##
**The general theory of Lie groupoids and Lie algebroids.**
*(English)*
Zbl 1078.58011

London Mathematical Society Lecture Note Series 213. Cambridge: Cambridge University Press (ISBN 0-521-49928-3/pbk). xxxvii, 501 p. (2005).

The book is a fine and up-to-date introduction to the theory of Lie groupoids and Lie algebroids written by a leading specialist in the field. The book consists of 12 chapters grouped in 3 parts, each dealing with a different aspect of the theory. Each chapter starts with a short introduction, presenting its contents and ends with very detailed and informative notes, where the history of the topic is sketched, credits for the presented results are given, and author’s choices explained and defended.

Groups appear naturally in many parts of mathematics, but one of the classical sources of groups is the study of symmetry, i.e., global transformations which preserve a given structure, in the modern mathematical language – global automorphisms of the given structure. For many well known geometrical structures these automorphisms groups have the additional structure of a smooth manifold compatible with the group structure which led to the notion of a Lie group. Groupoids seem to be the natural formulation of the symmetry of a fibred object. The automorphism group of a linear space is its general linear group. If there is a scalar product or a complex structure, then this new geometrical structure has as its automorphism group the orthogonal or complex linear group. A Riemannian structure or an almost complex structure on a manifold induces a scalar product or a complex structure, respectively, in each tangent space. As the tangent bundle at a point is a linear approximation of a small open neighbourhood of this point, and the study of geometry of the manifold involves the study of local transformations preserving the geometrical structure, it is important to study automorphisms of the corresponding tangent spaces with their geometrical structures. In this way one obtains the groupoid of all linear isomorphisms between tangent spaces, or its subgroupoid of isomorphisms which preserve a given geometrical structure, e.g., scalar products or complex structures. If we consider local automorphisms of a geometrical structure we obtain a pseudogroup. The germs or jets of elements of this pseudogroup form a groupoid. These examples clearly show that the study of local automorphisms of a geometrical structure lead directly to a groupoid.

Another source of examples of the groupoid theory is the fundamental groupoid of a topological space. The Lie theory of Lie groups and Lie algebras is one of the cornerstones of the 20th century mathematics, and Lie groupoids and Lie algebroids are good candidates for one of the cornerstones of the 21st century. There are many similarities between the theories, e.g., the three classical results of Lie theory generalize to meaningful questions in the framework of the theory of Lie groupoids and Lie algebroids, i.e., the integrability of morphisms, the integrability of subobjects, and the integrability of abstract Lie algebroids; the research programme and the results were first formulated by J. Pradines in a series of four CRAS notes in 1966–1968 [C. R. Acad. Sci., Paris, Sér. A 263, 907–910 (1966; Zbl 0147.41102), 264, 245–248 (1967; Zbl 0154.21704), 266, 1194–1196 (1968; Zbl 0172.03601), 267, 21–23 (1968; Zbl 0172.03502)].

The second part of the book is dedicated to a detailed exposition of these problems in the case of locally trivial Lie groupoids and transitive Lie algebroids. However, Lie groupoids which are not locally trivial appear naturally in differential geometry and topology, e.g., symplectic realizations of Poisson manifolds or in the theory of foliations. Most results presented in the first and third part apply to general Lie groupoids and Lie algebroids. Some of the most recent developments not covered in the book are discussed in the appendix, i.e., the relations with the foliation theory, the general integrability obstructions of Crainic and Fernandes, and double Lie groupoids and double Lie algebroids. The book contains an extensive and very up-to-date bibliography.

The first part dealing with the general theory consists of four chapters. In the first chapter the fundamental theory is presented. At the very beginning basic definitions and examples are given, including the notions of locally trivial Lie groupoid and bisection. The last two sections are concerned with actions of Lie groupoids on general maps. For groupoids there is a diagramatic characterization of actions in terms of the notion of action groupoid and action morphism. This feature of groupoid theory is not available for groups. It permits to give conceptually simple proofs of many fundamental results, especially those which involve passing from a groupoid action to the corresponding Lie algebroid action.

