively used, in the nontransitive case, in symplectic geometry). It is certainly not a definitive textbook which closes a classical subject [the author himself announces improvements and developments, Cah. Topologie Géom. Différ. Catégoriques 28, 283-302 (1987; Zbl 0646.22002), ibid., No.1, 29-52 (1987; Zbl 0626.57015) and in J. Pure Appl. Algebra 58, No.2, 181- 208 (1989)], but a stimulating guide for new researches.

{Reviewer’s remarks: The reviewer regrets that the Lie functorial correspondence between Lie groupoids and Lie algebroids is developed only with a fixed base: it is certainly illuminating to interpret conditions of Maurer-Cartan type as defining more general morphisms of Lie algebroids and to state Lie’s second theorem in this wider framework. On the other hand, the reader is warned that the assumption of paracompactness for manifolds is not sufficient for certain statements (notably Corollary 1.9, p. 89) and should be replaced by the second countability condition.}

##### MSC:

53C05 | Connections (general theory) |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

58H05 | Pseudogroups and differentiable groupoids |

55R15 | Classification of fiber spaces or bundles in algebraic topology |

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

17B56 | Cohomology of Lie (super)algebras |

22E60 | Lie algebras of Lie groups |