LIE ALGEBROIDS

BANACH CENTER PUBLICATIONS, VOLUME 54 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2001

ON THE GROUP OF LAGRANGIAN BISECTIONS OF A SYMPLECTIC GROUPOID

TOMASZ RYBICKI Department of Applied Mathematics, University of Mining and Metallurgy Mickiewicza 30, 30-059 Kraków, Poland E-mail: tomasz@uci.agh.edu.pl

Abstract. The group of lagrangian bisections of a symplectic groupoid extends the concept of the symplectomorphism group. The flux homomorphism is a basic invariant of this group. It is shown that this group is a regular Lie group. The group of exact (hamiltonian) bisections is also studied. The existence of the flux homomorphism enables a characterization of exact isotopies.

1. Introduction. The aim of this note is to prove that bisection groups related to a symplectic groupoid admit a structure of regular Lie group. In view of this fact we show the existence of the flux homomorphism and some further consequences, e.g. concerning exact bisection isotopies, quite similar to properties of the symplectomorphism group. We indicate also the integrability of some Lie algebras associated with the Poisson structure on the space of units of a symplectic groupoid.

Observe that the flux homomorphism for symplectic groupoids and some properties of the group of lagrangian bisections have been studied independently by P. Dazord [5] and P. Xu [22] without considering a Lie group structure on this group but rather exploiting some “weak” Lie theories (see below). However, by means of a chart at e we are able to obtain also the local form of the flux, which is essential for a characterization of exact isotopies.

We will use the definition of a Lie group from the convenient setting of the infinite dimensional Lie theory [8] due to A. Kriegl and P. Michor. The convenient setting is based on Boman’s theorem which states that a mapping f : ℝ^{n} → ℝ is smooth whenever f ∘ c is smooth for any smooth curve c : ℝ → ℝ^{n}. Consequently, a mapping between two possibly infinite dimensional manifolds is smooth if, by definition, it sends smooth

2000 Mathematics Subject Classification: Primary 22E65, 53D05.

Key words and phrases: symplectic groupoid, lagrangian bisection, group of bisections, flux homomorphism, Lie group.

Supported in part by the AGH grant n. 11.420.04.

The paper is in final form and no version of it will be published elsewhere.

curves to smooth curves. Furthermore, for modelling manifolds a special type of LCTVS is in use, namely so-called convenient vector spaces which fulfil some weak completeness condition.

Recall that a convenient Lie group is called regular if for 𝔤 = T_{e}G there exists a smooth bijective evolution map

evol_{G}^{r} : C^{∞} (ℝ, 𝔤) → C^{∞} ((ℝ, 0), (G, e)).

The right logarithmic derivative δ_{G}^{r} is then the inverse of evol_{G}^{r}. Frequently one encounters the situation that there is a closed subgroup H ⊂ G and a Lie algebra 𝔥 such that smooth curves with values in 𝔥 are sent bijectively by evol_{G}^{r} to isotopies with values in H. However we would like to emphasize that such a bijection does not yield a Lie group structure on H.

It is a characteristic feature of the convenient setting that Diff(M) for M open is still a regular Lie group, but its identity component consists only of compactly supported diffeomorphisms. So in our results the assumption on the compactness of the space of units (cf. [22]) is superfluous.

We would like also to indicate that there are some ”weak” settings of the infinite dimensional Lie theory and that they usually do not correspond strictly to each other. One example is the notion of diffeological groups due to J. M. Souriau [20]. A smooth structure is there defined by establishing sets of local smooth mappings from ℝ^{n} to G, n = 1, 2, …, and by imposing some conditions on them. As another example of a ”weak” setting can serve the concept of generalized Lie groups of H. Omori [11]. The definition is based on a continuous mapping exp : G → 𝔤 between a topological group G and a topological Lie algebra 𝔤 with some conditions which mimics essential properties of the exponential map. A common feature of such theories is that any closed subgroup of a Lie group is a Lie subgroup (which in a sense measures a lack of subtlety of them). Consequently, analogues of the presented results in those settings are rather trivial.

2. Preliminaries. We adopt the notation for groupoids from [3] rather than from [9].

Definition. A groupoid structure on a set Γ is given by two surjections (the source and target) α, β : Γ → M ⊂ Γ, by a multiplication m : Γ_{2} → Γ, where Γ_{2} = {(x, y) ∈ Γ × Γ : α(x) = β(y)}, and by an inversion i : Γ → Γ such that the following axioms are fulfilled:

(Ass) If one of the products m(x, m(y, z)) and m(m(x, y), z) is defined then so is the other and they are equal.

(Id) The products m(β(x), x), m(x, α(x)) are defined and equal to each other.

(Inv) m(x, i(x)) is defined and equal to β(x), and m(i(x), x) is defined and equal to α(x).

The elements of M are called unities. For simplicity we write x.y for m(x, y) and x^{−1} for i(x). We wil use the symbol Γ ⇉ M for the groupoid (Γ, M, α, β, m, i).

Next, a groupoid Γ is said to be a Lie groupoid if Γ is a smooth (C^{∞} ) manifold (not necessarily separated), M is a separated paracompact submanifold, α and β are submersions, m is a smooth mapping, and i is a diffeomorphism. Notice that Γ is separated iff M is closed in Γ.

For u ∈ M the set α(β^{−1} (u)) = β(α^{−1} (u)) is an orbit. The family of orbits forms a generalized foliation F_{Γ} of M (cf.[3]). Γ is called transitive if it has only one orbit M.

Examples. 1. Lie groups coincide with Lie groupoids with a unique unity.

2. Another extreme example are manifolds: Γ = M.

3. If α = β then for all u ∈ M the fiber α^{−1} (u) carries a Lie group structure. Any vector bundle is a Lie groupoid of this type.

