Camille Laurent-Gengoux

Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids

We study holomorphic Poisson manifolds and holomorphic Lie algebroids from the viewpoint of real Poisson geometry. We give a characterization of holomorphic Poisson structures in terms of the Poisson Nijenhuis structures of Magri-Morosi and describe a double complex that computes the holomorphic Poisson cohomology. A holomorphic Lie algebroid structure on a vector bundle A → X is shown to be equivalent to a matched pair of complex Lie algebroids (T 0,1 X, A 1,0 ), in the sense of Lu. The holomorphic Lie algebroid cohomology of A is isomorphic to the cohomology of the elliptic Lie algebroid T 0,1 X ⋈ A 1,0 . In the case when (X,π) is a holomorphic Poisson manifold and A = (T*X) π , such an elliptic Lie algebroid coincides with the Dirac structure corresponding to the associated generalized complex structure of the holomorphic Poisson manifold.

Universal lifting theorem and quasi-Poisson groupoids

We prove the universal lifting theorem: for an α-simply connected and α-connected Lie groupoid with Lie algebroid A, the graded Lie algebra of multi-differentials on A is isomorphic to that of multiplicative multi-vector fields on . As a consequence, we obtain the integration theorem for a quasi-Lie bialgebroid, which generalizes various integration theorems in the literature in special cases. The second goal of the paper is the study of basic properties of quasi-Poisson groupoids. In particular, we prove that a group pair (D, G) associated to a Manin quasi-triple (d, g, h) induces a quasi-Poisson groupoid on the transformation groupoid G ×D/G ⇉ D/G. Its momentum map corresponds exactly with the D/G-momentum map of Alekseev and Kosmann-Schwarzbach.

Non-abelian differentiable gerbes

Abstract We study non-abelian differentiable gerbes over stacks using the theory of Lie groupoids. More precisely, we develop the theory of connections on Lie groupoid G-extensions, which we call “connections on gerbes”, and study the induced connections on various associated bundles. We also prove analogues of the Bianchi identities. In particular, we develop a cohomology theory which measures the existence of connections and curvings for G-gerbes over stacks. We also introduce G-central extensions of groupoids, generalizing the standard groupoid S 1 -central extensions. As an example, we apply our theory to study the differential geometry of G-gerbes over a manifold.

Quantization of pre-quasi-symplectic groupoids and their Hamiltonian spaces

We study the prequantization of pre-quasi-symplectic groupoids and their Hamiltonian spaces using S 1-gerbes. We give a geometric description of the integrality condition. As an application, we study the prequantization of the quasi-Hamiltonian G-spaces of Alekseev-Malkin-Meinrenken.

Hom-Lie algebroids

Abstract We define hom-Lie algebroids, a definition that may seem cumbersome at first, but which is justified, first, by a one-to-one correspondence with hom-Gerstenhaber algebras, a notion that we also introduce, and several examples, including hom-Poisson structures.

Integration of holomorphic Lie algebroids

We prove that a holomorphic Lie algebroid is integrable if and only if its underlying real Lie algebroid is integrable. Thus the integrability criteria of Crainic–Fernandes (Theorem 4.1 in Crainic, Fernandes in Ann Math 2:157, 2003) do also apply in the holomorphic context without any modification. As a consequence we prove that a holomorphic Poisson manifold is integrable if and only if its real part or imaginary part is integrable as a real Poisson manifold.

Poisson–Lie Groups

In this chapter we discuss Poisson–Lie groups and their infinitesimal counterparts Lie bialgebras. A Poisson–Lie group is a Lie group G which is equipped with a Poisson structure π, having the property that the product map on G is a morphism of Poisson manifolds. A Lie bialgebra is a Lie algebra which is at the same time a Lie coalgebra, the algebra and coalgebra structures satisfying some compatibility relation. We show that there is a functor which associates to every Poisson–Lie group a Lie bialgebra and that every finite-dimensional Lie bialgebra is the Lie bialgebra of some Poisson–Lie group, which can be chosen to be connected and simply connected. Using dressing actions, we shortly discuss the symplectic leaves of Poisson–Lie groups.

Exponential map and L∞ algebra associated to a Lie pair

Abstract In this Note, we unveil homotopy-rich algebraic structures generated by the Atiyah classes relative to a Lie pair ( L , A ) of algebroids. In particular, we prove that the quotient L / A of such a pair admits an essentially canonical homotopy module structure over the Lie algebroid A, which we call Kapranov module.

Lectures on Poisson groupoids

In these lecture notes, we give a quick account of the theory of Poisson groupoids and Lie bialgebroids. In particular, we discuss the universal lifting theorem and its applications including integration of quasi-Lie bialgebroids, integration of Poisson Nijenhuis structures and Alekseev and Kosmann-Schwarzbach’s theory of D=G ‐ momentum maps. 53D17

Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles

Let K be a Lie group and P be a K-principal bundle on a manifold M. Suppose given furthermore a central extension