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Poisson 2-groups

We prove a 2-categorical analogue of a classical result of Drinfeld: there is a one-to-one correspondence between connected, simply-connected Poisson Lie 2-groups and Lie 2-bialgebras. In fact, we also prove that there is a one-to-one correspondence between connected, simply connected quasi-Poisson 2-groups and quasi-Lie 2-bialgebras. Our approach relies on a"universal lifting theorem"for Lie 2-groups: an isomorphism between the graded Lie algebras of multiplicative polyvector fields on the Lie 2-group on one hand and of polydifferentials on the corresponding Lie 2-algebra on the other hand.


Introduction
A Poisson group is a Lie group equipped with a compatible Poisson structure. Poisson groups are the classical limit of quantum groups and have been extensively studied in the past two decades. For instance, Drinfeld proved that there is a bijection between connected, simply connected Poisson groups and Lie bialgebras [7,8].
Lie 2-groups (also called strict Lie 2-groups in the literature) are Lie group objects in the category of Lie groupoids, or equivalently Lie groupoid objects in the category of Lie groups. More explicitly, a Lie 2-group is a Lie groupoid Γ 1 ⇒ Γ 0 , where both Γ 1 and Γ 0 are Lie groups and all the groupoid structure maps are group homomorphisms. Lie 2-groups are special instances of Mackenzie's double groupoids [14].
The recent categorification trend motivates the search for an appropriate notion of quantum 2-groups. Poisson 2-groups are a natural first step in that direction. By a Poisson 2-group, we mean a Lie 2-group equipped with a Poisson structure Π on Γ 1 , which is multiplicative with respect to both the group and the groupoid structures on Γ 1 . In other words, (Γ 1 , Π) is simultaneously both a Poisson group [7,8] and a Poisson groupoid [20].
Lie 2-algebras are Lie algebra objects in the category of Lie algebroids [1]. They can be identified with Lie algebra crossed modules: pairs of Lie algebras θ and g together with a linear map φ : θ → g and an action of g on θ by derivations satisfying a certain compatibility condition. Likewise, a Lie 2-bialgebra can be considered as a Lie bialgebra crossed module, i.e. a pair of Lie algebra crossed modules in duality: (θ φ → g) and (g * −φ * → θ * ) are both Lie algebra crossed modules, and (g ⋉ θ, θ * ⋉ g * ) is a Lie bialgebra.
We prove that, at the infinitesimal level, Poisson 2-groups induce Lie 2bialgebras. More precisely, we prove the following analogue of Drinfeld's theorem: Theorem A. There is a one-to-one correspondence between connected, simply-connected Poisson 2-groups and Lie 2-bialgebras.
We will also prove a more general result: There is a one-to-one correspondence between connected, simply-connected quasi-Poisson 2-groups and quasi-Lie 2-bialgebras.
A natural generalization of Lie 2-algebras (or Lie algebra crossed modules), weak Lie 2-algebras are two-term L ∞ algebras. They can be described concisely in terms of the shifted degree "big bracket," which is a Gerstenhaber bracket on S • (V [2] ⊕ V * [1]). Here V = θ ⊕ g is a graded vector space, where θ is of degree 1 and g is of degree 0. Identifying S • (V [2] ⊕ V * [1]) with the space Γ(∧ • T [4]M ) of polyvector fields on M = V * [−2] with polynomial coefficients, the big bracket can be simply described as the Schouten bracket of polyvector fields on M .
In [6], we developed a notion of weak Lie 2-bialgebras: objects that are simultaneously weak Lie 2-algebras as well as weak Lie 2-coalgebras, both structures being compatible with one another in a certain sense. In terms of the big bracket, a weak Lie 2-bialgebra on a graded vector space V is a degree-(−4) element t of S • (V [2]⊕V * [1]) satisfying {t, t} = 0. Quasi-Lie 2-bialgebras are a special instance of weak Lie 2-bialgebras.
Our proofs of Theorems A and B rely on the following "universal lifting theorem," which should be of independent interest: Theorem C. Given a Lie 2-group Γ 1 ⇒ Γ 0 , if both Γ 1 and Γ 0 are connected and simply connected, then the graded Lie algebras k≥0 X k mult (Γ 1 ) and k≥0 A k are isomorphic.
Here k≥0 X k mult (Γ 1 ) denotes the space of multiplicative polyvector fields on Γ 1 which, being closed with respect to the Schouten bracket, is naturally a graded Lie algebra. On the other hand, k≥0 A k denotes the graded Lie algebra formed by the polydifferentials on the associated Lie 2-algebra -the infinitesimal counterparts of the multiplicative polyvector fields on the Lie 2-group.
Theorems A and B are proved simply by expressing the algebraic data defining the weak Lie 2-bialgebra structure in terms of the graded Lie algebra k≥0 A k .
We refer to the recent papers [13,16,18] on integration of Courant algebroids to symplectic 2-groupoids, which may have a close connection to our work.

