Cohomology of skew-holomorphic lie algebroids

We introduce the notion of a skew-holomorphic Lie algebroid on a complex manifold and explore some cohomology theories that can be associated with it. We present examples and applications of this notion in terms of different types of holomorphic Poisson structures.     

A Poisson manifold of strong compact type

We construct a corank one Poisson manifold which is of strong compact type, i.e., the associated Lie algebroid structure on its cotangent bundle is integrable, and the source 1-connected (symplectic) integration is compact. The construction relies on the geometry of the moduli space of marked K3 surfaces.     

Rigidity of infinitesimal momentum maps

In this paper we prove rigidity theorems for Poisson Lie group actions on Poisson manifolds. In particular, we prove that close infinitesimal momentum maps associated to Poisson Lie group actions are equivalent using a normal form theorem for SCI spaces. When the Poisson structure of the acted manifold is integrable, this yields rigidity also for lifted actions to the symplectic groupoid.     

Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras

In a previous paper (C. R. Acad. Sci. Paris Sér. I 333 (2001) 763–768), the author introduced a notion of compatibility between a Poisson structure and a pseudo-Riemannian metric. In this paper, we introduce a new class of Lie algebras called pseudo-Riemannian Lie algebras. The two notions are closely related: we prove that the dual of a Lie algebra endowed with its canonical linear Poisson structure carries a compatible pseudo-Riemannian metric if and only if the Lie algebra is a pseudo-Riemannian Lie algebra. Moreover, the Lie algebra obtained by linearizing at a point a Poisson manifold with a compatible pseudo-Riemannian metric is a pseudo-Riemannian Lie algebra. We also give some properties of the symplectic leaves of such manifolds, and we prove that every Poisson manifold with a compatible Riemannian metric is unimodular. Finally, we study Poisson Lie groups endowed with a compatible pseudo-Riemannian metric, and we give the classification of all pseudo-Riemannian Lie algebras of dimension 2 and 3.     

On Filippov algebroids and multiplicative Nambu–Poisson structures

We discuss relations of linear Nambu–Poisson structures to Filippov algebras and define a Filippov algebroid—a generalization of a Lie algebroid. We also prove results describing multiplicative Nambu–Poisson structures on Lie groups. In particular, it is shown that simple Lie groups do not admit multiplicative Nambu–Poisson structures of order >2.     

Hopf cyclic cohomology and Hodge theory for proper actions on complex manifolds

We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem.     

Modular classes of skew algebroid relations

Skew algebroid is a natural generalization of the concept of Lie algebroid. In this paper, for a skew algebroid E, its modular class mod(E) is defined in the classical as well as in the supergeometric formulation. It is proved that there is a homogeneous nowhere-vanishing 1-density on E ^* which is invariant with respect to all Hamiltonian vector fields if and only if E is modular, i.e., mod(E)?=?0. Further, the relative modular class of a subalgebroid is introduced and studied together with its application to holonomy, as well as the modular class of a skew algebroid relation. These notions provide, in particular, a unified approach to the concepts of a modular class of a Lie algebroid morphism and of a Poisson map.     

Connections on the Total Space of a Holomorphic Lie Algebroid

The main purpose of the paper is the study of the total space of a holomorphic Lie algebroid E. The paper is structured in three parts. In the first section, we briefly introduce basic notions on holomorphic Lie algebroids. The local expressions are written and the complexified holomorphic bundle is introduced. The second section presents two approaches on the study of the geometry of the complex manifold E. The first part contains the study of the tangent bundle \(T_{\mathbb {C}}E=T'E\oplus T''E\) and its link, via the tangent anchor map, with the complexified tangent bundle \(T_{\mathbb {C}}(T'M)=T'(T'M)\oplus T''(T'M)\). A holomorphic Lie algebroid structure is emphasized on \(T'E\). A special study is made for integral curves of a spray on \(T'E\). Theorem 2.8 gives the coefficients of a spray, called canonical, obtained from a complex Lagrangian on \(T'E\). In the second part of section two, we study the holomorphic prolongation \(\mathcal {T}'E\) of the Lie algebroid E. In the third section, we study how a complex Lagrange (Finsler) structure on \(T'M\) induces a Lagrangian structure on E. Three particular cases are analysed by the rank of the anchor map, the dimensions of manifold M, and those of the fibres. We obtain the correspondent on E of the Chern–Lagrange nonlinear connection from \(T'M\).     

Lie algebroid structures on double vector bundles and representation theory of Lie algebroids

A VB-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB-algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in special cases reproduce characteristic classes constructed by Crainic and Fernandes. We give a complete classification of regular VB-algebroids, and in the process we obtain another characteristic class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids.     

Bi-Poisson Structures and Integrability of Geodesic Flow on Homogeneous Spaces

Let G/K be a semisimple orbit of the adjoint representation of a real connected reductive Lie group G. Let K₁ be any closed subgroup of K containing the commutant of the identity component of K. We prove that the geodesic flow on the symplectic manifold T*(G/K₁), corresponding to a G-invariant pseudo-Riemannian metric on G/K₁ which is induced by a bi-invariant pseudo-Riemannian metric on G, is completely integrable in the class of real analytic functions, polynomial in momenta. To this end we study the Poisson geometry of the space of G-invariant functions on T*(G/K) using a one-parameter family of moment maps.     

