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Courant algebroids and Poisson Geometry

Given a manifold M with an action of a quadratic Lie algebra d, such that all stabilizer algebras are co-isotropic in d, we show that the product M\times d becomes a Courant algebroid over M. If the bilinear form on d is split, the choice of transverse Lagrangian subspaces g_1, g_2 of d defines a bivector field on M, which is Poisson if (d,g_1,g_2) is a Manin triple. In this way, we recover the Poisson structures of Lu-Yakimov, and in particular the Evens-Lu Poisson structures on the variety of Lagrangian Grassmannians and on the de Concini-Procesi compactifications. Various Poisson maps between such examples are interpreted in terms of the behaviour of Lagrangian splittings under Courant morphisms.


Introduction
Let d be a (real or complex) Lie algebra, equipped with an invariant symmetric bilinear form of split signature. A pair g 1 , g 2 of transverse Lagrangian subalgebras of d defines a Manin triple (d, g 1 , g 2 ). Manin triples were introduced by Drinfeld [10], and are of fundamental importance in his theory of Poisson Lie groups and Poisson homogeneous spaces. In their paper [12], S. Evens and J.-H. Lu found that every Manin triple defines a Poisson structure on the variety X of all Lagrangian subalgebras d. If g is a complex semisimple Lie algebra, and d = g ⊕ g (where g carries the Killing form and g indicates the same Lie algebra with the opposite bilinear form), then one of the irreducible components of X is the de Concini-Procesi 'wonderful compactification' of the adjoint group G integrating g. More recently, J.-H. Lu and M. Yakimov [19] studied Poisson structures on homogeneous spaces of the form D/Q, where D is a Lie group integrating d, and Q is a closed subgroup whose Lie algebra is co-isotropic for the inner product on d. Their results show that if M is a D-manifold such that all stabilizer algebras are co-isotropic, then the Manin triple (d, g 1 , g 2 ) defines a Poisson structure on M .
In this paper, we put this construction into the framework of Courant algebroids. Suppose M is a manifold with a Lie algebra action a : M ×d → T M . We show that the trivial bundle M × d carries the structure of a Courant algebroid, with a as the anchor map, and with Courant bracket extending the Lie bracket on constant sections, if and only if all stabilizer algebras are co-isotropic. If this is the case, any Lagrangian subalgebra of d defines a Dirac structure, and Manin triples (d, g 1 , g 2 ) define pairs of transverse Dirac structures. In turn, by a result of Mackenzie-Xu [20], any pair of transverse Dirac structures determines a Poisson bivector π. In this way, we recover the Lu-Yakimov Poisson bivector. Various Poisson maps between examples of this type are interpreted in terms of the behaviour under Courant morphisms.
More generally, we give an explicit formula for the Schouten bracket [π, π] for the bivector resulting from any pair of transverse Lagrangian subspaces (not necessarily Lie subalgebras). If g is a Lie algebra with invariant inner product, then d = g ⊕ g with Lagrangian splitting given by the diagonal g ∆ = {(x, x)| x ∈ g} and the anti-diagonal g −∆ = {(x, −x)| x ∈ g} is a typical example. The obvious d-action on the Lie group G integrating g has co-isotropic (in fact Lagrangian) stabilizers, and the bivector defined by (d, g ∆ , g −∆ ) is the quasi-Poisson structure on G described in [2,3]. In a similar fashion, one obtains a natural quasi-Poisson structure on the variety of Lagrangian subalgebras of g ⊕ g.
Our theory involves some general constructions with Courant algebroids, which may be of independent interest. Suppose A → N is a given Courant algebroid. We introduce the notion of an action of a Courant algebroid A → N on a manifold M , similar to the Higgins-Mackenzie definition of a Lie algebroid action [15]. A Courant algebroid action is given by two maps Φ : M → N, ̺ : Φ * A → T M with a suitable compatibility condition, and we show that the vector bundle pull-back Φ * A acquires the structure of a Courant algebroid, provided all 'stabilizers' ker(̺ m ) are co-isotropic.
The organization of this article is as follows. Section 1 gives a review of Courant algebroids A → M , Dirac structures, and Courant morphisms. In Section 2 we discuss coisotropic reductions of Courant algebroids, which we then use to define pull-backs Φ ! A under smooth maps. Such a pull-back is different from the vector bundle pull-back Φ * A, and to define a Courant algebroid structure on the latter we need a Courant algebroid action (Φ, ̺) with co-isotropic stabilizers. In Section 3 we consider Lagrangian sub-bundles of Courant algebroids. We show that the Courant tensors of Lagrangian sub-bundles behave naturally under 'backward image', and give a simple construction of the bivector π for a Lagrangian splitting as the backward image under a 'diagonal morphism'. We give a formula for the rank of π and compute its Schouten bracket [π, π]. Finally, we present a compatibility condition for Lagrangian splittings relative to Courant morphisms, which guarantees that the underlying map of manifolds is a bivector map. Section 4 specializes the theory to Courant algebroids coming from actions of quadratic Lie algebras, and puts the examples mentioned above into this framework. In Section 5 we consider the Lagrangian splittings of M × d coming from Lagrangian splittings of d. In the final Section 6 we show how to interpret the basic theory of Poisson Lie groups from the Courant algebroids perspective.