Quasi-Lie 2-bialgebras
In this section, we recall some basic notions regarding quasi Lie 2bialgebras developed in [6].
1.1. The big bracket. We will introduce a graded version of the big bracket [11,12] involving graded vector spaces.
Let V = k∈Z V (k) be a graded vector space. Consider the Z-graded manifold M = V * [−2] and the shifted tangent space Consider the space of polyvector fields on M with polynomial coefficients: There is a standard way to endow S • = Γ(∧ • T [4]M ) with a graded Lie bracket, i.e. the Schouten bracket, which is denoted by { , }. It is a bilinear map { , } : S • ⊗ S • → S • satisfying the following properties: for all homogeneous e i ∈ S • , and the following graded Jacobi identity holds: Hence (S • , ⊙ , { , }) is a Schouten algebra, also known as an odd Poisson algebra, or a Gerstenhaber algebra [19].
Due to our degree convention, when V is an ordinary vector space considered as a graded vector space concentrated at degree 0, the bracket above is different from the usual big bracket in the literature [11].
These maps are required to satisfy the following compatibility conditions: for all w, x, y, z ∈ g and u, v ∈ θ, If h vanishes, we call it a strict Lie 2-algebra, or simply a Lie 2-algebra. Now consider the degree shifted vector spaces V [2] and V * [1]. Under such a degree convention, the degrees of g, θ, g * and θ * are specified as follows: We will maintain this convention throughout this section. We remind the reader that the abbreviation S • stands for S • (V * [1] ⊕ V [2]). Proposition 1.1 ([6]). Under the above degree convention, a weak Lie 2-algebra structure is equivalent to a solution to the equation (1) {s, s} = 0, where s =φ +b +ǎ +ȟ is an element in S (−4) such that Here the bracket in Eq. (1) stands for the big bracket as in Section 1.1.
In the sequel, we denote a weak Lie 2-algebra by (θ→g, s) in order to emphasize the map from θ to g. Sometimes, we will omit s and denote a weak Lie 2-algebra simply by (θ→g). If (g * → θ * ) is a weak Lie 2algebra, then (θ → g) is called a weak Lie 2-coalgebra. Equivalently, a weak Lie 2-coalgebra is a 2-term L ∞ -structure on g * ⊕ θ * , where g * has degree 1 and θ * has degree 0.
One can verify directly that these relations indeed define a weak Lie 2-bialgebra.
1.3. Lie bialgebra crossed modules. Definition 1.6. A Lie algebra crossed module consists of a pair of Lie algebras θ and g, a linear map φ : θ → g, and an action of g on θ by derivations satisfying, for all x, y ∈ g, u, v ∈ θ, where ⊲ denotes the g-action on θ.
Note that 1 and 2 imply that φ must be a Lie algebra homomorphism.
We write (θ φ → g) to denote a Lie algebra crossed module. The associated semidirect product Lie algebra is denoted by g ⋉ θ.
The following proposition indicates that crossed modules of Lie algebras are in one-to-one correspondence with Lie 2-algebras. We refer the reader to [1] for details. Proposition 1.7. Lie algebra crossed modules are equivalent to (strict) Lie 2-algebras. Definition 1.8. A Lie bialgebra crossed module is a pair of Lie algebra crossed modules in duality: (θ φ → g) and (g * φ T → θ * ), where φ T = −φ * , are both Lie algebra crossed modules such that (g ⋉ θ, θ * ⋉ g * ) is a Lie bialgebra.
Lie bialgebra crossed modules are symmetric as we see in the next The following result justifies our terminology.
The following theorem was proved in [6]. Theorem 1.12. There is a bijection between Lie bialgebra crossed modules and (strict) Lie 2-bialgebras. Example 1.13. Consider the Lie subalgebra u(n) ⊂ gl n (C) of n × n skew-Hermitian matrices. Let θ ⊂ gl n (C) be the Lie subalgebra consisting of upper triangular matrices whose diagonal elements are real numbers. It is standard that (θ, u(n)) is a Lie bialgebra. Indeed θ ⊕ u(n) ∼ = gl n (C), and both θ and u(n) are Lagrangian subalgebras of gl n (C) under the nondegenerate pairing X|Y = Im(Tr(XY )), for X, Y ∈ gl n (C). Hence (θ, u(n), gl n (C)) is a Manin triple, and thus (θ, u(n)) forms a Lie bialgebra. Let g denote the Lie algebra of traceless upper triangular matrices with real diagonal coefficients. It turns out that (θ φ → g), where φ is the map A → A − trA, is a Lie bialgebra crossed module. Lie groups and all the groupoid structure maps are group homomorphisms. A Lie 2-group is a special case of double groupoid in the sense of Mackenzie [14].
Definition 2.1 ( [21,22]). A Lie group crossed module consists of a Lie group homomorphism Φ : Θ → G and an action of G on Θ by automorphisms satisfying the following compatibility conditions: for all g ∈ G and α, β ∈ Θ. Here g⊲β denotes the action of g ∈ G on β ∈ Θ.
We write (Θ Φ → G) to denote a Lie group crossed module.
There is a bijection between Lie 2-groups and crossed modules of Lie groups.
Proof. This is standard. For instance, see [2,4,5,15,17]. Here we will sketch the construction of the Lie 2-group out of a crossed module, which will be needed later on.
The Lie 2-group corresponding to a Lie group crossed module (Θ Φ → G) will be denoted by G ⋉ Θ ⇒ G, or simply G ⋉ Θ, by abuse of notations.
In the sequel, we will use Lie 2-groups and crossed modules of Lie groups interchangeably.