The second chapter is dedicated to quotients, in the most general sense, which include the descend constructions, as well as pullbacks and semidirect products for Lie groupoids.

The third chapter presents the fundamental theory of Lie algebroids. In the first two sections the construction of the Atiyah sequence of a principal bundle is given; the first and one of the most important classes of examples of Lie algebroids. The following two sections contain basic definitions and examples. The fifth section contains the construction of the Lie algebroid of a Lie groupoid and the construction is illustrated with examples. Sections 6 and 7 are dedicated to the Lie groupoid version of the exponential map and adjoint formulas.

The forth chapter is dedicated to the general concept of morphism of Lie algebroids. One has to pay particular attention to this notion as the bracket of a Lie algebroid is defined not on the vector bundle but on the module of its sections, and a morphism of vector bundles does not generally induce a map of sections. The author deals with the problem as follows. First, he considers the case when the target Lie algebroid may be pulled back across the base map; in this case there is a natural definition of a morphism of Lie algebroids, and the general definition is a reformulation of this particular case. This approach requires that the pullbacks of Lie algebroids are defined before the notion of a morphism, which is defined in the third section. The remaining sections treat general quotients, generalized actions and notions of semidirect products of Lie algebroids.

The second part, consisting of four chapters, presents the theory of transitive groupoids and algebroids. Chapter 5 treats infinitesimal connections, that is connections in transitive Lie algebroids. A lot of the general connection theory, called by the author the infinitesimal theory can be naturally formulated in the language of abstract transitive Lie algebroids. Such an exposition, according to the author, maintaining a separation between those parts of the theory which require principal bundles or Lie groupoids and those which do not, clarifies the exposition and strengthen the results. The definition of an infinitesimal connection is presented in the second section, together with some basic properties and examples, including the notion of curvature and the existence theorem.

The third section, the longest one, is concerned with the translation between the principal bundle language and that of Lie algebroids. The last section is concerned with the local theory.

The sixth chapter presents another aspect of the connection theory, the process by which an infinitesimal connection in the Lie algebroid of a locally trivial Lie groupoid may be integrated to a law of path lifting, i.e., to a path connection. The first section is preparatory, there is a construction of the monodromy groupoid of a locally trivial and \(\alpha\)-connected Lie groupoid. Its role in the Lie groupoid theory is similar to the role played by the universal covering of a connected Lie group in the theory of Lie groups. The second section contains proofs of the first and second integrability theorems of Lie for locally trivial Lie groupoids and algebroids and transitive Lie algebroids. The proofs are based on those known for Lie groups and Lie algebras. In the next section the notion of path connection in a locally trivial Lie groupoid is formally defined, and the author establishes the correspondence between path connections in Lie groupoids and infinitesimal connections in the corresponding Lie algebroid. The author treats these two notions of connection separately as one transitive Lie algebroid may arise from several distinct Lie groupoids, which are only locally isomorphic. Some properties like curvature depend only on the algebroid, but, e.g., the holonomy depend on the connectivity properties of the Lie groupoid. The next section studies relations between the action of an infinitesimal connection and the action of its holonomy groupoid, e.g., a strong, Lie groupoid version of the Ambrose-Singer theorem is presented. The fifth section deals with abstract transitive Lie algebroids and their morphisms, it presents various refinements of some previously obtained results.

The seventh chapter is dedicated to cohomology and Schouten calculus for Lie algebroids. Most results are valid for arbitrary Lie algebroids. In the first section, the cohomology of an arbitrary Lie algebroid with coefficients in an arbitrary representation is constructed in terms of a standard resolution of de Rham or Chevalley-Eilenberg type. As the focus of geometric interest is usually on specific transversals or cocycles, the standard cochain complex is the best suited, according to the author, to the purpose. The first three sections present the definitions and the techniques. The forth one deals with the spectral sequence of a transitive Lie algebroid which provides an abstraction and algebraization of the Leray-Serre spectral sequence of a principal bundle in de Rham cohomology. In these sections, the author demonstrates how the standard identities of infinitesimal connection theory arise naturally in cohomological terms. The last section provides an exposition of the extension of the classical Schouten calculus of multi-vector fields to arbitrary Lie algebroids.