4. For any set M put Γ = M ×M, α((x, y)) = y, β((x, y)) = x, m((z, y), (y, x)) = (z, x) and i((x, y)) = (y, x). We get the coarse groupoid with the space of units M ≃ Δ_{M}.

5. Given a principal fiber bundle P (M, π, G) one defines the equivalence relation ∼ on P × P by (p_{1}, p_{2}) ∼ (q_{1}, q_{2}) iff ∃a ∈ G p_{i}a = q_{i}. Letting denote Γ = P × P/ ∼, α([(p_{1}, p_{2})]) = π(p_{2}), β([(p_{1}, p_{2})]) = π(p_{1}), we get the gauge groupoid (with obvious m and i). Γ is identified with the set of equivariant bundle morphisms over id_{M}.

6. Assume that a Lie group G acts on a manifold M. Then we set: Γ = G × M, α((g, x)) = g.x, β((g, x)) = x, (g′, x′).(g, x) = (g′g, x), and (g, x)^{−1} = (g^{−1} , g.x). We say that Γ is a transformation groupoid.

7. For any Lie groupoid Γ the tangent space T Γ = (T Γ ⇉ T M, T α, T β, ⊕, I) possesses a structure of Lie groupoid. Here the multiplication ⊕ is given by

X ⊕ Y = ( \frac{d}{dt}(x(t).y(t))) |_{t=0} ,

where X = \frac{dx}{dt} |_{t=0} , Y = \frac{dy}{dt} |_{t=0} , α(x(t)) = β(y(t)), and the inversion IX = \frac{dx}{dt} x(t)^{−1} |_{t=0} if X = \frac{dx}{dt} |_{t=0} .

A bisection of a Lie groupoid Γ is a submanifold B of Γ such that α|B and β|B are diffeomorphisms onto M. Let Bis(Γ) be the set of all bisections. It is a group endowed with the product law

B_{1}.B_{2} = {x_{1}.x_{2}|α(x_{1}) = β(x_{2})}.

Notice that bisections of the coarse groupoid Γ = M × M (Ex. 4) coincide with diffeomorphisms on M, i.e. groups of bisections constitute a generalization of diffeomorphism groups. Bis(Γ) has natural left and right representations in Γ given by

ψ^{l} : Bis(Γ) ∋ B ↦ ψ^{l}(B) := {x ↦ B.x} ∈ Diff(Γ),
ψ^{r} : Bis(Γ) ∋ B ↦ ψ^{r}(B) := {x ↦ x.B} ∈ Diff(Γ).

Next there are the left and right representations in the unit space (M, F_{Γ})

ϕ^{l} : Bis(Γ) ∋ B ↦ ϕ^{l}(B) := β ∘ ψ^{l}(B)|_{M} ∈ Diff(M, F_{Γ}),
ϕ^{r} : Bis(Γ) ∋ B ↦ ϕ^{r}(B) := α ∘ ψ^{r}(B)|_{M} ∈ Diff(M, F_{Γ}),

where Diff(M, F_{Γ}) is the group of leaf preserving diffeomorphisms. Bis(Γ)_{c} will stand for the subgroup of all compactly controlled elements, that is, all B such that ϕ^{l}(B), or equivalently ϕ^{r}(B), has compact support. In general, compactly controlled bisections need not have compact support, e.g. in Example 3.

It is well known (J. Pradines [14], [15]) that to any Lie groupoid Γ ⇉ M is assigned the associated algebroid A(Γ), namely A(Γ) = (kerT β, [[,]], T α), where [[,]] is a Lie algebra bracket on Sect(kerT β) introduced by means of left invariant vector fields and of the identification T Γ|_{M} /T M ≃ kerT β.

Theorem 2.1. The groups Bis(Γ) and Bis(Γ)_{c} are regular Lie groups with the same Lie algebra Sect_{c}(kerT β).

The proof follows that for diffeomorphisms in [8], and makes use of the Tubular Neighborhood Theorem and the identification T Γ|_{M} /T M ≃ kerT β. The topology of Bis(Γ) is the identification topology by charts of a Lie group structure of Bis(Γ). In particular, all bisections in the identity component are compactly controlled.

To show the regularity of Bis(Γ) for any u ∈ M and B ∈ Bis(Γ) we denote by B^{β}(u) the unique point in B such that β(B^{β}(u)) = u. Now given a smooth isotopy B_{t} in Bis(Γ) (which is a concept without problem) with B_{0} = M there is a unique time-dependent family of vector fields X̂_{t} along B_{t}^{β} corresponding to B_{t}, i.e. for all u ∈ M

(2.1)  X̂_{t}(B_{t}^{β}(u)) = \frac{d}{ds}B_{s}^{β}(u)|_{s=t} .

By definition, X̂_{t} are tangent to the fibers of β.

Let u ∈ M and B ∈ Bis(Γ). We have a diffeomorphism σ_{u}^{B} : β^{−1} (u) → β^{−1} (u) given by

σ_{u}^{B}(x) = x.B^{β}(α(x)).

Then clearly σ_{u}^{B}(u) = B^{β}(u). By gluing-up the tangent mappings T σ_{u}^{B} of diffeomorphisms σ_{u}^{B} we get the canonical identification

(2.2)  σ^{B} : kerT β ≃ T Γ|_{M} /T M ≃ T Γ|_{B}/T B.

We have also the canonical identification T B^{β} : T M ≃ T B. By combining it with (2.2) we get

(2.3)  σ̃^{B} : T Γ|_{M} ≃ T Γ|_{B},

for any B ∈ Bis(Γ). Now by using the identifications (2.2) for any smooth bisection isotopy B_{t} with B_{0} = M we get a unique smooth curve X_{t} in Sect_{c}(kerT β) such that

(2.4)  σ^{B_{t}} X_{t} = X̂_{t}.