2.2.
Multiplicative polyvector fields on Lie groupoids. We recall some standard results regarding multiplicative polyvector fields on a Lie groupoid. Let Γ ⇒ M be a Lie groupoid with source and target maps s and t, respectively. Consider the graph of the groupoid multiplication . In other words, Here A denotes the Lie algebroid of Γ, and ← − X denotes the left invariant vector field on Γ corresponding to X.
The following lemma gives a useful characterization of multiplicative polyvector fields.

k-differentials of a Lie algebroid.
It is known that the Schouten bracket of two multiplicative polyvector fields on a Lie groupoid is still multiplicative. Therefore, the space of multiplicative polyvector fields is a graded Lie algebra [9]. On the level of Lie algebroids, multiplicative k-vector fields correspond to k-differentials of the Lie algebroid, whose definition we recall below.
Given a Lie algebroid A, a k-differential is a linear map The commutator of a k 1 differential ∂ 1 and a k 2 -differential ∂ 2 is the Theorem 2.7 ([9]). Let Γ be a Lie groupoid, and A its Lie algebroid. Then every multiplicative k-vector field Σ induces a k-differential ∂ Σ by Moreover, the map Σ → ∂ Σ is a homomorphism of graded Lie algebras, which is an isomorphism of graded Lie algebras provided Γ is s-connected and s-simply connected.
In case of Lie groups, k-differentials of multiplicative k-vector fields can be described more explicitly.
Lemma 2.8. Let Σ be a multiplicative k-vector field on a Lie group G.
2) The Lie algebra 1-cocycle induced by σ is the k-differential ∂ Σ : 2.4. Infinitesimal data of multiplicative vector fields on Lie 2groups. This section is devoted to the description of infinitesimal data of multiplicative vector fields on Lie 2-groups.