Courant algebroids
1.1. Definition of a Courant algebroid. The notion of a Courant algebroid was introduced by Liu-Weinstein-Xu [18] to provide an abstract framework for Courant's theory of Dirac structures [8,9]. The original definition was later simplified by Roytenberg [22] (based on ideas ofŠevera [25]) and Uchino [27] to the following set of axioms. See Kosmann-Schwarzbach [17] for a slightly different version.
consisting of a vector bundle A → M , a non-degenerate bilinear form (inner product) ·, · on the fibers of A, a bundle map a : A → T M called the anchor, and a bilinear Courant bracket [[·, ·]] on the space Γ(A) of sections, such that the following three axioms are satisfied: for all sections x 1 , x 2 , x 3 ∈ Γ(A). Here a * is the dual map a * : T * M → A * ∼ = A, using the isomorphism given by the inner product.
Axioms c1) and c2) say that [[x, ·]] is a derivation of both the Courant bracket and of the inner product, while c3) relates the symmetric part of the bracket with the inner product. From the axioms, one can derive the following properties of Courant algebroids: for f ∈ C ∞ (M ), µ ∈ Ω 1 (M ), and with L v , ι v denoting Lie derivative, contraction relative to a vector field v. The properties p1) and p2) are sometimes included as part of the axioms; their redundancy was first observed by Uchino [27].
for vector fields v i ∈ X(M ) and 1-forms µ i ∈ Ω 1 (M ). More generally, given a closed 3-form η ∈ Ω 3 (M ) one defines a Courant algebroid TM (η) , by adding an extra term (0, ι v 2 ι v 1 η) to the Courant bracket. By a result ofSevera, a Courant algebroid A → M is isomorphic to T M (η) if and only if it is exact, in the sense that the sequence and if its space Γ(E) of sections is closed under the Courant bracket. This then implies that E, with the restriction of the Courant bracket and the anchor map, is a Lie algebroid. For any Dirac structure, the generalized distribution a(E) ⊂ T M is integrable in the sense of Stefan-Sussmann, hence it defines a generalized foliation of M .
The lack of integrability of a given Lagrangian subbundle E ⊂ A is measured by the (Using the Courant axioms, one checks that the right hand side is tensorial and antisymmetric in x 1 , x 2 , x 3 ∈ Γ(E).) Example 1.3. Let π ∈ X 2 (M ) = Γ(∧ 2 T M ) be a bivector on M . Then the graph of π, is a Lagrangian subbundle. Conversely, a Lagrangian subbundle of TM is of the form Gr π if and only if its intersection with T M is trivial. Using the pairing of T M and Gr π given by the inner product, we may identify Gr * π ∼ = T M and hence view the Courant tensor of Gr π as a section of ∧ 3 T M . A direct calculation using the definition of the Courant bracket on TM shows [8] (1) Υ Grπ = 1 2 [π, π] where [π, π] is the Schouten bracket of multi-vector fields. Thus Gr π is a Dirac structure if and only if π is a Poisson bivector.
with the following properties: The composition of Courant morphisms is given as the fiberwise composition of relations: (One imposes the usual transversality conditions to ensure that this composition is a smooth subbundle.) For a Courant morphism R Φ : A A ′ , and elements From now on, we will often describe the subbundle R Φ in terms of the relation ∼ R Φ . and R ⊂ A ′ ×Ā is a Dirac structure whose foliation is tangent to the graph of Φ. Then the restriction R Φ = R| Graph(Φ) is a Courant morphism. However, not every Courant morphism arises in this way: For instance, the bracket on Γ(R Φ ) need not be skew-symmetric. (d) Associated to any smooth map Φ : (e) If g, g ′ are quadratic Lie algebras (viewed as Courant algebroids over a point), a Courant morphism R : g g ′ is a Lagrangian subalgebra of g ′ ⊕ g.
Another example is described in the following Proposition.
Using p3), we see that the image of f (x, µ) under the anchor map for A × A × TM is (a(x), a(x), a(x)) which is tangent to the graph of diag. Now let x 1 , x 2 ∈ Γ(A) and µ 1 , Hence, the Courant bracket again restricts to a section of R diag . This completes the proof (cf. [7, Remark 2.5]).

Constructions with Courant algebroids
In this section we describe three constructions involving Courant algebroids. We begin by discussing a reduction procedure relative to co-isotropic subbundles. We then use reduction to define restrictions and pull-backs Φ ! A of Courant algebroids. Finally, we introduce the notion of a Courant algebroid action, and describe conditions under which the usual pullback as a vector bundle Φ * A acquires the structure of a Courant algebroid.
2.1. Coisotropic reduction. The reduction procedure is frequently used to obtain new Courant algebroids out of old ones. See [6] and [28] for similar constructions for the case of exact Courant algebroids. Then the anchor map, bracket and inner product on C descend to A C = C/C ⊥ , and make A C into a Courant algebroid over S. The inclusion Φ : S ֒→ M lifts to a Courant morphism where q : C → A C is the quotient map.
Proof. By r1),r2) the inner product and the anchor map descend to A C . We want to define the bracket on A C by [[q(x| S ), q(y| S )]] = q([[x, y]]| S ), for any sections x, y ∈ Γ(A) that restrict to sections of C. To see that this is well-defined, we must show that the right hand side vanishes if q(x| S ) = 0 or if q(y| S ) = 0. Equivalently, letting x, y, z ∈ Γ(A) with x| S , y| S , z| S ∈ Γ(C), we must show that [[x, y]], z vanishes on S if one of the two sections x| S , y| S takes values in C ⊥ . (i) Suppose y| S ∈ Γ(C ⊥ ). Then vanishes on S, since y, z | S = 0 and a(x) is tangent to S. (ii) Suppose x| S ∈ Γ(C ⊥ ). Then vanishes on S: The first term vanishes by (i), and the second term vanishes since x, y | S = 0 and consequently d x, y | S ∈ (Γ(ann(T S)). The Courant axioms for A C follow from the Courant axioms for A.
Example 2.2. Suppose ker(a) = C is a smooth subbundle of A, as happens for example if the anchor map a is surjective. Then C ⊥ = ran(a * ), and A C is a Courant algebroid with trivial anchor map. That is, A C is simply a bundle of quadratic Lie algebras.
. This is parallel to a construction in symplectic geometry, see Guillemin-Sternberg [14]. Of course, the restriction A S as a Courant algebroid is different from the restriction A| S as a vector bundle.
Condition r3) follows from p2), and since the Lie bracket of two vector fields tangent to S is again tangent to S. To describe C ⊥ , note that a * µ is orthogonal to C if and only if µ annihilates all elements of a(C) = T S ∩ ran(a). Hence C ⊥ = a * (ann(T S ∩ ran(a))). Since C is the kernel of a surjective map A| S ⊕ T S → T M | S , one obtains rk(C) = rk(A) − dim M + dim S. Consequently rk(C ⊥ ) = dim M − dim S, and the formula for rk(A C ) follows.
Remark 2.5. The transversality condition (2) may be replaced by the weaker assumption that C is a smooth subbundle. Note also that the transversality condition is automatic if ran(a) = T M , e.g. for exact Courant algebroids. For this case, restriction of Courant algebroids is discussed in [6,Lemma 3.7] and in [13,Appendix].
Restriction to submanifolds generalizes to pull-back under maps: Explicitly, we have the following description: Proof. This is just a special case of Proposition 2.4. Let us nevertheless give some details on the computation of C ⊥ . As in the proof of Proposition 2.4, C ⊥ is contained the range of the dual of the anchor map. Hence, elements of C ⊥ are of the form (a * λ, (0, ν)) with λ ∈ T * M and ν ∈ T * S. Pairing with (x; v, µ) ∈ C we obtain the condition Note that for ran(a) = T M (e.g. exact Courant algebroids) the description of C ⊥ simplifies to the condition (dΦ) * λ + ν = 0.
takes values in C, with ψ(a −1 (T S))∩C ⊥ = ψ(a * (ann(T S∩ran(a)))). The resulting inclusion A S → Φ ! A is an isomorphism of vector bundles by dimension count, and it clearly preserves inner products. It is an isomorphism of Courant algebroids since for all sections Proposition 2.9. For any smooth map Φ : S → M , one has a canonical isomorphism The bundle TS is embedded in C as the subbundle defined by λ = 0, and it defines a complement to the subbundle C ⊥ given by the conditions v = 0, µ = −(dΦ) * λ. Since the inclusion TS → C preserves Courant brackets, this shows C/C ⊥ = TS as Courant algebroids.