Then evol_{Bis(Γ)}^{r} (X_{t}) = B_{t}, and δ_{Bis(Γ)}^{r} (B_{t}) = X_{t}. Clearly X̂_{t} = X_{t} = X̃_{t} on M.

3. Symplectic groupoids and algebroids

Definition. A Lie groupoid Γ equipped with a symplectic form ω is called symplectic if the graph of multiplication graph(m) is a lagrangian submanifold of (−Γ)×Γ×Γ. Here −Γ means the symplectic manifold (Γ, −ω).

Let us recall that a Poisson structure on M can be introduced by a bivector Λ such that [Λ, Λ] = 0, where [., .] is the Schouten-Nijenhuis bracket (cf. [19]). Then the rank of Λ_{x} may vary but it is even everywhere. We have the ’musical’ bundle homomorphism Λ^{♯} associated with Λ by

Λ^{♯} : T^{∗}M → T M, β(Λ^{♯}α) = Λ(α, β).

In case Λ is nondegenerate (i.e. rank(Λ) = dim(M)), we get a symplectic structure ω, and Λ^{♯} is an isomorphism, denoted by ω^{♯}. The distribution Λ^{♯}(T_{x}^{∗}M), x ∈ M, integrates to a generalized foliation such that Λ restricted to any leaf induces a symplectic structure. This foliation is called symplectic and denoted by F_{Λ} . If the dimension of its leaves is constant, the Poisson structure Λ is called regular.

Proposition 3.1 [3]. If (Γ, ω) is a symplectic groupoid then:

(i) the inversion i is an antisymplectomorphism (i.e. i^{∗}ω = −ω), and M is a lagrangian submanifold;

(ii) the foliations by fibers of α and β are ω-orthogonal;

(iii) the space of units M admits a Poisson structure Λ such that its symplectic foliation F_{Λ} coincides with F_{Γ}.

(iv) α (resp. β) is a Poisson morphism (resp. anti-morphism).

Such a groupoid will be usually denoted by (Γ, ω) ⇉ (M, Λ).

Observe that the set of all lagrangian bisections Bis(Γ, ω) is a sugroup of Bis(Γ). This group has natural left and right representations in (Γ, ω):

ψ^{l} : Bis(Γ, ω) ∋ C ↦ ψ^{l}(C) = {x ↦ C.x} ∈ Symp(Γ, ω),
ψ^{r} : Bis(Γ, ω) ∋ C ↦ ψ^{r}(C) = {x ↦ x.C} ∈ Symp(Γ, ω).

Now the corresponding representations ϕ^{l} and ϕ^{r} take their values in Diff(M, Λ), the automorphism group of (M, Λ).

Examples. 8. The coarse groupoid Γ = X×X with (−ω)⊕ω is a symplectic groupoid. Then Bis(Γ, ω) = Symp(X).

9. If M is a manifold then T^{∗}M, where m is the addition in fibers and π_{M} = α = β, is a Lie groupoid (Ex. 3). T^{∗}M endowed with the canonical symplectic form ω_{M} = −dλ_{M} is also a symplectic groupoid. In fact, the graph of m

graph(m) = {(x_{3}, x_{2}, x_{1}) : x_{1} + x_{2} − x_{3} = 0}.

This is the image of N Δ_{M}, the normal bundle of the diagonal Δ_{M} ⊂ M^{3} in (T^{∗}M)^{3}, into (T^{∗}M)^{3} by the mapping (x_{3}, x_{2}, x_{1}) ↦ (−x_{3}, x_{2}, x_{1}). Notice that N Δ_{M} is lagrangian in (T^{∗}M)^{3}, and the mapping is symplectic. So T^{∗}M is indeed a symplectic groupoid.

10. Let Γ ⇉ M be a Lie groupoid. Then the cotangent space T^{∗}Γ equipped with ω_{Γ} = −dλ_{Γ} carries a structure of symplectic groupoid with N^{∗}M, the conormal bundle of M in Γ, being the space of units. Here the multiplication, denoted ⊕ as in Ex. 7, is determined by the equality

< ξ ⊕ η, X ⊕ Y >=< ξ, X > + < η, Y >, for X, Y ∈ T Γ, ξ, η ∈ T^{∗}Γ,

where <, > is the canonical pairing. Furthermore, the canonical projection p : T^{∗}Γ → Γ is an epimorphism of groupoids.

11. If G is a Lie group, T^{∗}G admits two symplectic groupoid structures. The first one is given as above, and the second is the structure of transformation groupoid (Ex. 6), where G acts on g^{∗} by the coadjoint action. As these structures obey a compatibility condition, T^{∗}G carries a structure of double groupoid, cf. [3].

Let us recall the following concept:

Definition. A Lie algebroid (T^{∗}M, {, }, ρ) over M is called symplectic if the following conditions are fulfilled:

(i) the (2,0)-tensor Λ given by Λ(α, β) = β(ρ(α)), ∀α, β ∈ Ω_{c}^{1}(M), is skew-symmetric; (ii) the space of all closed forms is a Lie subalgebra of (Sect(T^{∗}M), {, }).

Clearly there is a one-to-one correspondence between symplectic algebroids over M and Poisson structures on M. Here ρ = Λ^{♯} and

{α, β} = ı_{Λ^{♯}(α)} dβ − ı_{Λ^{♯}(β)} dα + dΛ(α, β).