Multiplicative polyvector fields on
For k ≥ 1, a k-vector field V is called multiplicative if it is multiplicative with respect to both the group and the groupoid structure on G ⋉ Θ. In other words, the graph of the group multiplication and the graph of the groupoid multiplication Denote the space of multiplicative k-vector fields by X k mult (G ⋉ Θ). The following lemma follows immediately.

The infinitesimal data.
It is natural to ask what is the infinitesimal data of a multiplicative k-vector field (with k ≥ 1) on a Lie 2-group. To answer this question, we need, as a first step, to describe the Lie algebroid A of the groupoid G ⋉ Θ ⇒ G.

It is simple to see that
Here the superscript ← − stands for the left invariant vector field on G associated to a Lie algebra element in g. It follows from Theorem 2.7 that a multiplicative k-vector field V ∈ X k mult (G ⋉ Θ) induces a k-differential of the Lie algebroid A. In particular, we have a map , for all f 1 , f 2 ∈ C ∞ (G), ∂ gpd induces a ∧ k−1 θ-valued vector field on G, which in turn can be identified with a g ⊗ (∧ k−1 θ)-valued function on G. Here we identify the tangent bundle T G with G × g by left translations. By skew symmetrization, we thus obtain a g ∧ (∧ k−1 θ)-valued function on G, denoted by δ. More explicitly, we have For k ≥ 1, let We have Here G acts on g by the adjoint action, and on θ by the induced action from the crossed module structure.
Proof. Proposition 2.6 describes how a multiplicative vector field on a Lie groupoid looks like along the base manifold. Now we apply this theorem to the groupoid G ⋉ Θ ⇒ G. Identify T G with G × g via left translations. We write V| g for the value of V at (g, 1 Θ ). For all g ∈ G, we have , ∀ x ∈ g, u ∈ θ, and, by abuse of notation, L g * denotes the tangent map of the left translation by (g, 1 Θ ) on the group G ⋉ Θ.
Since V is multiplicative with respect to the group structure on G ⋉ Θ, it follows that V| gh = L g * V| h + R h * V| g , where R h * denotes the tangent map of the right translation by (h, 1 Θ ) in the group G ⋉ Θ. Substituting Eq. (17) into the equation above, we see that δ is indeed a Lie group 1-cocycle.
Moreover, Proposition 2.6 implies that As a consequence, δ takes values in W k . This concludes the proof. q.e.d.
Taking the derivative of δ at the unit: we obtain the following Corollary 2.13. Any multiplicative k-vector field on a Lie 2-group G ⋉ Θ induces a Lie algebra 1-cocycle δ : g → g ∧ (∧ k−1 θ).
Lemma 2.14. Identify Θ with the Lie subgroup {1 G } × Θ of G ⋉ Θ, where 1 G is the unit element of G. Then any multiplicative k-vector field V (k ≥ 1) is tangent to Θ, and therefore defines a multiplicative k-vector field V| Θ on Θ.

It is clear that
To prove the lemma, it suffices to prove that, for any function Here 1 G is the unit element of G, and L stands for the left translations with respect to the groupoid structure. The fact that ∂ gpd (f )| 1 G = 0 is due to Eq. (16) and Lemma 2.12. q.e.d.