Proposition 2.10. There is a canonical Courant morphism
Note that as a space, P Φ is the fibered product of A and T S over T M . It is a smooth vector bundle over S ∼ = Graph(Φ) since a is transverse to dΦ by assumption.
A × TS as explained in Proposition 2.4. The Lagrangian subbundle R Φ 2 consists of elements of the form (x; v, µ; y) with (x; v, µ) ∈ C mapping to y ∈ C/C ⊥ . On the other hand, the projection map Φ 1 : M × S → M lifts to a Courant morphism R Φ 1 : A × TS A (given as the direct product of the identity morphism A A with the standard morphism TS → T(pt) = pt). Thus R Φ 1 is given by elements of the form (x; x; v, 0) with x ∈ A and v ∈ T S. Composition gives the Courant morphism P Φ : Φ ! A A as described above.
Our construction of a pull-back Courant algebroid is similar to the notion of pull-back Lie algebroid, due to Higgins-Mackenzie [15]. Suppose E → M is a Lie algebroid with anchor map a : E → T M , and Φ : S → M is a smooth map such that a is transverse to dΦ. One defines By the transversality assumption, this is a subbundle of rank rk(Φ ! E) = rk(E) − dim(M ) + dim(S). Given two sections of E × T S whose restriction to Graph(Φ) takes values in Φ ! E, then so does their Lie bracket. Furthermore, if one of the two sections vanishes along Graph(Φ), then so does their Lie bracket. This defines a Lie bracket on Γ(Φ ! E), making Φ ! E into a Lie algebroid. If E is a Lagrangian sub-bundle of A → M , and such that dΦ Clearly, if E is a Dirac structure, then the Courant bracket on Φ ! A restricts to the Lie algebroid bracket on Φ ! E, and in particular Φ ! E is a Dirac structure in Φ ! A.

2.3.
Actions of Courant algebroids. The pull-back Φ ! A of a Courant algebroid is different from the pull-back as a vector bundle Φ * A. In order to define a Courant algebroid structure on Φ * A, one needs the additional structure of an action of A on the map Φ, such that all stabilizers of the action are co-isotropic. Here actions of Courant algebroids may be defined in analogy with the actions of Lie algebroids [15]: , for all x, y ∈ Γ(A) and f ∈ C ∞ (M ).
The last condition shows that ̺ defines a vector bundle map (denoted by the same letter) For each s ∈ S, the kernel of the map ̺ s : A Φ(s) → T s S is called the stabilizer at s.

Proof. Consider the direct product Courant algebroid
The image of f (x, µ) under the anchor map is the vector field (a(x), ̺(x)), which is tangent to the graph of Φ since dΦ(̺(x)) = a(x). The Courant bracket between two sections of this be the subbundle spanned by the restrictions of f (x, µ) to the graph of Φ. This is a co-isotropic subbundle, since its fiber at (Φ(s), s) contains the co-isotropic subspace ker(ρ s ) × T * s S. Its orthogonal is given by (Indeed, it is easy to check that the elements on the right hand side lie in C ⊥ . Equality follows since rk(C) = rk(A) + dim S, hence rk( Since the co-isotropic subbundle satisfies all the conditions from Proposition 2.1, the reduced Courant algebroid A C over Graph(Φ) ∼ = S is defined. The sections of the form f (x, 0) span a complement to C ⊥ in C, and identify A C = Φ * A as a vector bundle. This identification also preserves inner products, since f ( then it is obvious that the action ̺ restricts to a Lie algebroid action of E (i.e. the map Γ(E) → Γ(T S) defined by ̺ preserves brackets). Also, since the Lie algebroid bracket on Example 2.13. Suppose A → M is a Courant algebroid, and Φ : S ֒→ M is an embedded submanifold, such that ran(a) is tangent to S. Then the map Γ(A) → Γ(T S), given by restriction to S followed by a, satisfies the axioms. Hence, Φ * A is a well-defined Courant algebroid.
Example 2.14. Let g be a quadratic Lie algebra, viewed as a Courant algebroid over a point. Let M be a manifold with a g-action whose stabilizer algebras are coisotropic. Then the product M × g (viewed as the vector bundle pull-back of g → pt by the map M → pt) acquires the structure of a Courant algebroid. This example will be explored in detail in Section 4.
Since the Courant bracket on Φ * A was defined by reduction, there is a Courant morphism Φ * A A × TS lifting the inclusion S ֒→ M × S as the graph of Φ. Its composition with A × TS A (lifting the projection to the first factor) is a Courant morphism