Proposition 3.2. If (Γ, ω) ⇉ (M, Λ) is a symplectic groupoid then its associated algebroid A(Γ) is identified with (T^{∗}M, {, }, Λ^{♯}), the symplectic algebroid of (M, Λ). In particular, T^{∗}M ≃ T Γ|_{M} /T M ≃ kerT β.

It is remarkable that any Poisson manifold can be represented as the space of units of a local symplectic groupoid, and that Proposition 3.2 still holds for local symplectic groupoids. Consequently, there is a bijection between symplectic algebroids and local symplectic groupoids [4], a local non-linear version of the third Lie theorem.

4. Bis(Γ, ω) as a Lie group. The construction of a chart for Bis(Γ, ω) at e = M starts by an observation that there are no topological restrictions in a small neighborhood of M in Γ.

Lemma 4.1 [3]. Let N be a not necessarily separated manifold, and let M ⊂ N be a separated paracompact manifold such that a submersion p : N → M exists. Then there is a neighborhood U of M in N which is separated and with p-connected fibers.

By a local addition we mean a diffeomorphism μ : T^{∗}M ⊃ U → V ⊂ Γ such that μ(0_{u}) = u, ∀u ∈ M. In our case the existence of a local addition is ensured by the identification T Γ|_{M} /T M ≃ T^{∗}M, and by the exponential mapping coming from a Riemannian metric.

Lemma 4.2. Let M ⊂ N be a closed submanifold, and let ω_{0}, ω_{1} be symplectic forms on N which are equal along M. Then there is a diffeomorphism ϕ : U → V, where U, V are open neighborhoods of M in N such that ϕ^{∗}ω_{1} = ω_{0}, ϕ|_{M} = id_{M} and ϕ_{∗}|_{T N|_{M}} = id_{T N|_{M}} .

The proof uses Moser’s argument and the relative Poincaré lemma. For details, see [8, 43.11].

Any θ ∈ Ω^{1}(M) can be regarded as a section θ : M → T^{∗}M, and T^{∗}M is endowed with ω_{M} = −dλ_{M}, where λ_{M} is the canonical 1-form on T^{∗}M. The following is well-known.

Proposition 4.3. Under the above identification, θ^{∗}λ_{M} = θ. Moreover, θ(M) is lagrangian in T^{∗}M iff dθ = 0.

For the regularity of Bis(Γ, ω) the following is needed.

Lemma 4.4 [8, 38.7]. Let H be a topological Lie subgroup of a regular Lie group G. If there are an open neighborhood U ⊂ G of e and a smooth mapping p : U → E, where E is a convenient vector space, such that p^{−1} (0) = U ∩ H and p is constant on left cosets Hg ∩ U, then H is regular.

Theorem 4.5. Given a symplectic groupoid (Γ, ω) ⇉ (M, Λ), the group Bis(Γ, ω) is a closed subgroup of Bis(Γ) and a regular Lie group. Its Lie algebra coincides with ZΩ_{c}^{1}(M), the subalgebra of closed 1-forms on M.

Proof. (See also [8], 43.12.) We fix a local addition μ : T^{∗}M ⊃ U_{0} → V_{0} ⊂ Γ. We have two symplectic structures on U_{0}, namely the canonical one ω_{0} = ω_{M} |_{U}, where ω_{M} = −dλ_{M}, and ω_{1} = μ^{∗}ω. Each of them has the vanishing pullback on the zero section 0_{T^{∗} M}. In view of Lemma 4.2 we have to show that, by shrinking U_{0} and V_{0} and modifying the ω_{i}, we may get ω_{0}|_{0_{T^{∗} M}} = ω_{1}|_{0_{T^{∗} M}} .

To this end we observe that there is a vector bundle isomorphism ψ_{0} : T (T^{∗}M)|_{0_{T^{∗} M}} → T (T^{∗}M)|_{0_{T^{∗} M}} over id_{0_{T^{∗} M}} such that ψ_{0} = id_{T (0_{T^{∗} M} )} and ψ_{0} sends ω_{0} to ω_{1} on each fiber. Due to the partition of unity it suffices to construct ψ_{0} locally, and this is accomplished by considering lagrangian subbundles L_{i} complementary to T (0_{T^{∗} M}) with respect to ω_{i}. When having ψ_{0} we define a diffeomorphism ψ : U_{1} → U_{2}, where U_{1}, U_{2} are open neighborhoods of 0_{T^{∗} M} in T^{∗}M, such that T ψ|_{0_{T^{∗} M}} = ψ_{0}. This is done by using a tubular neighborhood of 0_{T^{∗} M}.

Now, in view of 4.2, we get a diffeomorphism ϕ of an open neighborhood of 0_{T^{∗} M} in T^{∗}M onto another such a neighborhood which satisfies ϕ^{∗}ω_{1} = ω_{0}, ϕ|_{M} = id_{M} and ϕ_{∗}|_{T (T^{∗} M)|_{0_{T^{∗} M}}} = id_{T (T^{∗} M)|_{0_{T^{∗} M}}} . We set

ρ := μ ∘ ϕ : T^{∗}M ⊃ U → V ⊂ Γ.

Let U be a neighborhood of e = M in Bis(Γ) consisting of all submanifolds B ⊂ Γ such that B^{β} : M → Γ is compactly supported, B^{β}(M) ⊂ V, and so small that μ^{−1} (B) is still the image of a β-section. We define a chart at e = M as follows

Φ : Bis(Γ) ⊃ U → V ⊂ Ω_{c}^{1}(M)

(4.1) Φ(B) := ρ^{−1} ∘ B^{β} ∘ (π ∘ ρ^{−1} ∘ B^{β})^{−1} .