Compatibility conditions.
This section is devoted to exploring the compatibility condition between the infinitesimal data ω and δ. The main theorem is the following: be a crossed module of Lie groups. A multiplicative k-vector field V on the Lie 2-group G ⋉ Θ associated to this crossed module determines a pair of linear maps ω : θ → ∧ k θ, which satisfy the following three properties: commutes; ID2: δ is a Lie algebra 1-cocycle valued in W k ; ID3: for all x ∈ g and u ∈ θ, where the bracket is taken in g ⋉ θ.
First of all, we prove that the k-differentials of V with respect to both the groupoid and the group structures can be expressed in terms of the infinitesimal data (ω, δ). Since A ∼ = G× θ, ∂ gpd is completely determined by two R-linear operators: ∂ gpd : C ∞ (G) → C ∞ (G, ∧ k−1 θ) and ∂ gpd : . The latter is determined by its value on constant functions due to the Leibniz rule.
Let ∂ gp : g ⋉ θ → ∧ k (g ⋉ θ) be the k-differential with respect to the group structure on G ⋉ Θ.
Proposition 2.16. The map ω : θ → ∧ k θ satisfies Proof. Every u ∈ θ ⊂ g ⋉ θ (considered as a constant section of the Lie algebroid A ∼ = G × θ) determines two vector fields on G ⋉ Θ: a vector field ← − u gp invariant under left translations relatively to the group structure and a vector field ← − u invariant under left translations relatively to the groupoid structure G ⋉ Θ ⇒ G. It is simple to see that Therefore, By definition, we have ω(u) = ∂ gp (u). The conclusion follows. q.e.d.
Proof of Theorem 2.15. According to Lemma 2.12, ID2 holds. It suffices to prove ID1 and ID3.
On the other hand, the k-differential ∂ gp satisfies where the brackets stand for the Lie algebra bracket on g ⋉ θ. Applying Proposition 2.17, and comparing the ∧ k θ-terms of both sides of the equation above, ID3 follows immediately. q.e.d.
Proposition 2.18. The map ω : θ → ∧ k θ is a Lie algebra 1-cocycle, Proof. Using ID1 and ID3 from Theorem 2.15, we have Here, in the last equality, we have used the identity which can be verified by a straightforward computation. q.e.d.
Now we extend the two maps ω and δ to degree-(k−1) derivations (which we denote by the same symbols by abuse of notation) on the exterior algebra ∧ • (g ⋉ θ) by setting ω(g) = 0 and δ(θ) = 0.
Corollary 2.20. When endowed with the bracket defined by Eqs. (24) and (25), the direct sum k≥0 A k is a graded Lie algebra. Theorem 2.21. Let G ⋉ Θ be a Lie 2-group, where both G and Θ are connected and simply connected. Given any (ω, δ) ∈ A k , there exists a unique multiplicative k-vector field V on G ⋉ Θ, whose infinitesimal corresponds to (ω, δ).
An immediate consequence is the following main result of the paper: Theorem 2.22 (universal lifting theorem). Let G ⋉ Θ be a Lie 2-group, where both G and Θ are connected and simply connected Lie groups with Lie algebras g and θ, respectively. There is a canonical isomorphism of graded Lie algebras 2.5.2. From infinitesimal data to k-differentials of the Lie algebra g ⋉ θ. The k = 0 case is obvious, so we will assume k ≥ 1 below. We will divide the proof of Theorem 2.21 into several steps.
First, since δ is a Lie algebra 1-cocycle, it integrates into a Lie group 1-cocycle Clearly the map δ takes values in W k .
Let ω : Θ → ∧ k θ be the group 1-cocycle integrating ω. As a direct consequence of property ID1 from Theorem 2.15, we have Define a linear map ∂ : Proposition 2.23. The operator ∂ defines a Lie algebra k-differential for the Lie algebra g ⋉ θ.
Proof. It suffices to prove that ∂ is a Lie algebra 1-cocycle for the Lie algebra g ⋉ θ. In fact, Proposition 2.18 implies that On the other hand, it follows from a direct verification that .
As a consequence, applying the operator 1−e −D φ Using the definitions of δ and D φ , we obtain the following identity: .
), u contains only the two terms and thus reduces to To prove that it vanishes, we need a couple of lemmas.
Lemma 2.24. For any v ∈ ∧ k−1 θ and 1 ≤ l ≤ k − 1, we have Proof. This follows from a straightforward computation, which is left to the reader. q.e.d.
From Lemma 2.5, it follows that, for any w ∈ W k and j ≥ 1, we have Now we return to the proof of Proposition 2.23. It remains to prove that (29) vanishes. Indeed, for any ζ ∈ g * , we have Here in the last two steps, we have used Eq. (30) and Lemma 2.24. This concludes the proof of the proposition. q.e.d.

