Mathematica Aeterna, Vol. 5, 2015, no. 1, 63 - 81 Comorphisms of Lie algebroids and groupoids: a short introduction Krzysztof Drachal Faculty of Mathematics and Information Science at the Warsaw University of Technology ul. Koszykowa 75, 00 – 662 Warszawa, Poland Abstract The aim of this paper is to present a short (and a bit informal) exposition to comorphisms of Lie algebroids and comorphisms of Lie groupoids. We briefly present the motivation to study such a concept. We also review some recent works done in this direction. Mathematics Subject Classification: 22A22, 53D17, 58H05. Keywords: comorphism, Lie algebroid, Lie groupoid, morphism. 1 Introduction The concept of comorphism in case of Lie algebroids and Lie groupoids was first fully described by Higgins and Mackenzie [16]. However, such a concept was tentatively discussed in some earlier works (see, for example, references in [16]). Recently, this notion was described with a help of suitable graphs and Lie pseudo–algebras by Chen and Liu [6]. One can be interested in the question why we do introduce comorphisms. First of all, notice that it is well–known that a linear map of Lie algebras, g → h, is a morphism if and only if its dual, i.e., h∗ → g∗, is a Poisson morphism. The comorphism concept allows us to extend this result to Poisson bundles and Lie algebroids. Secondly, the module of sections of a Lie algebroid is a Lie pseudo–algebra, but the concepts of a Lie algebroid morphism and a Lie pseudo–algebra morphism do not correspond. And thirdly, notice that for a Lie algebroid the dual vector bundle possesses the structure of a Poisson bundle and the dual of a Poisson bundle has a Lie algebroid structure. However, these dualities are on objects only. Whereas, with a help of the comorphism concept, they can be expanded on the whole categories. 64 Krzysztof Drachal Moreover, according to Weinstein, comorphisms can be regarded as a re- formulation of canonical relations of the symplectic geometry (see [16]). We do not give detailed proofs in this papers. Indeed, most of the content of this paper is heavily based on [16, 6] and [10], from which we extracted the most important (in our opinion) theorems. The Reader interested in deeper understanding should refer to the cited literature and references therein. On the other hand, this paper is a try to present some recently developed ideas in a reasonably simple and short draft. 2 Fundamental concepts Now, we will remind the fundamental concepts from the theory of Lie alge- broids and Lie groupoids. In case of any doubts, [19] is a very well–known (and good) reference. Definition 2.1. A morphism of Lie algebras, g and h, is a linear map ϕ : g → h, such that ϕ([a, b]) = [ϕ(a),ϕ(b)] for every a, b ∈ g, where [·, ·] denotes the Lie bracket. q∗ Definition 2.2. A dual vector bundle is denoted in the following way E∗ → q M, where E → M is the initial bundle. Fibers of the dual vector bundle are, by the definition, the dual spaces of the fibers of E (i.e., all linear functionals on fibers of E). q Definition 2.3. Let E → M be a vector bundle and let f : N → M be q! a mapping. Then, f !E → N is called an inverse image bundle, if f !E := {(x, p) | f(x)= q(p)} and q!((x, p)) := p. ! Definition 2.4. A canonical morphism is the following mapping f! : f E → E. E q f M N ! ∞ Sections of f E can be regarded as sums Pi αi ⊗Xi, where αi ∈ C (N) and ∞ Xi ∈ ΓE. Notice also, that there exists an isomorphism of C (N)–modules: C∞(N) ⊗ ΓE Γ(f !E), α ⊗ X → (α ◦ f)X!, where X! is a pullback of X. Definition 2.5. A groupoid is a small category, in which every morphism is an isomorphism. Comorphisms of Lie algebroids and groupoids... 