Due to Proposition 4.3 B ∈ U ∩ Bis(Γ, ω) if and only if dΦ(B) = 0. Therefore (U, Φ) is a submanifold chart at e = M for Bis(Γ, ω) modelled on the subspace ZΩ_{c}^{1}(M) of all closed forms on M with compact support.

Next for arbitrary C ∈ Bis(Γ, ω) we get a submanifold chart at C as follows: U_{C} := {B : B.C^{−1} ∈ U} and Φ_{C}(B) := Φ(B.C^{−1} ). Thus Bis(Γ, ω) is a closed submanifold of Bis(Γ) and a Lie group.

Let C_{t} be a smooth isotopy in Bis(Γ, ω) and X_{t} ∈ Sect_{c}(kerT β) be defined by (3.4). By definition there is a time-dependent 1-form θ_{t} on M such that

(4.2)  ı(X_{t})ω|_{T Γ|_{M}} = θ_{t}.

Now if we set

c_{t} := ψ^{r}(C_{t}) and X̃_{t} := δ_{Symp (Γ,ω)}^{r} (c_{t}),

the smooth curve of vector fields X̃_{t} on Γ satisfies (cf. (3.3) and (3.4)) σ̃^{C_{t}} X_{t} = X̃_{t}. Clearly, X̃_{t} are tangent to the β-fibers, and c_{t} are β-fibers preserving. Also, c_{t}|_{M} = C_{t}^{β}, ∀t.

We get the equality

(4.3)  ı(X̃_{t})ω = β^{∗}θ_{t}

on the whole Γ. Consequently, θ_{t} is closed, as β^{∗}θ_{t} is closed.

Therefore, for any B_{t} ∈ C^{∞} ((ℝ, 0), (Bis(Γ), e)) and X_{t} = evol_{Bis(Γ,ω)}^{r} (B_{t}) we get by the regularity of Symp(Γ, ω), and by Prop. 3.2, the equivalence

B_{t} ∈ C^{∞} ((ℝ, 0), (Bis(Γ, ω), e)) ⇔ θ_{t} = ω^{♯}X_{t} ∈ C^{∞} (ℝ, ZΩ_{c}^{1}(M)),

which ensures that the Lie algebra of Bis(Γ, ω) is ZΩ_{c}^{1}(M), that is, that the restriction of evol_{Bis(Γ)}^{r} to ZΩ_{c}^{1}(M) identifies with evol_{Bis(Γ,ω)}^{r} .

Finally, let us consider p : Bis(Γ) → Ω^{2}(Γ) given by p(B) = ψ^{r}(B)^{∗}ω − ω. It follows from Lemma 4.4 that Bis(Γ, ω) is regular.

Since the neighborhood Φ(U) above can be chosen convex we can state the following

Corollary 4.6. The group Bis(Γ) is locally contractible and, consequently, locally arcwise connected.

Remark. Plausibly the monomorphisms ψ^{l} and ψ^{r} enable to introduce a Lie group structure on Bis(Γ, ω) in another way, namely as a Lie subgroup of Symp(Γ, ω). This would make easier all reasonings concerning the flux and lagrangian bisection isotopies. However, this is impossible unless Γ is compact. The reason is that no isotopy of the form c_{t} = ψ_{t}^{l}(C_{t}) or c_{t} = ψ_{t}^{r}(C_{t}) is contained in Symp(Γ, ω)_{0} (in our framework). Nonetheless, these representations are indispensable in section 5 and 6.

5. The flux homomorphism. The concept of the flux homomorphism was introduced by E. Calabi [2]. It is a basic invariant not only of the symplectomorphism groups, but also of some other transformation groups, cf. [17], [18]. The flux is still meaningful for the group of Lagrangian bisections (cf. [5], [22]). In light of Theorem 4.5 one can obtain the flux also in its local form which is necessary for a characterization of exact isotopies.

Let us fix the notation. For any locally arcwise connected topological group G its universal cover group G̃ is the set of all pairs (g, {g_{t}}), where {g_{t}} is the homotopy rel. endpoints class of the path g_{t} in G such that g_{0} = e and g_{1} = g, endowed with the pointwise multiplication. By G_{0} we denote the component of e in G. Clearly G̃_{0}/π_{1}(G_{0}) = G_{0}, where π_{1} is the first homotopy group.

The multiplication in G̃ can be also thought of as the juxtaposition of representants. The latter means that {g_{t}}.{f_{t}} = {g_{t} ∗ f_{t}} where

g_{t} ∗ f_{t} = \{ \begin{matrix}f_{2t}, & for 0 ≤ t ≤ \frac{1}{2} \\ g_{2t−1} ∘ f_{1}, & for \frac{1}{2} ≤ t ≤ 1. \end{matrix}

These multiplications are the same on the homotopy level.

Given a symplectic groupoid (Γ, ω) ⇉ (M, Λ), by using (4.2) and (4.3) we have the mapping

C^{∞} (ℝ, Bis(Γ, ω)) ∋ C_{t} ↦ θ_{t} ∈ C^{∞} (ℝ, ZΩ_{c}^{1}(M))

which induces the flux homomorphism.

Theorem 5.1. The mapping S̃_{Γ} : B̃is(Γ, ω)_{0} → H_{c}^{1}(M) defined by

S̃_{Γ} ({C_{t}}) = [ ∫_{0}^{1} θ_{t}dt]

is a well-defined continuous epimorphism (with the abelian structure on H^{1}(M)), called the flux homomorphism for the symplectic groupoid Γ. If Γ, in addition, is compact then the relation between S̃_{Γ} and S̃, the flux for the symplectomorphism group Symp(Γ, ω), can be visualized in the following commutative diagram

B̃is(Γ, ω)_{0}  S̃ymp(Γ, ω)_{0}

H_{c}^{1}(M)  H_{c}^{1}(Γ)

ψ^{r} [figure]

S̃_{Γ}

[figure]

S̃

[figure] β^{∗} [figure]

The proof is similar to that for symplectomorphisms. We emphasize the existence of S̃_{Γ} for an arbitrary symplectic groupoid (not necessarily with M compact as in [22]). However, the diagram makes sense for Γ compact only, since otherwise no c_{t} = ψ^{r}(C_{t}) lies in Symp(Γ, ω)_{0}.