2.5.3.
Multiplicative with respect to the groupoid structure. As a consequence of Proposition 2.23, we obtain a k-vector field Σ on G ⋉ Θ, which is multiplicative with respect to the group structure and whose induced k-differential with respect to the group structure on G ⋉ Θ is ∂. Now we need to prove that Σ is also multiplicative with respect to the groupoid structure on G ⋉ Θ ⇒ G. For this purpose, we need an explicit expression of Σ. Since Σ is multiplicative with respect to the group structure on G ⋉ Θ, it suffices to find an explicit expression of Σ along the subgroups {1 G } × Θ and G × {1 Θ }, respectively. The next two lemmas are devoted to this investigation.
The following lemma is immediate.
Proposition 2.27. The k-vector field Σ is also multiplicative with respect to the groupoid structure on G ⋉ Θ ⇒ G.
Proof. We divide the proof into three steps.
(1) The base manifold G is coisotropic with respect to Σ. For every g ∈ G, we have T (g,1 Θ ) (G ⋉ Θ) ∼ = T g G ⊕ θ. The conormal space of T g G can thus be canonically identified with θ * . It follows that G is coisotropic with respect to Σ since Σ| g does not contain any (∧ k θ)components according to Lemma 2.26.
Since ω| α takes values in ∧ k θ, we have Let m be the point t(g, α) = gΦ(α) of G. Choose a ζ ∈ g * and set ξ| m = L * m −1 ζ ∈ T * m G. We have, for all u ∈ θ, which follows from the identity Also note that, for all V ∈ T (g,1 Θ ) (G ⋉ Θ), To prove this identity, we observe that which implies that Thus Eq. (34) follows from a straightforward verification.
Applying Eq. (33), we obtain Using Eq. (34) and Lemma 2.26), we have where, in the last step, we used the fact that δ is a Lie group 1-cocycle. In particular, we have and therefore This proves that ι t * (ξ) Σ is indeed left-invariant with respect to the groupoid structure.
is left-invariant with respect to the groupoid structure. It suffices to consider X = f u, where f ∈ C ∞ (G) and u ∈ θ being considered as a constant section of A ∼ = G × θ. Then, which is clearly left-invariant according to Claim (2). Finally, Claims (1), (2), and (3) imply that Σ is indeed multiplicative by Lemma 2.3. q.e.d.
Proof of Theorem 2.21. From the infinitesimal data (ω, δ), we have constructed a multiplicative k-vector field Σ on the 2-group G ⋉ Θ. Assume that the infinitesimal data corresponding to Σ is (ω ′ , δ ′ ). Proposition 2.17 implies that ω ′ and δ ′ can be recovered from ∂ gp , the kdifferential of Σ with respect to the group structure on G ⋉ Θ, by the following relations: Since ∂ is defined by Eq. (28) and Σ integrates ∂, ∂ gp must coincide with ∂. Hence it follows that ω ′ = ω and δ ′ = δ.
Since both G and Θ are connected and simply connected, so must be G ⋉ Θ. Hence the multiplicative vector field Σ that integrates ∂ must be unique. q.e.d.

