Lie Algebroids and Lie Pseudoalgebras

Lie algebroids and Lie pseudoalgebras arise from a wide variety of constructions in differential geometry; they have been introduced repeatedly into the geometry, physics and algebra literatures since the 1950s, under some 14 different terminologies. The first main part (Sections 2-5) of this survey describes the four principal classes of examples, emphazising that each arises by means of a generalization of the Lie theory of Lie groups and Lie algebras; this part is addressed primarily to those interested in differential geometry. The second part (Sections 7-8), addressed equally to algebraists, describes various algebraic constructions currently known for Lie algebroids and Lie pseudoalgebras, and their geometric significance. Lie algebroids and Lie pseudoalgebras are an unrecognized part of the folklore of differential geometry. They have been introduced repeatedly into differential geometry since the early 1950s, and also into physics and algebra, under a wide variety of names, chiefly as infinitesimal invariants associated to geometric structures: in connection theory (see, for example, [75, 161, 168,169,173, 143, 147]), as a means of treating de Rham cohomology by algebraic methods [163, 172, 100, 117, 143, 107, 108,109], as invariants of foliations and pseudogroups of various types [157,158, 74, 159, 88], in symplectic and Poisson geometry [177, 84, 119, 118, 91, 106, 178] and, in a more algebraic setting, as algebras of differential operators associated with vector bundles and with infinitesimal actions of Lie groups [168, 169, 97, 153, 76]. A list of original sources, and the various terminologies used, is given in Section 1. A Lie pseudoalgebra j / is a Lie algebra over a commutative ring M, which is at the same time a module over a commutative ^-algebra #, the two structures being related by a suitable action of stf on # by derivations (see 1.2). Although much of the elementary theory of Lie pseudoalgebras follows that of finite-dimensional Lie algebras, this is by and large the least valuable part of the theory. The results of greatest interest, from both the algebraic and the geometric points of view, arise by considering certain generalized Lie theories which associate Lie pseudoalgebras as infinitesimal invariants to geometric structures. In fact, these generalized Lie theories usually produce Lie algebroids rather than Lie pseudoalgebras. Lie algebroids are a smooth, C, form of Lie pseudoalgebras, and stand in relation to them as smooth vector bundles of finite rank do to modules over commutative algebras; the module of sections of a Lie algebroid is a Lie pseudoalgebra over the algebra of smooth functions on the base manifold. Lie algebroids provide a great widening of the ambit of Lie theory. Lie groups owe their central importance to two factors above all others: to the richness of their Lie theory, and to their crucial role in any consideration of symmetry. Here we are concerned with Lie theories for objects possessing symmetry in a much broader sense of the term. We deal with four main examples. First, in Section 2 we describe the relationship between principal bundles and their Atiyah sequences in terms of Lie theory, with particular reference to connection theory: several of the basic results of Received 29 July 1992; revised 2 November 1993. 1991 Mathematics Subject Classification 58-02. Bull. London Math. Soc. 27 (1995) 97-147 98 KIRILL C. H. MACKENZIE elementary connection theory may be seen as analogues of fundamental results of Lie theory [143]. In Section 4 we describe the relationship between symplectic groupoids and Poisson manifolds, as developed by Weinstein, Karasev and others [177, 84,118, 119], in terms of 'nonlinear Lie theory'; from this perspective, Poisson manifolds stand in a similar relationship to symplectic groupoids as do Lie algebras to Lie groups. Behind these two theories stands the Lie theory of Lie groupoids as envisaged by Pradines [48, 165-167], and we consider this in Section 3. Lastly, in Section 5 we outline Molino's [157, 159] theory of transversally complete and transversally parallelizable foliations in terms of their associated Lie algebroids. It will be clear that none of Sections 2-5 is intended as a comprehensive survey of the areas which they treat; we deal only with aspects of the theories which are related to Lie algebroid and Lie pseudoalgebra constructions (and not always with all of those). An explanation of what we mean by the term 'Lie theory' is given in Section 2. The Lie theory of Lie groups and Lie algebras owes its utility to the fact that the Lie functor preserves many fundamental algebraic properties: it maps subgroups to subalgebras, normal subgroups to ideals, the centre and commutator subgroup to the corresponding Lie subalgebras and so on, and, most importantly, it is exact. It is clear that the algebraic