Examples. 1. If Γ = M × M is the coarse groupoid then S̃_{Γ} is the usual flux for Symp(M, ω).

2. If Γ = T^{∗}M then Bis(Γ) identifies with the space of all closed 1-forms, and S̃_{Γ} assigns to 1-form its cohomology class.

In analogy with the symplectic case (cf. [2,10]) Theorem 4.5 enables us to define the local flux homomorphism.

Corollary 5.2. If ω is exact, i.e. ω = −dλ, then S̃_{Γ} ({C_{t}}) = [λ|_{T M} −(C_{1}^{β})^{∗}λ] = [(λ− c_{1}^{∗}λ)|_{T M}], where c_{t} = ψ^{r}(C_{t}). In particular, the group of periods Ξ_{Γ} := S̃_{Γ} (π_{1}(Bis(Γ, ω)_{0})) is zero.

Proof. Let X_{t} = evol_{Bis(Γ,ω)}^{r} (C_{t}), X_{t} = ω^{♯}θ_{t}, c_{t} = ψ^{r}(C_{t}), and X̃_{t} := δ_{Symp (Γ,ω)}^{r} (c_{t}). By composing each side of (4.2) with β^{∗} we get and

\begin{aligned}β^{∗}θ_{t} & = c_{t}^{∗}β^{∗}θ_{t} = c_{t}^{∗}β^{∗}ı_{X_{t}} ω|_{T Γ|_{M}} = c_{t}^{∗}ıX̃_{t} ω \\ & = −c_{t}^{∗}ıX̃_{t} dλ = −c_{t}^{∗}LX̃_{t} λ = −\frac{d}{dt}c_{t}^{∗}λ. \end{aligned}

Hence

∫_{0}^{1} β^{∗}θ_{t} = λ − c_{1}^{∗}λ

holds on Γ. By restricting this equality to T M we get

∫_{0}^{1} θ_{t} = (λ − c_{1}^{∗}λ)|_{T M} ,

as β^{∗}θ|_{T M} = θ. In view of C_{t}^{β} = c_{t}|_{M} this yields the required equalities.

Definition. Let Φ : U → V be a chart for Bis(Γ, ω) at e = M. If C_{t} is a lagrangian isotopy such that C_{t} ∈ U for any t then

S̃_{Γ}^{Φ} ({C_{t}}) := −[θ_{1}] ∈ H_{c}^{1}(M),

where θ_{t} = Φ(C_{t}). S̃_{Γ}^{Φ} is called the local flux homomorphism for Bis(Γ, ω).

Proposition 5.3. S̃_{Γ}^{Φ} does not depend on the choice of Φ and extends S̃_{Γ} .

Proof. Let C_{t} be a lagrangian bisection isotopy. Then θ_{t} is a lagrangian bisection isotopy in U ⊂ T^{∗}M with respect to the canonical symplectic form ω_{M} = −dλ_{M}. Let S̃_{M} be the flux homomorphism for the symplectic groupoid T^{∗}M (Ex. 9). Then

S̃_{M} ({θ_{t}}) = S̃_{M} ({Φ(C_{t})}) = ϕ_{C_{t}}^{∗} S̃_{Γ} ({C_{t}}) = S̃_{Γ} ({C_{t}}),

where ϕ_{C_{t}} = π_{M} ∘ ρ ∘ C_{t}^{β}, by the homotopy invariance. But in view of Cor. 5.2 and Prop. 4.3 we get

S̃_{M} ({θ_{t}}) = [λ_{M} |_{T (0_{T^{∗} M} )} − θ_{1}^{∗}λ_{M}] = [0_{T^{∗} M}^{∗} λ_{M} − θ_{1}^{∗}λ_{M}] = −[θ_{1}],

as required.

6. A characterization of exact isotopies. A lagrangian bisection isotopy C_{t} is called exact if (C_{t}) = ω^{♯}du_{t} for some smooth curve u_{t} in C_{c}^{∞} (M). Let Bis^{ex} (Γ, ω) be the set of all exact lagrangian bisections. That is, by definition C ∈ Bis^{ex} (Γ, ω) iff it can be joined with e = M by an exact lagrangian bisection isotopy.

Proposition 6.1. A lagrangian isotopy C_{t} is exact iff δ_{Symp (Γ,ω)}^{r} (c_{t}) = ω^{♯}β^{∗}(du_{t}), where c_{t} = ψ^{r}(C_{t}).

Indeed, it follows from the fact that β is a Poisson anti-morphism.

Proposition 6.2. Bis^{ex} (Γ, ω) is a path-connected normal subroup of Bis(Γ, ω).

Proof. First we check that Bis^{ex} (Γ, ω)) is a group. Let B_{t}, C_{t} be exact isotopies, that is (B_{t}) = ω^{♯}(dv_{t}), (C_{t}) = ω^{♯}(du_{t}) for some smooth families of C^{∞} -functions u_{t} and v_{t}. If ψ^{r}(B_{t}) = b_{t}, and ψ^{r}(C_{t}) = c_{t} then ψ^{r}(B_{t}.C_{t}) = c_{t} ∘ b_{t}. Is is apparent that

δ_{Symp (Γ,ω)}^{r} (c_{t} ∘ b_{t}) = ω^{♯}d(β^{∗}u_{t} + β^{∗}v_{t} ∘ c_{t}^{−1} ) = ω^{♯}d.(β^{∗}u_{t} + β^{∗}v_{t}).