Manin pairs and Manin triples
A pair (A, E) of a Courant algebroid A → M together with a Dirac structure is called a Manin pair (over M ). Given a second Dirac structure F ⊂ A such that A = E ⊕ F , the triple (A, E, F ) is called a Manin triple (over M ). For M = pt, this reduces to the classical notion of a Manin triple (d, g 1 , g 2 ) as introduced by Drinfeld [ Then the backward image where α * is the map dual to α, extended to the exterior algebras.
Proof. The fact that the backward image is a Lagrangian subbundle is parallel to Guillemin-Sternberg's construction in symplectic geometry [14], hence we will be brief. We have It is easy to see that E is isotropic, and hence Lagrangian by dimension count. The map α associates to each x ∈ E m the unique x ′ ∈ E Φ(m) with x ∼ R Φ x ′ . To prove (9) we must show that vanishes along the graph of Φ: Indeed [[σ 2 , σ 3 ]] restricts to a section of R Φ , hence its inner product with σ 1 restricts to 0. Choosing the σ i such that σ i | (Φ(m),m) = (x ′ i , x i ) this gives (10).
Similarly, given a Courant algebroid action (Φ, ̺) of A as in Section 2.3, the backward image of a Lagrangian sub-bundle E under the Courant morphism U Φ is just the pull-back bundle Φ * E.

3.2.
The bivector associated to a Lagrangian splitting. Let A → M be a Courant algebroid. By a Lagrangian splitting of A, we mean a direct sum decomposition where E, F are Lagrangian subbundles. Given a Lagrangian splitting A = E ⊕ F , let pr E , pr F be the projections onto the two summands. Thus x = pr E (x) + pr F (x) for all x ∈ A. Note x, x = 2 pr E (x), pr F (x) , so that pr E (x) and pr F (x) are orthogonal if x is isotropic. This applies in particular to elements of the form x = a * (µ). Define a bi-vector field π ∈ X 2 (M ) by the identity This is well-defined since ι µ a(pr F (a * (µ)) = a * (µ), pr F (a * (µ) = 0. Since a(a * (µ)) = 0, the formula for π may also be written ι µ π = −a(pr E (a * (µ)). If e i is a local frame for E, and f i the dual frame for F , i.e. e i , f j = δ j i , we find, Note that π changes sign if the roles of E, F are reversed.
Example 3.4. If π is any bivector on M , with graph Gr π ⊂ TM , the splitting TM = T M ⊕ Gr π is Lagrangian splitting. The bivector associated to this splitting is just π itself.
The following Proposition gives an alternative description of the bivector field π.
Proposition 3.5. Let A = E ⊕ F be a Lagrangian splitting with associated bivector π. Then Gr −π ⊂ TM is the backward image of E × F under the diagonal morphism R diag from Proposition 1.6. Furthermore, 0 ∼ R diag (x, y) ∈ E × F implies x = 0, y = 0.
Proof. By definition of the diagonal morphism, we have (v, µ) ∼ R diag (x, y) if and only if v = a(x), x − y = a * (µ). In particular, for µ = 0 and x ∈ E, y ∈ F we obtain x = y = 0. Hence, Proposition 3.2 shows that the backward image of E × F under R diag is a smooth Lagrangian subbundle of TM , transverse to T M . The backward image is thus of the form Gr −π for some bivector π. The relation with x ∈ E and y ∈ F means by definition of R diag that a * µ = x − y, thus y = pr F a * µ, and −ι µ π = a(y). Thus ι µ π = a(pr F (a * (µ))) proving the Formula (11).

3.3.
Rank of the bivector. The rank of the map π ♯ : T * M → T M, µ → ι µ π at m ∈ M is called the rank of π at m. If π is integrable (i.e. Poisson), the range of π ♯ m is the tangent space to the symplectic leaf, and so rk(π m ) is its dimension. By definition of π, (12) ran π ♯ = a(pr F ran a * ) = a(pr E ran a * ).
For each m, the subspace L m = ran(a * m ) + (ker(a m ) ∩ F m ) is Lagrangian, as one verifies by taking its orthogonal. We have: Lemma 3.6. At any point m ∈ M , the rank of the bivector π is given by the formula, Proof. Formula (12) shows that π m has rank rk(π m ) = dim(pr E (ran(a * m ))) − dim(ker(a m ) ∩ pr E (ran(a * m ))).

3.4.
Integrability of the bivector. Theorem 3.9. Let A = E ⊕ F be a Lagrangian splitting, with associated bivector π ∈ X 2 (M ), and let Υ E ∈ Γ(∧ 3 F ), Υ F ∈ Γ(∧ 3 E) be the Courant tensors, where we are using the isomorphisms E * ∼ = F, F * ∼ = E given by the pairing between E, F . Then   (15) they are equal to these components.
To calculate the dual map On the other hand, as remarked in Section 1.2 we also have Υ Gr −π = 1 2 [π, π] using the identification Gr * −π ∼ = T M . Hence (17) translates into (16). If both E, F are Dirac structures the tensors Υ E , Υ F vanish, and (16) shows that [π, π] = 0. The description of the symplectic leaves follows from Proposition 3.7.
The fact that pairs of transverse Dirac structures define Poisson structures is due to Mackenzie-Xu [20]. Their paper expresses this result in terms of Lie bialgebroids (cf. Remark 3.1).