In the next chapter the author studies the cohomological obstructions to integrability of transitive Lie algebroids, that is Lie’s third theorem for transitive Lie algebroids. There is a single cohomological obstruction in this case which gives a complete solution. For general Lie algebroids the problem was recently solved by M. Crainic and R. L. Fernandes [Ann. Math. (2) 157, 575–620 (2003; Zbl 1037.22003)]. The first section of the chapter presents some interesting examples. In the second one the author presents his classification of transitive Lie algebroids in terms of ‘system of transition data’. This classification is an exact analogue of the classification principal bundles by transition functions. On the way, a strong form of local integrability for transitive Lie algebroids is proved. For the Lie algebroid of a locally trivial Lie groupoid, the system of transitional data is obtained from a groupoid cocycle by differentiation. Trying to reverse the process one encounters the cohomological obstruction class, this is presented in the third and last section of this chapter and part.

The last part of the book is concerned with Poisson and symplectic theories. The ninth chapter is dedicated to double vector bundles. The importance of double vector bundles reveals itself when one studies the relations between Lie algebroids and Poisson structures. A lot of the basic theory is a more or less straightforward generalization of results well known for ordinary vector bundles. The real differences appear in the theory of duality, which is presented in the second section. The next three sections are dedicated to the study of the two duals of the tangent bundle of a vector bundle. In the sixth section, relations between bracket structures on a double tangent bundle and its duals are studied in view of the canonical isomorphisms defined earlier. The following two sections present applications of the general results to Lie algebroids and their prolongations. The formalism used in this chapter is global and intrinsic, the author avoids local coordinate formulations of the traditional type.

The second chapter of the third part is dedicated to the explanation of relations between Poisson structures and Lie algebroids. A Poisson structure on a manifold \(M\) gives rise to a Lie algebra structure on \(C^{\infty}(M)\), which is infinite dimensional, and its properties are not easily translated into the properties of the Poisson manifold \(M\). However, the same structure induces a Lie algebroid structure on the cotangent bundle \(T^{\ast}M\). This cotangent Lie algebroid of \(M\) is a major tool in the study of Poisson manifolds and an important source of examples of Lie algebroids. Many important problems of the Poisson manifold theory can be reformulated in the terms of properties of the cotangent Lie algebroid where one can apply the whole machinery of the Lie theory of Lie groupoids and Lie algebroids.

The penultimate chapter of the book is concerned with Poisson and symplectic groupoids. The author does not pretend to present the theories of these groupoids in their generality as they are well developed and have very rich literature. The aim of the chapter is to show how the general features of Poisson groupoids, and thus of symplectic groupoids may be obtained by a systematic application of the so called ‘diagramatic’; methods to the cotangent bundles. The account of Poisson is new and appears for the first time in print. As an immediate consequence of this approach to Poisson groupoids is the fact that, according to the author, the basics of symplectic groupoids theory may be developed without any use of genuine symplectic geometry.

The last chapter is dedicated to elements of the theory of Lie bialgebroids, which are the infinitesimal form of Poisson groupoids; applying the Lie functor to a Poisson groupoid one gets a standard Lie algebroid with a Lie algebroid structure on its vector bundle dual. The pair obtained in such a way constitutes the Lie bialgebroid of the initial Poisson groupoid. The abstract definition of a Lie bialgebroid is not related to Poisson manifolds and is presented in the first section. The second section provides a criterion for a vector bundle and its dual to be a Lie bialgebroid. In the following section, the author provides additional information to clarify even more the notion under consideration and its relation to Poisson groupoids. The last section gives a brief introduction to moments maps in the language of Lie algebroids and Lie groupoids.

The book can serve both as a reference source and a manual for advanced students. The presentation and the choice of the topics are very impressive. One can easily discern a long fascination by and research involvement in the subject. Numerous notions and results presented are due to the author and his collaborators and have been mostly available in research papers. The second part has a lot in common with the previous book of the author [Lie groupoids and Lie algebroids in differential geometry. London Mathematical Society Lecture Note Series, 124. (Cambridge University Press). (1987; Zbl 0683.53029)], but numerous improvements and corrections have been made.

