LIE GROUPOIDS AND LIE ALGEBROIDS LECTURE NOTES, FALL 2017 ECKHARD MEINRENKEN Abstract. These notes are under construction. They contain errors and omissions, and the references are very incomplete. Apologies! Contents 1. Lie groupoids 4 1.1. Definitions 4 1.2. Examples 6 1.3. Exercises 9 2. Foliation groupoids 10 2.1. Definition, examples 10 2.2. Monodromy and holonomy 12 2.3. The monodromy and holonomy groupoids 12 2.4. Appendix: Haefliger’s approach 14 3. Properties of Lie groupoids 14 3.1. Orbits and isotropy groups 14 3.2. Bisections 15 3.3. Local bisections 17 3.4. Transitive Lie groupoids 18 4. More constructions with groupoids 19 4.1. Vector bundles in terms of scalar multiplication 20 4.2. Relations 20 4.3. Groupoid structures as relations 22 4.4. Tangent groupoid, cotangent groupoid 22 4.5. Prolongations of groupoids 24 4.6. Pull-backs and restrictions of groupoids 24 4.7. A result on subgroupoids 25 4.8. Clean intersection of submanifolds and maps 26 4.9. Intersections of Lie subgroupoids, fiber products 28 4.10. The universal covering groupoid 29 5. Groupoid actions, groupoid representations 30 5.1. Actions of Lie groupoids 30 5.2. Principal actions 31 5.3. Representations of Lie groupoids 33 6. Lie algebroids 33 1 2 ECKHARD MEINRENKEN 6.1. Definitions 33 6.2. Examples 34 6.3. Lie subalgebroids 36 6.4. Intersections of Lie subalgebroids 37 6.5. Direct products of Lie algebroids 38 7. Morphisms of Lie algebroids 38 7.1. Definition of morphisms 38 7.2. Morphisms and sections 41 7.3. Fibered products, pre-images 42 7.4. Pull-backs 42 7.5. Further Constructions 43 7.6. Lie algebroid actions, representations of Lie algebroids 45 8. The generalized foliation of a Lie algebroid 46 8.1. Integral submanifolds 46 8.2. Normal bundles and tubular neighborhoods 47 8.3. Euler-like vector fields 49 8.4. Some applications of Theorem 8.9 51 8.5. The splitting theorem for Lie algebroids 53 8.6. The generalized foliation 56 9. The Lie functor 56 9.1. The Lie algebra of a Lie group 56 9.2. The Lie algebroid of a Lie groupoid 58 9.3. Left-and right-invariant vector fields 58 9.4. The Lie functor from Lie groupoids to Lie algebroids 61 9.5. Examples 62 9.6. Groupoid multiplication via σL; σR 62 10. Integrability of Lie algebroids: The transitive case 64 10.1. The Almeida-Molino counter-example 64 10.2. Transitive Lie algebroids 65 10.3. Splittings 66 10.4. Gauge transformations of transitive Lie algebroids 67 10.5. Classification of Lie algebroids over 2-spheres 68 10.6. The monodromy groups 70 10.7. Construction of an integration 71 11. Integrability of non-transitive Lie algebroids 74 12. Lie algebroid cohomology, Lie groupoid cohomology 76 12.1. The de Rham complex of a Lie algebroid 76 12.2. The Lie algebroid structure 78 12.3. Examples 80 12.4. The Lie groupoid complex 81 12.5. Weinstein-Xu's van Est map 83 12.6. The Crainic double complex 84 12.7. Perturbation lemma 86 12.8. Construction of the van Est map 87 Appendix A. Deformation to the normal cone 90 LIE GROUPOIDS AND LIE ALGEBROIDS 3 A.1. Basic properties 90 A.2. Charts on D(M; N) 92 A.3. Euler-like vector fields 93 A.4. Vector bundles 94 A.5. Lie groupoids 94 A.6. Lie algebroids 96 References 97 4 ECKHARD MEINRENKEN 1. Lie groupoids Symmetries in mathematics, as well as in nature, are often defined to be invariance properties under actions of groups. Lie groupoids are given by a manifold M of `objects' together with a type of symmetry of M that is more general than those provided by group actions. For example, a foliation of M provides an example of such a generalized symmetry, but foliations need not be obtained from group actions in any obvious way. 1.1. Definitions. The groupoid will assign to any two objects m0; m1 2 M a collection (possibly empty) of arrows from m1 to m0. These arrows are thought of as `symmetries', but in contrast to Lie group actions this symmetry need not be defined for all m 2 M { only pointwise. On the other hand, we require that the collection of all such arrows (with arbitrary end points) fit together smoothly to define a manifold, and that arrows can be composed provided the end point (target) of one arrow is the starting