The second equality holds by β ∘ c_{t} = β. In view of Prop. 6.1 B_{t}.C_{t} is still exact. To get that C_{t}^{−1} is exact we use the equality

δ_{Symp (Γ,ω)}^{r} (c_{t}^{−1} ) = ω^{♯}d(−u_{t} ∘ c_{t}) = ω^{♯}d(−u_{t}).

It follows that Bis^{ex} (Γ, ω) is a group. Finally, Bis^{ex} (Γ, ω) is normal since

δ_{Symp (Γ,ω)}^{r} (b^{−1} ∘ c_{t} ∘ b) = ω^{♯}d(u_{t} ∘ b).

Now by repeating an argument for the symplectomorphism group (cf. [10]) it is possible to relate lagrangian bisection isotopies and S̃_{Γ} . Consequently, under some assumption it is introduced a regular Lie group structure on Bis^{ex} (Γ, ω) (Theorem 6.8).

Proposition 6.3. C_{t} is an exact isotopy iff S̃_{Γ} ({C_{τ}}_{0≤τ≤t} ) = 0, ∀t.

Proof. (⇒) It follows by the definition of S̃_{Γ} . (⇐) Notice that C_{t} is exact for t small due to the local definition of S̃_{Γ} . Next we extend t by replacing successively ϕ_{t} by C_{t}.C_{t_{0}}^{−1} with t − t_{0} small enough.

Proposition 6.4. If C ∈ Bis(Γ, ω)_{0} then C is an exact bisection if and only if there is a lagrangian bisection isotopy C_{t} with C_{0} = M and C_{1} = C such that S̃_{Γ} ({C_{t}}) = 0. Furthermore, if S̃_{Γ} ({B_{t}}) = 0, B_{t} ∈ Bis(Γ, ω), then B_{t} is homotopic with fixed endpoints to an exact isotopy.

Proof. If C is exact then there is an exact isotopy C_{t} joining C with M. Then ı(X_{t})ω = dv_{t} for v_{t} ∈ C_{c}^{∞} (M), where X_{t} = δ_{Bis(Γ,ω)}^{r} (C_{t}). Hence S̃_{Γ} ({C_{t}}) = 0 as [dv_{t}] = 0.

In order to show the converse let C_{t} = evol_{Bis(Γ,ω)}^{r} (X_{t}) be such that S̃_{Γ} ({C_{t}}) = 0, that is ∫_{0}^{1} ι(X_{t})ωdt = dv, v ∈ C_{c}^{∞} (M). Here v can be chosen compactly supported as all suppX_{t} are in a fixed compact subset. Let C_{t}^{v} be the isotopy of X_{v} = ω^{♯}dv. It suffices to consider (C_{1}^{v})^{−1} .C instead of C. Thus, reparametrizing if necessary, we may assume that ∫_{0}^{1} X_{t}dt = 0 as ı(∫_{0}^{1} X_{t}dt)ω = 0.

Next we set Y_{t} = − ∫_{0}^{t} X_{τ}dτ. Then if s ↦ B_{t}^{s} be the isotopy of Y_{t} we get for Ĉ_{t} = B_{t}^{1}.C_{t} that C_{1} = Ĉ_{1} and S̃_{Γ} ({Ĉ_{τ}}_{0≤τ≤t} ) = 0 for any t. In fact,

S̃_{Γ} ({B_{τ}^{1}}_{0≤τ≤t} ) = S̃_{Γ} ({B_{t}^{s}}_{0≤s≤1} ) = [ı(Y_{t})ω]

in view of the homotopy rel. endpoints invariance. Therefore

\begin{aligned}S̃_{Γ} (\{Ĉ_{τ}\}_{0≤τ≤t} ) & = S̃_{Γ} (\{B_{τ}^{1}\}_{0≤τ≤t} ) + S̃_{Γ} (\{C_{τ}\}_{0≤τ≤t} ) \\ & = [ı(Y_{t})ω] + [ ∫_{0}^{t} ı(X_{τ})ωdτ] = 0. \end{aligned}

By Prop. 6.3 this proves the first assertion, and the proof of the second can be deduced from the above argument.

Let Ξ_{Γ} = S̃_{Γ} (π_{1}(Bis(Γ, ω))). In the symplectic case Ξ_{Γ} is called the group of periods. Observe that Ξ_{Γ} is countable since π_{1}(Bis(Γ, ω)) is countable. This follows from the fact that Bis(Γ, ω) has the homotopy type of a countable simplicial complex ([12]).

Notice that the form θ_{1} = Φ(C_{1}), where Φ : U → V is a chart for Bis(Γ, ω) at e, need not be exact even if C_{t} is an exact isotopy and C_{1} ∈ U. However we have the following

Proposition 6.5. C ∈ Bis(Γ, ω) ∩ U if and only if [Φ(C)] ∈ Ξ_{Γ}.

Proof. (⇒) We define the lagrangian bisection isotopy B_{t} by Φ(B_{t}) = tθ, where θ = Φ(C) (B_{1} = C). But there is an exact isotopy C_{t} joining C = C_{1} with M = C_{0}. Then for the loop C_{1−t} ∗ B_{t}, one gets [θ] = −S̃_{Γ} ({B_{t}}) = S̃_{Γ} ({C_{t}}) − S̃_{Γ} ({B_{t}}) = −S̃_{Γ} ({C_{1−t} ∗ B_{t}}) ∈ Ξ_{Γ}.