Groups appear naturally in many parts of mathematics, but one of the classical sources of groups is the study of symmetry, i.e., global transformations which preserve a given structure, in the modern mathematical language – global automorphisms of the given structure. For many well known geometrical structures these automorphisms groups have the additional structure of a smooth manifold compatible with the group structure which led to the notion of a Lie group. Groupoids seem to be the natural formulation of the symmetry of a fibred object. The automorphism group of a linear space is its general linear group. If there is a scalar product or a complex structure, then this new geometrical structure has as its automorphism group the orthogonal or complex linear group. A Riemannian structure or an almost complex structure on a manifold induces a scalar product or a complex structure, respectively, in each tangent space. As the tangent bundle at a point is a linear approximation of a small open neighbourhood of this point, and the study of geometry of the manifold involves the study of local transformations preserving the geometrical structure, it is important to study automorphisms of the corresponding tangent spaces with their geometrical structures. In this way one obtains the groupoid of all linear isomorphisms between tangent spaces, or its subgroupoid of isomorphisms which preserve a given geometrical structure, e.g., scalar products or complex structures. If we consider local automorphisms of a geometrical structure we obtain a pseudogroup. The germs or jets of elements of this pseudogroup form a groupoid. These examples clearly show that the study of local automorphisms of a geometrical structure lead directly to a groupoid.

Another source of examples of the groupoid theory is the fundamental groupoid of a topological space. The Lie theory of Lie groups and Lie algebras is one of the cornerstones of the 20th century mathematics, and Lie groupoids and Lie algebroids are good candidates for one of the cornerstones of the 21st century. There are many similarities between the theories, e.g., the three classical results of Lie theory generalize to meaningful questions in the framework of the theory of Lie groupoids and Lie algebroids, i.e., the integrability of morphisms, the integrability of subobjects, and the integrability of abstract Lie algebroids; the research programme and the results were first formulated by J. Pradines in a series of four CRAS notes in 1966–1968 [C. R. Acad. Sci., Paris, Sér. A 263, 907–910 (1966; Zbl 0147.41102), 264, 245–248 (1967; Zbl 0154.21704), 266, 1194–1196 (1968; Zbl 0172.03601), 267, 21–23 (1968; Zbl 0172.03502)].

The second part of the book is dedicated to a detailed exposition of these problems in the case of locally trivial Lie groupoids and transitive Lie algebroids. However, Lie groupoids which are not locally trivial appear naturally in differential geometry and topology, e.g., symplectic realizations of Poisson manifolds or in the theory of foliations. Most results presented in the first and third part apply to general Lie groupoids and Lie algebroids. Some of the most recent developments not covered in the book are discussed in the appendix, i.e., the relations with the foliation theory, the general integrability obstructions of Crainic and Fernandes, and double Lie groupoids and double Lie algebroids. The book contains an extensive and very up-to-date bibliography.

The first part dealing with the general theory consists of four chapters. In the first chapter the fundamental theory is presented. At the very beginning basic definitions and examples are given, including the notions of locally trivial Lie groupoid and bisection. The last two sections are concerned with actions of Lie groupoids on general maps. For groupoids there is a diagramatic characterization of actions in terms of the notion of action groupoid and action morphism. This feature of groupoid theory is not available for groups. It permits to give conceptually simple proofs of many fundamental results, especially those which involve passing from a groupoid action to the corresponding Lie algebroid action.

The second chapter is dedicated to quotients, in the most general sense, which include the descend constructions, as well as pullbacks and semidirect products for Lie groupoids.

The third chapter presents the fundamental theory of Lie algebroids. In the first two sections the construction of the Atiyah sequence of a principal bundle is given; the first and one of the most important classes of examples of Lie algebroids. The following two sections contain basic definitions and examples. The fifth section contains the construction of the Lie algebroid of a Lie groupoid and the construction is illustrated with examples. Sections 6 and 7 are dedicated to the Lie groupoid version of the exponential map and adjoint formulas.