point (source) of the next. The formal definition of a Lie groupoid G ⇒ M involves a manifold G of arrows, a submanifold i: M,!G of units (or objects), and two surjective submersions s; t: G! M called source and target such that t ◦ i = s ◦ i = idM : One thinks of g as an arrow from its source s(g) to its target t(g), with M embedded as trivial arrows. g (1) t(g) s(g) Using that s; t are submersion, one finds (cf. Exercise 1.1 below) that for all k = 1; 2;::: the set of k-arrows (k) k G = f(g1; : : : ; gk) 2 G j s(gi) = t(gi+1)g g gk g1 g2 g3 k−1 (2) m0 m1 m2 ··· ··· mk−1 mk k (k) is a smooth submanifold of G , and the two maps G ! M taking (g1; : : : ; gk) to s(gk), (0) respectively to t(g1), are submersions. For k = 0 one puts G = M. The definition of a Lie groupoid also involves a smooth multiplication map, defined on composable arrows (i.e., 2-arrows) (2) MultG : G !G; (g1; g2) 7! g1 ◦ g2; such that s(g1 ◦ g2) = s(g2); t(g1 ◦ g2) = t(g1). It is thought of as a concatenation of arrows. Note that when picturing this composition rule, it is best to draw arrows from the right to the left. g1◦g2 g1 g2 (3) m0 m1 m2 m0 m2 LIE GROUPOIDS AND LIE ALGEBROIDS 5 Definition 1.1. The above data define a Lie groupoid G ⇒ M if the following axioms are satisfied: (3) 1. Associativity: (g1 ◦ g2) ◦ g3 = g1 ◦ (g2 ◦ g3) for all (g1; g2; g3) 2 G . 2. Units: t(g) ◦ g = g = g ◦ s(g) for all g 2 G. 3. Inverses: For all g 2 G there exists h 2 G such that s(h) = t(g); t(h) = s(g), and such that g ◦ h; h ◦ g are units. The inverse of an element is necessarily unique (cf. Exercise). Denoting this element by g−1, we have that g ◦ g−1 = t(g); g−1 ◦ g = s(g). Inversion is pictured as reversing the direction of arrows. From now on, when we write g = g1 ◦ g2 we implicitly assume that g1; g2 are composable, i.e. s(g1) = t(g2). Let 3 Gr(MultG) = f(g; g1; g2) 2 G j g = g1 ◦ g2g: be the graph of the multiplication map; we will think of MultG as a smooth relation from G ×G to G. Remark 1.2. A groupoid structure on a manifold G is completely determined by Gr(MultG), i.e. by declaring when g = g1 ◦ g2. Indeed, the units are the elements m 2 G such that m = m ◦ m. Given g 2 G, the source s(g) and target t(g) are the unique units for which g = g◦s(g) = t(g)◦g. The inverse of g is the unique element g−1 such that g ◦ g−1 is a unit. Remark 1.3. In the definition above, our manifolds are always assumed to satisfy the Hausdorff separation axiom. For a (possibly) non-Hausdorff Lie groupoid, we allow the space G to be a non-Hausdorff manifold, but still require that the fibers of the source and target maps, as well as the units M, are Hausdorff. 1 Non-Hausdorff Lie groupoids are very common in the theory of foliations; see below. Remark 1.4. One may similarly consider `set-theoretic' groupoids G ⇒ M, by taking s; t, and MultG to be set maps (with s; t surjective). Such a set-theoretic groupoid is the same as a category for which the objects M and arrows G are sets, and with the property that every arrow is invertible. −1 Remark 1.5. Let InvG : G!G; g 7! g be the inversion map. As an application of the implicit function theorem, it is automatic that InvG is a diffeomorphism. Definition 1.6. A morphism of Lie groupoids F : H!G is a smooth map such that F (h1 ◦ h2) = F (h1) ◦ F (h2) (2) for all (h1; h2) 2 H . If F is an inclusion as a submanifold, we say that H is a Lie subgroupoid of G. 1One of the consequences of the Hausdorff property is the uniqueness of flows of vector fields. But in the theory to be developed below, the vector fields that we integrate are all tangent to source fibers, the target fibers, or the units. 6 ECKHARD MEINRENKEN By Remark 1.2, it is automatic that such a morphism takes units of H to units of G, and that it intertwines the source, target, and inversion maps. We will often present Lie groupoid homomorphisms by diagrams, as follows: G // M H / N If G ⇒ M is a Lie groupoid, and m 2 M, the intersection of the source and target fibers −1 −1 Gm = t (m) \ s (m) is a Lie group, with group structure induced by the groupoid multiplication. (We will prove later that it is a submanifold.) It is called the isotropy group of G at m.