65 Unfortunately, except aesthetic elegance, this definition has few advantages. In practical usage it gives no help. More illustrative is the reformulation in terms of arrows. Indeed, we can also see a groupoid, denoted by G ⇉ M, as the pair consisting of a collection of arrows G, and a base M (consisting of objects). Then, for every x ∈ M there exists ½x ∈ G. Of course, arrows are composable, associative and invertible. We strongly refer the Reader to the alternative definition of a groupoid in terms of relations, which was proposed by Zakrzewski [33, 34]. This formulation is also discussed in [26] and [3]. Definition 2.6. A Lie groupoid is a groupoid, such that: • its source and target maps, s, t : G → M, are surjective submersions (i.e., differentiable surjections, which differentials are also surjections), • its inclusion map, M ∋ x → ½x ∈ G is smooth, • the multiplication in G is smooth. Now, we will present some examples of Lie groupoids. Example 2.7. Let M be an arbitrary manifold. Let s = idM = t and let every element be a unity. Example 2.8. Let G × M → M be a smooth action of a Lie group on a manifold M. Let s := prM and t : G × M → M. Let i : x → ½x = (1, x) −1 −1 and (g2,y)(g1, x):=(g2g1, x), where y = g1x. Let (g, x) := (g , gx). Then, G × M is so called an action groupoid. q Example 2.9. Let E → M be a vector bundle. Consider Φ(E), the set of all vector space isomorphisms ξ : Ex → Ey, where x, y ∈ M. Let s(ξ) := x, t(ξ) := y and i : x → idEx . Let the multiplication be defined as a composition of mappings and let the inverse be defined as an inverse of mappings. Such a Lie groupoid is called a frame groupoid. Example 2.10. Consider U ⊂ M and V ⊂ M, where M is a manifold. 1 Consider also a diffeomorphism ϕ : U → V . Denote by jxϕ the first order jet of ϕ at x and denote by J 1(M, M) the set of all first order jets. Then, J 1(M, M) 1 1 can be given a Lie groupoid structure. Indeed, let s(jxϕ) := x, t(jxϕ) = ϕ(x) 1 1 1 and jϕ(x)ψ · jxϕ = jx(ψ ◦ ϕ). (It is isomorphic to Φ(T M), i.e., consider the 1 following mapping jxϕ → Txϕ.) Definition 2.11. A morphism of Lie groupoids, (ϕ, f) : G ⇉ M → H ⇉ N, is a pair consisting of smooth mappings ϕ : G → H and f : M → N, such that • sH ◦ ϕ = f ◦ sG and tH ◦ ϕ = f ◦ tG, where s, t are source and target maps respectively, 66 Krzysztof Drachal • ϕ(hg)= ϕ(h)ϕ(g) for every h, g ∈ G that can be composed (multiplied). Definition 2.12. If the assumption of a smoothness in the above definition is dropped, one obtains the definition of a morphism of groupoids. Definition 2.13. A Lie algebroid is defined as triple consisting of q • a vector bundle E → M, • an anchor mapping a : E → T M, • and a Lie bracket [·, ·]:ΓE × ΓE → ΓE, such that – [X,fY ] = f[X,Y ]+(a(X)f)Y for every f ∈ C∞(M) and X,Y ∈ ΓE, – a([X,Y ]) = [a(X), a(Y )]. We remind that a Lie bracket means an operator, which is R–bilinear, alternating and satisfying the Jacobi identity. The Jacobi identity is usually written in the following way: [a, [b, c]] + [c, [a, b]] + [b, [c, a]] = 0. However, requiring adab := [a, b] to be a derivation, i.e., to fulfil the following rule ada[b, c] = [adab, c] + [b, adac], gives a conceptually better definition [13]. Now, we will present two examples of Lie algebroids. Example 2.14. Every Lie algebra over a point. Example 2.15. A tangent bundle, T M, with the identity as the anchor. q1 q2 Definition 2.16. A morphism of vector bundles, E1 → M and E2 → N, is defined as a pair (f,g), such that • g ◦ q1 = q2 ◦ f, −1 −1 • for every p ∈ M the mapping q1 ({p}) → q2 ({g(x)}) induced by f is a linear mapping of vector spaces. f E1 E2 q1 q2 M g N Notice, that a morphism of vector bundles induces a morphism of modules of sections only in base–preserving case (i.e., if g = idM ). Comorphisms of Lie algebroids and groupoids... 67 Definition 2.17. A morphism of Lie algebroids is defined as a pair (ϕ, f) satisfying the following conditions. In particular, let E1 be a Lie algebroid on a base M and let E2 be a Lie algebroid on a base N. Now, consider a vector bundles morphism, such that • a1 ◦ϕ = T (f)◦a2, i.e., the anchor is preserved, where T (f) : T N → T M, ! ! • for X,Y ∈ ΓE2, if ϕ (X) = Pi ui ⊗ Xi and ϕ (Y ) = Pj vj ⊗ Yj, then ! ϕ ([X,Y ]) = Pi,j uivj ⊗ [Xi,Yj]+ Pj a2(X)(vj) ⊗ Yj − Pi a2(Y )(ui) ⊗ Xi, i.e., the bracket is preserved. ϕ E2 E1 q2 q1 N M f As a result, we obtain a category of Lie algebroids, which we will denote by LA. Definition 2.18. A Lie pseudo–algebra is defined as an A–module C, such that • A is a commutative and unitary k–algebra, • there exists a Lie bracket [·, ·] on C, • there exists an anchor a : C → DerA, • [X,fY ]= f[X,Y ]+(a(X)f)Y , • a([X,Y ]) = [a(X), a(Y )], for every X,Y ∈ C and every f ∈ A. Sometimes, (if k = R) a Lie pseudo–algebra is called a Lie–Rinehart algebra (especially in cohomology theories).