The forth chapter is dedicated to the general concept of morphism of Lie algebroids. One has to pay particular attention to this notion as the bracket of a Lie algebroid is defined not on the vector bundle but on the module of its sections, and a morphism of vector bundles does not generally induce a map of sections. The author deals with the problem as follows. First, he considers the case when the target Lie algebroid may be pulled back across the base map; in this case there is a natural definition of a morphism of Lie algebroids, and the general definition is a reformulation of this particular case. This approach requires that the pullbacks of Lie algebroids are defined before the notion of a morphism, which is defined in the third section. The remaining sections treat general quotients, generalized actions and notions of semidirect products of Lie algebroids.

The second part, consisting of four chapters, presents the theory of transitive groupoids and algebroids. Chapter 5 treats infinitesimal connections, that is connections in transitive Lie algebroids. A lot of the general connection theory, called by the author the infinitesimal theory can be naturally formulated in the language of abstract transitive Lie algebroids. Such an exposition, according to the author, maintaining a separation between those parts of the theory which require principal bundles or Lie groupoids and those which do not, clarifies the exposition and strengthen the results. The definition of an infinitesimal connection is presented in the second section, together with some basic properties and examples, including the notion of curvature and the existence theorem.

The third section, the longest one, is concerned with the translation between the principal bundle language and that of Lie algebroids. The last section is concerned with the local theory.

The sixth chapter presents another aspect of the connection theory, the process by which an infinitesimal connection in the Lie algebroid of a locally trivial Lie groupoid may be integrated to a law of path lifting, i.e., to a path connection. The first section is preparatory, there is a construction of the monodromy groupoid of a locally trivial and \(\alpha\)-connected Lie groupoid. Its role in the Lie groupoid theory is similar to the role played by the universal covering of a connected Lie group in the theory of Lie groups. The second section contains proofs of the first and second integrability theorems of Lie for locally trivial Lie groupoids and algebroids and transitive Lie algebroids. The proofs are based on those known for Lie groups and Lie algebras. In the next section the notion of path connection in a locally trivial Lie groupoid is formally defined, and the author establishes the correspondence between path connections in Lie groupoids and infinitesimal connections in the corresponding Lie algebroid. The author treats these two notions of connection separately as one transitive Lie algebroid may arise from several distinct Lie groupoids, which are only locally isomorphic. Some properties like curvature depend only on the algebroid, but, e.g., the holonomy depend on the connectivity properties of the Lie groupoid. The next section studies relations between the action of an infinitesimal connection and the action of its holonomy groupoid, e.g., a strong, Lie groupoid version of the Ambrose-Singer theorem is presented. The fifth section deals with abstract transitive Lie algebroids and their morphisms, it presents various refinements of some previously obtained results.

The seventh chapter is dedicated to cohomology and Schouten calculus for Lie algebroids. Most results are valid for arbitrary Lie algebroids. In the first section, the cohomology of an arbitrary Lie algebroid with coefficients in an arbitrary representation is constructed in terms of a standard resolution of de Rham or Chevalley-Eilenberg type. As the focus of geometric interest is usually on specific transversals or cocycles, the standard cochain complex is the best suited, according to the author, to the purpose. The first three sections present the definitions and the techniques. The forth one deals with the spectral sequence of a transitive Lie algebroid which provides an abstraction and algebraization of the Leray-Serre spectral sequence of a principal bundle in de Rham cohomology. In these sections, the author demonstrates how the standard identities of infinitesimal connection theory arise naturally in cohomological terms. The last section provides an exposition of the extension of the classical Schouten calculus of multi-vector fields to arbitrary Lie algebroids.