properties of principal bundles, Lie groupoids and symplectic structures will be much less straightforward than those of Lie groups. However, if one restricts attention to constructions which leave invariant the base manifold, the algebra remains recognizably close to that of Lie groups and Lie algebras. We do not dwell on this aspect of the theory, which was dealt with at length in [143], but have incorporated what we need into Sections 2-5. In Sections 7-8 we deal with constructions which vary the base manifold or algebra. Here there are substantial features which have no analogue in the classical theory of Lie algebras. We are concerned with quite basic algebraic concepts and constructions—actions, semi-direct products, quotients and various forms of exactness—but these already encompass a great variety of geometric phenomena. Most of the material in Sections 7-8 is quite recent in origin, and arose by linearizing known results from Lie groupoid theory, but we do not take that approach here. Rather, we aim to show how these results arise directly from algebraic and geometric considerations. Typical of this is the case of actions: infinitesimal actions of Lie algebroids and Lie pseudoalgebras are characterized by morphisms of a specific type, the so-called action morphisms, and the integrability problem for infinitesimal actions is thereby reduced to the integrability of morphisms. This process requires the notion of Lie algebroid, or Lie pseudoalgebra, even if the original action comes from a Lie algebra; see Section 7. In Lie pseudoalgebra terms, the action morphism is defined on the crossed product Lie pseudoalgebra [97, 153]. Other basic concepts also need extensive reworking to function adequately for Lie algebroids and Lie pseudoalgebras : for example, one needs to regard an involutive distribution corresponding to a simple foliation as an ideal of the tangent bundle of the manifold, with the tangent bundle to the quotient manifold being the quotient Lie algebroid. Accordingly, the notions of ideal and quotient Lie algebroid require a more sophisticated formulation than suffices when the base manifold is preserved; see Section 8. Much of the algebra discussed in Sections 7-8 was developed in order to handle multiple structures based on the category of Lie algebroids—for example, a Poisson Lie group gives rise to, and may be identified with, a groupoid object in the category of Lie algebroids. However, here we mention such applications only very briefly. LIE ALGEBROIDS AND LIE PSEUDOALGEBRAS 99 In this survey, we concentrate on the case of Lie algebroids, since these are the primary invariants of the Lie theories. Lie pseudoalgebras may arise as invariants of infinite-dimensional and singular systems, but it is far from clear how to develop a systematic Lie theory in that generality. Nonetheless, much of the algebra of Lie algebroids treated here extends to general Lie pseudoalgebras, and some constructions are more readily expressed in Lie pseudoalgebra terms. After Section 1, which gives the basic definitions and history, this survey divides into two main parts: Sections 2-5 which present the main geometrical constructions by which Lie algebroids and Lie pseudoalgebras arise—from principal bundles, Lie groupoids, symplectic groupoids and Poisson manifolds, and foliations of various types—and consider the algebraic constructions which are possible over a fixed base, and Sections 7-8 which deal with the general algebraic theory. The short Section 6 is a brief account of graded structures related to Lie pseudoalgebras and Lie algebroids. Readers who are primarily interested in the algebra may prefer to pass directly from Section 1 to Section 7. The final section contains comments on material which has had to be omitted, and some indications of possible future developments. A bibliography is included. I am grateful to a number of people for comments on various drafts of this article: specifically, to R. Brown, S. Chase, M. Gerstenhaber, P. J. Higgins, M. V. Karasev, Y. Kosmann-Schwarzbach, M.-P. Malliavin, S. Sternberg and A. Weinstein. I particularly wish to thank Madame Malliavin for the opportunity to speak in the Seminaire Dubreil-Malliavin, which gave the original impetus to this survey, and for her hospitality there. Lastly, I warmly thank both referees for their comments, which have materially improved the article. 1. Lie algebroids and Lie pseudoalgebras DEFINITION 1.1. Let B be a manifold. A Lie algebroid on B, or with base B, is a vector bundle A -* B, together with a bracket [,]: FA x FA -> FA on the module TA of global sections of A, and a vector bundle morphism a.A -*• TB from A to the tangent bundle TB of B, called the anchor of A, such that (i) the bracket on FA is R-bilinear, alternating, and satisfies the Jacobi identity; (ii) [X,uY] = u[X, Y] + a(X)(u)Y for all X,YeFA and all smooth functions