Relations of Lagrangian splittings. Suppose
Lagrangian splittings of Courant algebroids over M, M ′ , with associated bivectors π, π ′ . Assume also that R Φ : A A ′ is a Courant morphism. Our goal in this Section is to formulate a sufficient condition on the two splittings such that the map Φ : i.e. (dΦ) m (π m ) = π ′ Φ(m) . Since the condition will only involve linear algebra we will temporarily just deal with vector spaces W with split bilinear form ·, · . For instance, if V is any vector space then V = V ⊕ V * carries a natural split bilinear form given by the pairing. Suppose W 1 ⊂ W is a co-isotropic subspace, and W 0 = W ⊥ 1 its orthogonal. Then W red = W 1 /W 0 is again a vector space with split bilinear form. For any Lagrangian subspace E ⊂ W , the quotient A bivector Π ∈ ∧ 2 (W ) descends to a bivector Π red on W 1 /W 0 if and only if it lies in the subspace ∧ 2 (W 1 ) + W 0 ∧ W , or equivalently (18) w ∈ W 1 ⇒ ι(w)Π ∈ W 1 .
Here ι(w) : ∧ W → ∧W is the derivation extension of w 1 → w 1 , w . Note that (18) implies w ∈ W 0 ⇒ ι(w)Π ∈ W 0 . Letting w red ∈ W red be the image of w ∈ W 1 , the bivector Π red ∈ ∧ 2 W red is then given by Lemma 3.10. Suppose that W = E ⊕ F is a Lagrangian splitting, and let Π ∈ ∧ 2 W be the bivector defined by Taking orthogonals, this is equivalent to W 0 = (E + W 0 ) ∩ (F + W 0 ). We claim that this in turn is equivalent to W 0 = E 0 ⊕ F 0 . The implication ⇐ is clear. For the opposite implication ⇒ let w ∈ W 0 , and write w = e + f with e = pr E (w) and f = pr F (w). Then Suppose then that W 0 = E 0 ⊕ F 0 . For w ∈ W 1 , the decomposition w = x + y with x = pr E (w) ∈ E 1 , y = pr F (w) ∈ F 1 descends to the decomposition w red = x red + y red with x ∈ E red and y ∈ F red . Hence ι(w red )Π red = (ι(w)Π) red = 1 2 (y − x) red = 1 2 (y red − x red ) which shows that Π red is the bivector for the decomposition W red = E red ⊕ F red .
Remark 3.11. The property E red ∩ F red = 0 is not automatic. Suppose e.g. that w is an isotropic vector not contained in E, F , and let W 0 be the 1-dimensional subspace spanned by w. Then x = pr E (w) ∈ E 1 and y = − pr F (w) ∈ F 1 descend to non-zero elements x red ∈ E red , y red ∈ F red , with x red − y red = w red = 0. On the other hand W 0 = E 0 ⊕ F 0 is not a necessary condition to ensure E red ∩ F red = 0, as one can see by taking W 0 to be any Lagrangian subspace transverse to E, F . (In this case the transversality condition is trivial since W red = 0, while E 0 + F 0 = 0.) A morphism of vector spaces W, W ′ with split bilinear forms is a Lagrangian subspace R ⊂ W ′ × W , where W is the vector space W with opposite bilinear form. As before we write R : The transpose of the morphism R : W W ′ is the morphism R t : W ′ W , given by Lemma 3.12. We have ker(R) = ran(R t ) ⊥ and ran(R) = ker(R t ) ⊥ . Furthermore, The morphism R : W W ′ gives a well-defined linear isomorphism Here w red , w ′ red are the images of w ∈ ran(R t ) and w ′ ∈ ran(R) in the two quotient spaces. Proof. We will write The dimension formula (21) follows since the spaces W 0 and W ′ 1 are the kernel and range of the projection of R onto W ′ . Taking the sum and the difference with a similar equation for R t , we obtain The second equation shows that dim Indeed we already know the inequality ≤ in these equations, and the equality follows by the first equation in (22). Finally, if w ∼ R w ′ , and w −w ∈ W 0 , w ′ −w ′ ∈ W ′ 0 , theñ w ∼ Rw ′ . This shows that the map R red : W red → W ′ red , w red → w ′ red is a well-defined isomorphism.
Definition 3.13. Given a morphism R : W W ′ and Lagrangian subspaces E ⊂ W, E ′ ⊂ W ′ , we write Remark 3.14. The relation (24) In general, (24) implies that R is an f-Dirac morphism followed by a b-Dirac morphism: Proof. This follows since Π red , Π ′ red are the bivectors defined by the splittings We now apply these results to Courant morphisms if this relation holds pointwise, at any m ∈ M . The main result of this Section is as follows: Proof. We have to show π Φ(m) = (dΦ) m (π m ) for all m ∈ M . To simplify notation, we omit base points in the following discussion. Let Π, Π ′ be the bivectors defined by the splittings, so that π = a(Π) and π ′ = a ′ (Π ′ ). We define , we see that a ′ (ker(R Φ ) ⊥ ) = 0 and a ′ (ran(R Φ )) ⊂ ran(dΦ). Similarly a(ker(R Φ )) ⊂ ker(dΦ). Hence a, a ′ descend to define the vertical maps of the following commutative diagram, The bottom map takes π red = a red (Π red ) to π ′ red = a ′ red (Π ′ red ). On the other hand, Lemma 3.10 shows that π ′ is the image of π ′ red under the inclusion ran(dΦ) → T M ′ , while π red is the image of π under the projection T M → T M/ ker(dΦ). This proves that dΦ : T M → T M ′ takes π to π ′ .

Courant algebroids of the form M × g
An action of a Lie algebra g on a manifold M is a Lie algebra homomorphism g → X(M ). The range of the action map a : M × g → T M defines an integrable generalized distribution; its leaves are called the g-orbits. The action map makes M × g into a Lie algebroid over M , with anchor map a and Lie bracket [·, ·] X extending the Lie bracket on constant sections. Explicitly, where [x 1 , x 2 ] g is the pointwise bracket.
4.1. g-actions. Let g be a quadratic Lie algebra, with inner product ·, · . Given a g-action on a manifold M , all of whose stabilizer are co-isotropic, Example 2.14 describes a Courant algebroid structure on the product A Proof. The Courant bracket is uniquely determined by its property of extending the Lie bracket on constant sections. It hence suffices to note that the right hand side restricts to the pointwise bracket on constant sections, and that it satisfies c3) and p1) from 1.1.