(⇐) We use Prop. 6.4. There is a lagrangian bisection loop C_{t} such that S̃_{Γ} ({C_{t}}) = −[Φ(C)]. Then S̃_{Γ} ({D_{t}}) = 0, where D_{t} = B_{t} ∗ C_{t} and Φ(B_{t}) = [(1 − t)Φ(C)]. Therefore D_{t} is homotopic to an exact bisection isotopy joining M with C. Thus C ∈ Bis^{ex} (Γ, ω).

Proposition 6.6. Every smooth curve C_{t} in Bis^{ex} (Γ, ω) is an exact bisection isotopy.

Proof. For t ≤ ϵ one has C_{t} ∈ U. It follows by Prop. 6.5 that [θ_{t}] ∈ Ξ_{Γ} for small t, where θ_{t} = Φ(C_{t}). This means that [θ_{t}] = 0 as θ_{0} = 0 and Ξ_{Γ} is countable. Therefore we have S̃_{Γ} ({C_{τ}}_{0≤τ≤t} ) = 0 for t ≤ ϵ. Thanks to Prop. 6.3 the C_{t} is an exact isotopy for t ≤ ϵ. By taking C_{t}.C_{ϵ}^{−1} instead of C_{t} this procedure extends the argument for all t.

Since Bis(Γ, ω)_{0} = B̃is(Γ, ω)_{0}/π_{1}(Bis(Γ, ω)_{0}) the flux S̃_{Γ} induces another homomorphism S_{Γ} : Bis(Γ, ω)_{0} → H_{c}^{1}(M)/Ξ_{Γ}.

Corollary 6.7. Ker(S_{Γ}) coincides with Bis^{ex} (Γ, ω).

Indeed, it follows from the local form of the flux and Prop. 6.5 for any C sufficiently near M. This can be extended to the whole Bis(Γ, ω)_{0} by the same argument as in Prop. 6.4.

Now we are in a position to define a Lie group structure on Bis^{ex} (Γ, ω).

Theorem 6.8. If Ξ_{Γ} ⊂ H_{c}^{1}(M) is discrete (or 0 is an isolated point in Ξ_{Γ}) then Bis^{ex} (Γ, ω) is a regular Lie group with BΩ_{c}^{1}(M), the space of exact compactly supported 1-forms on M, as its Lie algebra.

Proof. Assume the notation of the proof of Theorem 4.5, and let

Φ : Bis(Γ) ⊃ U → V ⊂ Ω_{c}^{1}(M)

be a chart at e given by (4.1). If C ∈ Bis^{ex} (Γ, ω) ∩ U then θ = Φ(C) ∈ ZΩ_{c}^{1}(M) and [θ] ∈ Ξ_{Γ} (Prop. 6.5). By the assumption, taking possibly a smaller U, we have C ∈ Bis^{ex} (Γ, ω) ∩ U if and only if θ = Φ(C) is exact. Proceeding as in the proof of 4.5 we see that Bis^{ex} (Γ, ω) is a submanifold of Bis(Γ, ω), and since the composition and inversion are smooth by restriction, it is a Lie group.

By definition of Bis^{ex} (Γ, ω) the restriction of evol_{Bis(Γ,ω)}^{r} to Bis^{ex} (Γ, ω) is evol_{Bis^{ex} (Γ,ω)}^{r} . Thus BΩ_{c}^{1}(M) is identified as the Lie algebra of Bis^{ex} (Γ, ω). Finally, setting p = S_{Γ} it follows from Cor. 6.7 and Lemma 4.4 that Bis^{ex} (Γ, ω) is regular.

7. Final remarks. The third theorem of Lie asserts that any finite dimensional Lie algebra is actually the Lie algebra of a Lie group. Since the paper by van Est and Korthagen [6] it is well known that, in general, this theorem is no longer true in the infinite dimensional case. Nevertheless, there are several generalizations of the third Lie theorem. First of all, some basic facts can be carried over from Lie groups to Lie or symplectic groupoids, cf. [14], [15], [4]. Next, the third Lie theorem still holds for remarkable infinite dimensional Lie algebras, e.g. the Poisson algebra of any symplectic or locally conformal symplectic manifold ([1], [7]).

Notice that Theorem 4.5 and 6.8 can be interpreted as follows. A Poisson manifold (M, Λ) is called integrable if it can be realized as the space of units of a symplectic groupoid. Given a symplectic groupoid (Γ, ω) ⇉ (M, Λ) such that Ξ_{Γ} = π_{1}(Bis(Γ, ω)_{0}) is discrete, the exact sequence of Lie algebras

0 → BΩ_{c}^{1}(M) → ZΩ_{c}^{1}(M) → H_{c}^{1}(M) → 0,

where H_{c}^{1}(M) is regarded as an abelian Lie algebra, can be integrated to the exact sequence of regular Lie groups

1 → Bis^{ex} (Γ, ω) → Bis(Γ, ω)_{0} → H_{c}^{1}(M)/Ξ_{Γ} → 0.

In [19] it has been shown that for any (regular) foliation or any Poisson structure with regular symplectic foliation the related Lie algebras of vector fields (tangent to the foliation) can be integrated analogously as in the transitive case. Unfortunately these results do not encompass the singular case. Let us also remark that the regular transitive case cannot be treated in terms of groupoid bisections, i.e. the leaf preserving diffeomorphisms cannot be expressed, in general, in terms of bisections. The reason is that in view of the ”holonomic imperative” of a groupoid over a foliated manifold ([16], [13]) to any bisection of such a groupoid is attached its holonomy class. Consequently, Lie groups considered in [19] are modelled on the space of foliated 1-forms, rather than on the space of ordinary 1-forms.

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