Quasi-Poisson Lie 2-groups
Throughout this section, (Θ Φ → G) denotes a Lie group crossed module, and G ⋉ Θ its associated Lie 2-group. By (θ φ → g), we denote its corresponding Lie algebra crossed module, and by g ⋉ θ the semidirect product Lie algebra.
3.1. Quasi-Poisson Lie 2-groups. and Here η is considered as a section in Γ Γ Γ(∧ 3 A). When η is zero, Π defines a Poisson structure on G ⋉ Θ. In this case, we say that (G ⋉ Θ, Π) is a Poisson 2-group.
The main result of this section is the following: Theorem 3.2. Any quasi-Poisson Lie 2-group (Π, η) on G ⋉ Θ naturally induces a quasi-Lie 2-bialgebra. Conversely, given a quasi-Lie 2-bialgebra (θ, g, t) as in Definition 1.3, if both G and Θ are connected and simply-connected Lie groups with Lie algebras θ and g, respectively, then G ⋉ Θ admits a quasi-Poisson Lie 2group structure whose infinitesimal is isomorphic to the given quasi-Lie 2-bialgebra.
The proof is deferred to Section 3.3. In fact, from its proof, it is clear that exactly the same conclusion holds between Poisson Lie 2-groups and Lie 2-bialgebras. Thus, as an immediate consequence, we obtain the following analogue of a classical theorem of Drinfeld in the context of 2-groups. 1) There is a one-to-one correspondence between connected and simply connected quasi-Poisson Lie 2-groups and quasi-Lie 2-bialgebras.
2) There is a one-to-one correspondence between connected and simplyconnected Poisson Lie 2-groups and Lie 2-bialgebras.

3.2.
Multiplicative k-vector fields generated by group 1-cocycles. Lemma 3.4. For any u ∈ θ, we have Here id θ denotes the identity map on θ.
1) The l-vector field Here λ is considered as a section in Γ Γ Γ(∧ l A), and ← − λ and − → λ , respectively, denote the left-and right-invariant l-vector fields on the groupoid G ⋉ Θ ⇒ G.
Proof. 1) It is clear that C λ is multiplicative with respect to the groupoid structure. It suffices to show that C λ is also multiplicative with respect to the group structure on G ⋉ Θ. Define c : G ⋉ Θ → ∧ l (g ⋉ θ) by , where L stands for the group left translations. It is well known that C λ is multiplicative with respect to the group structure if and only if c is a group 1-cocycle, i.e. ,α) ).
3) We first prove the following formula: where δ : G → g ∧ (∧ k−1 θ) is the Lie group 1-cocycle corresponding to δ as in Eq. (26). To prove it, assume that where f i , h j ∈ C ∞ (G), u i ∈ ∧ l θ, x j ∈ g, and w j ∈ ∧ k−1 θ. According to Eq. (16) and Proposition 2.16, we have Here in the second from the last equality, we used the identity λ| g exp tx j = λ| exp tx j + (exp tx j ) −1 * λ| g .

Proof of the main theorem.
The following result describes the infinitesimal data of a quasi-Poisson structure on the 2-group G ⋉ Θ.
This concludes the proof. q.e.d.
Proof. The proof is standard, and is left to the reader. q.e.d.
Example 3.12. Let θ = gl(2) ∼ = R id ⊕ sl(2) and g = sl (2). Then the projection φ : gl(2) → sl(2) is a Lie algebra crossed module. It is easy to check that any r ∈ ∧ 2 θ is indeed an r-matrix. Example 3.13. Let g be a Lie algebra and θ ⊆ g an ideal. Consider the Lie algebra crossed module ι : θ → g, where ι is the inclusion. Assume that r ∈ ∧ 2 θ such that [r, r] ∈ ∧ 3 θ is g-invariant. Then r is clearly an r-matrix. For example, we take g = gl(2) and θ = sl (2). Then any bivector in ∧ 2 θ is indeed an r-matrix.