Examples.
Example 4.2. Let g be a quadratic Lie algebra, and g the same Lie algebra with the opposite bilinear form. Then d = g ⊕ g is a quadratic Lie algebra. Let G be a Lie group with Lie algebra g, and denote by u L resp. u R the left-invariant resp. right-invariant vector fields on G defined by u ∈ g. The action of D = G × G on G, given by (a, b).g = agb −1 , defines an action of d, a : d → T G, (u, v) → v L − u R The stabilizers for the infinitesimal action are co-isotropic: In fact, ker(a g ) = d g = {(u, v)| u = Ad g (v)} is Lagrangian. Thus A G = G × d is a Courant algebroid. This is extensively studied in [1], where an explicit isomorphism A G ∼ = TG (η) is constructed (cf. Definition b), with η = 1 12 θ L · [θ L , θ L ] ∈ Ω 3 (G) the Cartan 3-form (where θ L ∈ Ω 1 (G, g) is the left-invariant Maurer-Cartan form.  Concini-Procesi compactification). Let G be a complex semisimple Lie group of adjoint type, and d = g ⊕ g as above. Let M be the complex manifold given as its de Concini-Procesi 'wonderful compactification' [11]. The action of D = G × G on G (given by (a, b).g = agb −1 ) extends to an action on M , and A M = M × d is a Courant algebroid. (The fact that elements in ran(a * ) are isotropic follows by continuity.) Example 4.4 (Homogeneous spaces G/H). Let g be a quadratic Lie algebra, with corresponding Lie group G, and suppose that H ⊂ G is a closed subgroup whose Lie algebra h is coisotropic. Then the stabilizer algebras for the induced G-action on G/H are coisotropic. (The stabilizer algebra at the coset of g ∈ G is Ad g (h).) Hence we obtain a Courant algebroid A G/H = G/H × g. In case h is a Lagrangian subalgebra, this example is discussed by P.Ševera in [25, #2] and by Alekseev-Xu in [5], see also [7]. .
Proof. Since R is a Lagrangian subalgebra, the direct product (M ′ × M )× R ⊂ A M × A M ′ is a Dirac structure. Condition (28) guarantees that (a × a ′ )(R) is tangent to the graph of Φ. The statements about Lagrangian splittings are obvious since the relation (29) is defined pointwise.
We stress that the relation R : g g ′ need not come from a Lie algebra homomorphism.
Remark 4.8. The Proposition 4.7 may be generalized to the set-up from Section 2.2, replacing R : g g ′ with a Courant morphism A A ′ .
Example 4.9 (The Courant algebroid A G ). Let g be a quadratic Lie algebra, with invariant inner product B g . The space d = g ⊕ g may be viewed as a pair groupoid over g, with source and target map s(x, y) = y, t(x, y) = x and with groupoid multiplication Then R is a Lagrangian subalgebra, with ran(R) = d and ker(R) = {(z ′ , z ′′ )| z ′ • z ′′ = 0}. Let G be a Lie group integrating g, with the d-action a : d → X(G) and Courant algebroid A G from Example 4.2. Let Mult : G × G → G be group multiplication. One easily checks (cf. [1, Proposition 3.8]) that Hence the conditions from Proposition 4.7 hold, and we obtain a Courant morphism There is also a Courant morphism U Inv : A G A G lifting the inversion map Inv : G → G, g → g −1 . It is given by the direct product of Graph(Inv) × U , where The relation S : d ⊕ g g, ((z, z ′ ), z ′ ) ∼ S z. defines a Lagrangian subalgebra of g ⊕ d ⊕ g, and it has the required property (28). The action property Φ •(Mult G × id M ) = Φ • (id G ×Φ) lifts to a similar property of the Courant morphisms, so that one might call this a Courant Lie group action. Example 4.11. Let g be a quadratic Lie algebra, with corresponding Lie group G. Suppose H ⊂ G is a closed subgroup whose Lie algebra h ⊂ g is Lagrangian, and let A G/H = G/H ×g be the resulting Courant algebroid. Then the quotient map Ψ : G → G/H lifts to a Courant morphism T Ψ : A G A G/H , by taking T : d g, (z, u) ∼ T z, z ∈ g, u ∈ h. If the subalgebra h ⊂ g is only co-isotropic, but contains a Lagrangian subalgebra k ⊂ h, one obtains a Courant morphism T Ψ by replacing u ∈ h with u ∈ k in the definition of T .