In the next chapter the author studies the cohomological obstructions to integrability of transitive Lie algebroids, that is Lie’s third theorem for transitive Lie algebroids. There is a single cohomological obstruction in this case which gives a complete solution. For general Lie algebroids the problem was recently solved by M. Crainic and R. L. Fernandes [Ann. Math. (2) 157, 575–620 (2003; Zbl 1037.22003)]. The first section of the chapter presents some interesting examples. In the second one the author presents his classification of transitive Lie algebroids in terms of ‘system of transition data’. This classification is an exact analogue of the classification principal bundles by transition functions. On the way, a strong form of local integrability for transitive Lie algebroids is proved. For the Lie algebroid of a locally trivial Lie groupoid, the system of transitional data is obtained from a groupoid cocycle by differentiation. Trying to reverse the process one encounters the cohomological obstruction class, this is presented in the third and last section of this chapter and part.

The last part of the book is concerned with Poisson and symplectic theories. The ninth chapter is dedicated to double vector bundles. The importance of double vector bundles reveals itself when one studies the relations between Lie algebroids and Poisson structures. A lot of the basic theory is a more or less straightforward generalization of results well known for ordinary vector bundles. The real differences appear in the theory of duality, which is presented in the second section. The next three sections are dedicated to the study of the two duals of the tangent bundle of a vector bundle. In the sixth section, relations between bracket structures on a double tangent bundle and its duals are studied in view of the canonical isomorphisms defined earlier. The following two sections present applications of the general results to Lie algebroids and their prolongations. The formalism used in this chapter is global and intrinsic, the author avoids local coordinate formulations of the traditional type.

The second chapter of the third part is dedicated to the explanation of relations between Poisson structures and Lie algebroids. A Poisson structure on a manifold \(M\) gives rise to a Lie algebra structure on \(C^{\infty}(M)\), which is infinite dimensional, and its properties are not easily translated into the properties of the Poisson manifold \(M\). However, the same structure induces a Lie algebroid structure on the cotangent bundle \(T^{\ast}M\). This cotangent Lie algebroid of \(M\) is a major tool in the study of Poisson manifolds and an important source of examples of Lie algebroids. Many important problems of the Poisson manifold theory can be reformulated in the terms of properties of the cotangent Lie algebroid where one can apply the whole machinery of the Lie theory of Lie groupoids and Lie algebroids.

The penultimate chapter of the book is concerned with Poisson and symplectic groupoids. The author does not pretend to present the theories of these groupoids in their generality as they are well developed and have very rich literature. The aim of the chapter is to show how the general features of Poisson groupoids, and thus of symplectic groupoids may be obtained by a systematic application of the so called ‘diagramatic’; methods to the cotangent bundles. The account of Poisson is new and appears for the first time in print. As an immediate consequence of this approach to Poisson groupoids is the fact that, according to the author, the basics of symplectic groupoids theory may be developed without any use of genuine symplectic geometry.

The last chapter is dedicated to elements of the theory of Lie bialgebroids, which are the infinitesimal form of Poisson groupoids; applying the Lie functor to a Poisson groupoid one gets a standard Lie algebroid with a Lie algebroid structure on its vector bundle dual. The pair obtained in such a way constitutes the Lie bialgebroid of the initial Poisson groupoid. The abstract definition of a Lie bialgebroid is not related to Poisson manifolds and is presented in the first section. The second section provides a criterion for a vector bundle and its dual to be a Lie bialgebroid. In the following section, the author provides additional information to clarify even more the notion under consideration and its relation to Poisson groupoids. The last section gives a brief introduction to moments maps in the language of Lie algebroids and Lie groupoids.

The book can serve both as a reference source and a manual for advanced students. The presentation and the choice of the topics are very impressive. One can easily discern a long fascination by and research involvement in the subject. Numerous notions and results presented are due to the author and his collaborators and have been mostly available in research papers. The second part has a lot in common with the previous book of the author [Lie groupoids and Lie algebroids in differential geometry. London Mathematical Society Lecture Note Series, 124. (Cambridge University Press). (1987; Zbl 0683.53029)], but numerous improvements and corrections have been made.

Reviewer: Robert A. Wolak (Kraków)

### MSC:

58H05 | Pseudogroups and differentiable groupoids |

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

53D17 | Poisson manifolds; Poisson groupoids and algebroids |

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