Lagrangian splittings
Throughout this Section, we assume that d is a split quadratic Lie algebra. with the Lagrangian subalgebra l m = ran(a * m ) + ker(a m ) ∩ g 2 ⊂ d. If e = g 1 and f = g 2 are Lagrangian subalgebras, then (A M , E, F ) is a Manin triple over M and hence π is a Poisson structure. The symplectic leaves of π are contained in the intersections of the g 1 -orbits and g 2 -orbits, with equality if and only if ker(a) = ran(a * ) + (ker(a 1 ) ⊕ ker(a 2 )). Here a i : g i → T M are the restrictions of the action.
As remarked in the introduction, the Poisson structure π on M given by Theorem 5.2, in the case that g i are Lagrangian sub-algebras, is due to Lu-Yakimov [19].
Remark 5.3. Note that for g 2 a Lagrangian subalgebra, m → l m = ran(a * m ) + ker(a m ) ∩ g 2 gives a map from M into the variety of Lagrangian subalgebras of d -a generalization of the Drinfeld map. In general, this map need not be smooth.
Remark 5.4. If the stabilizer algebras for the d-action are Lagrangian, then necessarily ker(a) = ran(a * ). In this case, if (d, g 1 , g 2 ) is a Manin triple, the symplectic leaves of the Manin triple (A M , E, F ) are the intersections of g 1 -orbits with g 2 -orbits.
Example 5.5. If d = g ⊕ g as in Example 4.2, the diagonal g ∆ ⊂ d = g ⊕ g is a Lagrangian subalgebra, and the anti-diagonal g −∆ = {(x, −x)| x ∈ g} is a Lagrangian complement. The anti-diagonal is not a subalgebra unless g is Abelian. In fact, letting Ξ ∈ ∧ 3 g be the structure constants tensor for g, with normalization Ξ(x 1 , x 2 , , the Courant tensor of the anti-diagonal is . Given a d-action on a manifold M , with co-isotropic stabilizers, the resulting Lagrangian splitting of the Courant algebroid A M = M × d defines a bi-vector field π ∈ X 2 (M ) with 1 2 [π, π] = a(Ξ). Up to an irrelevant constant this is the definition of a quasi-Poisson manifold as in [3]. In particular, the group G, and the variety of Lagrangian subalgebras in d = g ⊕ g carry natural quasi-Poisson structures.
Example 5.6. Suppose g is complex semi-simple, with triangular decomposition g = n − ⊕ h ⊕ n + , and let d = g ⊕ g. Then another choice of a complement to the diagonal g ∆ is Since l is a Lie subalgebra, the Manin triple (d, g ∆ , l) defines a Poisson structure on M . The Poisson structure on G obtained in this way is due to Semenov-Tian-Shansky [24]. Its extension to a Poisson structure on the variety of Lagrangian subalgebras (and hence in particular the de Concini-Procesi compactification) is due to Evens-Lu [12].
Example 5.7. Let (d, g 1 , g 2 ) be a Manin triple, and D a Lie group integrating d. Given a closed subgroup Q ⊂ D whose Lie algebra q is co-isotropic in D (cf. Example 4.4). The resulting Poisson structure π ∈ X 2 (D/Q) was studied in detail in the work of Lu-Yakimov [19]. At the coset [d] = dQ ∈ D/Q, we have ker(a [d] ) = Ad d (q), hence ran(a * [d] ) = Ad d (q ⊥ ). Furthermore, as shown in [19,Proposition 2.5] the space l d = Ad d (q ⊥ ) + Ad d (q) ∩ g 2 is the Lagrangian subalgebra given by the Drinfeld homomorphism for the Poisson homogeneous space D/Q. Hence (31) reduces to the following formula [19, (2.11)] 5.2. Lagrangian splittings of A D . Suppose d is a split quadratic Lie algebra, and D a Lie group integrating d. Then A D = D × (d ⊕ d) is a Courant algebroid, as a special case of Example 4.2 (with d playing the role of g in that example). If (d, g 1 , g 2 ) is a Manin triple, we obtain two Manin triples for d ⊕ d: Manin triples over D, defining Poisson structures π + , π − ∈ X 2 (D). Since the Courant algebroid D×(d⊕d) is exact, Theorem 5.2 gives a simple description of the symplectic leaves: Let G 1 , G 2 be subgroups of D integrating g 1 , g 2 , then the symplectic leaves of π + are the components of the intersections of the G 1 − G 2 double cosets with the G 2 − G 1 double cosets, while the leaves of π − are the components of the intersections of the G 1 − G 1 double cosets with the G 2 − G 2 double cosets. Note that the symplectic leaf of the group unit e ∈ D for the Poisson structure π + is an open neighbourhood of e, whereas relative to the Poisson structure π − the leaf of e is a point. Hence only π − is a possible candidate for a Poisson Lie group structure on D. Under the morphism R : Note in particular that in each case, ker(R) is a direct sum of its intersections with the two Lagrangian subspaces, as required in Definition 3.13.
Hence, we obtain similar relations ( , and therefore the following relations of the Poisson bivector fields: In particular (D, π − ) is a Poisson Lie group, and its action on (D, π + ) is a Poisson action.
Remark 5.8. The Poisson Lie group (D, π − ) is the well-known Drinfeld double (of the Poisson Lie group G 1 , see below), while (D, π + ) is known as the Heisenberg double. One has the explicit formulas π ± = r R ± r L , where the r-matrix r ∈ ∧ 2 d is given in dual bases of g 1 , g 2 by r = 1 2 i e i ∧ f i , and the superscripts L, R indicate the extensions to left, rightinvariant vector fields. The description of the symplectic leaves of (D, π + ), as intersections of double cosets, was obtained by Alekseev-Malkin [4] using a direct calculation.
Remark 5.9. Given a Manin triple (d, g 1 , g 2 ), one may also consider the Manin triples (d ⊕ d, d ∆ , e + ) and (d ⊕ d, d ∆ , f + ). These define yet other Poisson structures on D, whose symplectic leaves are the intersections of conjugacy classes in D with G 2 −G 1 double cosets, respectively the G 1 − G 2 double cosets.
Consider now a D-manifold M as in Example 4.10. Let π ∈ X 2 (M ) be the Poisson bivector defined by the Manin triple (d, g 1 , g 2 ). Under the morphism S : (d ⊕ d) ⊕ d d from that example, It follows that relative to the action map Φ : In particular, the (D, π − )-action is a Poisson Lie group action. Finally, let G ⊂ D be a closed subgroup such that g ⊂ d is a Lagrangian subalgebra, and consider the quotient map Ψ : D → D/G and the morphism T : d ⊕ d d as in Example 4.11. Let D/G carry the Poisson structure π defined by a Manin triple (d, g 1 , g 2 ). Assuming that (32) g = (g ∩ g 1 ) ⊕ (g ∩ g 2 ) (which is automatic if g = g 1 or g = g 2 ) one finds and consequently π + ∼ Ψ π, π − ∼ Ψ π. Here (32) ensures that ker(T ) = 0 ⊕ g is the direct sum of its intersections with e + , f + , respectively with e − , f − .

Poisson Lie groups
In the last Section, we explained how the choice of a Manin triple (d, g 1 , g 2 ) defines Poisson structures π ± on the Lie group D corresponding to d, where π − is multiplicative. Let Φ : G 1 → D be a map exponentiating the inclusion g 1 → d to the level of Lie groups. In this Section, we show that the pull-back Courant algebroid Φ ! A D has the form G 1 × d for a suitable d-action on G 1 (the dressing action). We fix the following notation: For any Lie group G and any ξ ∈ C ∞ (G, g), we denote by ξ L ∈ X(G) the vector field corresponding to ξ under left trivialization G × g ∼ = T G, and by ξ R the vector field corresponding to ξ under right trivialization. If ξ ∈ g (viewed as a constant function G → g), this agrees with our earlier notation for left-invariant and right-invariant vector fields.
6.1. The Courant algebroid G 1 × d. Let p 1 , p 2 : d → g i be the projection to the two summands of the Manin triple (d, g 1 , g 2 ).
Theorem 6.1. Each of the following two maps d → X(G 1 ) defines a Lie algebra action of d, with co-isotropic stabilizers. Let A R = G 1 ×d, A L = G 1 ×d be the Courant algebroids for these actions. Then Proof. Let C R be the subbundle of A D consisting of elements x with a(x) tangent to the foliation g R 1 ⊂ T D spanned by the right-invariant vector fields ξ R , ξ ∈ g 1 . Then C R contains the Lagrangian subbundle ker a, and hence is co-isotropic. We will show that C R /(C R ) ⊥ is a Courant algebroid D × d defined by the following d-action ζ → p 1 (Ad d ζ) R ∈ X(D). The isomorphism A R ∼ = Φ ! A D and the description of the resulting action on G 1 are then immediate. Since a(x) = a(u, v) = v L − u R = (Ad d v − u) R for x = (u, v), the fiber of C R at d ∈ D is given by We read off that C R has rank dim d + dim g 1 , and hence (C R ) ⊥ has rank dim g 1 . If x ∈ C R , i.e. ι a(x) θ R ∈ g 1 where θ R ∈ Ω 1 (D, d) is the right-invariant Maurer Cartan form, it follows that for all ξ ∈ g 1 x, a * θ R , ξ = ι a(x) θ R , ξ = 0.
Hence (C R ) ⊥ contains all a * θ R , ξ = −(ξ, Ad d −1 ξ) with ξ ∈ g 1 , and for dimensional reasons such elements span all of (C R ) ⊥ | d . Hence, (C R ) ⊥ | d consists of all those elements (u, v) ∈ C R | d for which u ∈ g 1 and Ad d v − u = 0. The set of elements (u, v) ∈ C R | d with u ∈ g 2 is hence a complement to (C R ) ⊥ | d in C R | d . Note that for any element of this form, u = p 2 (u) = p 2 (Ad d v). We conclude that the trivial bundle D × d, embedded in A D = D × (d ⊕ d) by the map (33) ζ → φ R (ζ) = (p 2 (Ad d ζ), ζ), is a complement to (C R ) ⊥ in C R . To compute the Courant bracket of two such sections let ζ,ζ ∈ d and put u = p 2 (Ad d ζ),ũ = p 2 (Ad dζ ) ∈ C ∞ (D, d). We have = [u,ũ] + L a(φ R (ζ))ũ − L a(φ R (ζ)) u, [ζ,ζ] + a * du,ũ . But du,ũ = 0 since u,ũ take values in the isotropic subalgebra g 2 . Hence the Courant bracket takes on the form (?, [ζ,ζ]) where '?' indicates some function with values in g 2 . Since this Courant bracket is a section of C R , we know without calculation that ? = p 2 (Ad d [ζ,ζ]). We have thus shown Finally, the anchor map for C R /(C R ) ⊥ ∼ = D × d is obtained from This gives the desired description of C R /(C R ) ⊥ . Similarly, to prove A L = Φ ! A D we define C L to be the set of all x with a(x) ∈ g L 1 . Arguing as above, one finds that C L /(C L ) ⊥ is isomorphic to D × d, where the isomorphism is defined by sections φ L (ζ) = (ζ, p 2 (Ad d −1 ζ)). Again, we find that φ L is a Lie algebra homomorphism relative to the Courant bracket. The anchor map for D × d is given by ζ → −p 1 (Ad d −1 ζ) L .
Remark 6.3. The fact that (35) defines d-actions holds true for any Lie algebra d, with Lie algebras g 1 , g 2 such that d = g 1 ⊕ g 2 as vector spaces. If d carries an invariant inner product for which g 2 is co-isotropic, the d-space G 1 (for each of these actions) defines a Courant algebroid G 1 × d, and similarly D × d.
with C = C R as in the proof of Theorem 6.1. Recall that R Mult consists of elements of the form (x, z; x, y, y, z) ⊂ d ⊕ d ⊕ d ⊕ d ⊕ d ⊕ d, while C consists of elements (u, v) ∈ d ⊕ d with p 2 (Ad d v − u) = 0. Taking the intersection of R Mult with C × C × C imposes conditions, at Since the quotient map C/C ⊥ takes (u, v) to ζ = v − Ad d −1 p 1 (u), the quotient on the right hand side of (39) consists of elements (ζ, ζ ′ , ζ ′′ ) where with x, y, z ∈ d subject to the conditions (40). It is straightforward to check that these are related by (38) if d ′ , d ′′ , d are the images of g ′ , g ′′ , g = g ′ g ′′ ∈ G 1 . Suppose now that (ζ ′ , ζ ′′ ) ∈ ker(Q Mult ) (at a given point (g ′ , g ′′ ) ∈ G 1 × G 1 ). The first equation in (38) (with ζ = 0) shows that ζ ′′ ∈ g 1 , and then the second equation shows that ζ ′ ∈ g 1 also, with ζ ′′ + Ad Φ((g ′′ ) −1 ) p 1 (ζ ′ ) = 0. This gives the desired description of ker(Q Mult ), while ran(Q Mult ) = d follows by dimension count using (21).