Λ-modules and holomorphic Lie algebroid connections

Let X be a complex smooth projective variety, and G a locally free sheaf on X. We show that there is a one-to-one correspondence between pairs (Λ, Ξ), where Λ is a sheaf of almost polynomial filtered algebras over X satisfying Simpson’s axioms and \( \equiv :Gr\Lambda \to Sym \bullet _{\mathcal{O}_X } \mathcal{G}\) is an isomorphism, and pairs (L, Σ), where L is a holomorphic Lie algebroid structure on \(\mathcal{G}\) and Σ is a class in F ¹ H ²(L, ?), the first Hodge filtration piece of the second cohomology of L.As an application, we construct moduli spaces of semistable flat L-connections for any holomorphic Lie algebroid L. Particular examples of these are given by generalized holomorphic bundles for any generalized complex structure associated to a holomorphic Poisson manifold.     

A subelliptic Taylor isomorphism on infinite-dimensional Heisenberg groups

Let G denote an infinite-dimensional Heisenberg-like group, which is a class of infinite-dimensional step 2 stratified Lie groups. We consider holomorphic functions on G that are square integrable with respect to a heat kernel measure which is formally subelliptic, in the sense that all appropriate finite-dimensional projections are smooth measures. We prove a unitary equivalence between a subclass of these square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the “Cameron–Martin” Lie subalgebra. The isomorphism defining the equivalence is given as a composition of restriction and Taylor maps.     

Classification of four-dimensional real Lie bialgebras of symplectic type and their Poisson–Lie groups

We classify all four-dimensional real Lie bialgebras of symplectic type and obtain the classical r-matrices for these Lie bialgebras and Poisson structures on all the associated four-dimensional Poisson–Lie groups. We obtain some new integrable models where a Poisson–Lie group plays the role of the phase space and its dual Lie group plays the role of the symmetry group of the system.     

Nijenhuis structures on Courant algebroids

We study Nijenhuis structures on Courant algebroids in terms of the canonical Poisson bracket on their symplectic realizations. We prove that the Nijenhuis torsion of a skew-symmetric endomorphism N of a Courant algebroid is skewsymmetric if N ² is proportional to the identity, and only in this case when the Courant algebroid is irreducible. We derive a necessary and sufficient condition for a skewsymmetric endomorphism to give rise to a deformed Courant structure. In the case of the double of a Lie bialgebroid (A, A*), given an endomorphism N of A that defines a skew-symmetric endomorphism N of the double of A, we prove that the torsion ofN is the sum of the torsion of N and that of the transpose of N.     

Dirac structures on protobialgebroids

Protobialgebroids include several kinds of algebroid structures such as Lie algebroid, Lie bialgebroid, Lie quasi-bialgebroid, etc. In this paper, the Dirac theories are generalized from Lie bialgebroid to protobialgebroid. We give the integrable conditions for a maximally isotropic subbundle being a Dirac structure for a protobialgebroid by the notion of a characteristic pair. From the integrable conditions, we found out that the Dirac structure has closed relations with the twisting of a protobialgebroid. At last, some special cases of the Dirac structures for protobialgebroids are discussed.     

Dirac submanifolds and Poisson involutions

Dirac submanifolds are a natural generalization in the Poisson category of the symplectic submanifolds of a symplectic manifold. They correspond to symplectic subgroupoids of the symplectic groupoid of the given Poisson manifold. In particular, Dirac submanifolds arise as the stable loci of Poisson involutions. In this paper, we make a general study of these submanifolds including both local and global aspects.In the second part of the paper, we study Poisson involutions and the induced Poisson structures on their stable loci. In particular, we discuss the Poisson involutions on a special class of Poisson groups, and more generally Poisson groupoids, called symmetric Poisson groups, and symmetric Poisson groupoids. Many well-known examples, including the standard Poisson group structures on semi-simple Lie groups, Bruhat Poisson structures on compact semi-simple Lie groups, and Poisson groupoid structures arising from dynamical r-matrices of semi-simple Lie algebras are symmetric, so they admit a Poisson involution. For symmetric Poisson groups, the relation between the stable locus Poisson structure and Poisson symmetric spaces is discussed. As a consequence, we prove that the Dubrovin Poisson structure on the space of Stokes matrices U₊ (appearing in Dubrovin's theory of Frobenius manifolds) is a Poisson symmetric space.     

The Euler-Poincaré-Hopf theorem for flat connections in some transitive Lie algebroids

This paper is a continuation of [19], [21], [22]. We study flat connections with isolated singularities in some transitive Lie algebroids for which either ℝ or sl(2, ℝ) or so(3) are isotropy Lie algebras. Under the assumption that the dimension of the isotropy Lie algebra is equal to n + 1, where n is the dimension of the base manifold, we assign to any such isolated singularity a real number called an index. For ℝ-Lie algebroids, this index cannot be an integer. We prove the index theorem (the Euler-Poincaré-Hopf theorem for flat connections) saying that the index sum is independent of the choice of a connection. Multiplying this index sum by the orientation class of M, we get the Euler class of this Lie algebroid. Some integral formulae for indices are given.     

Anti-Kählerian manifolds

An anti-Kählerian manifold is a complex manifold with an anti-Hermitian metric and a parallel almost complex structure. It is shown that a metric on such a manifold must be the real part of a holomorphic metric. It is proved that all odd Chern numbers of an anti-Kählerian manifold vanish and that complex parallelisable manifolds (in particular the factor space G/D of a complex Lie group G over the discrete subgroup D ) are anti-Kählerian manifolds. A method of generating new solutions of Einstein equations by using the theory of anti-Kählerian manifolds is presented.     

The localized longitudinal index theorem for Lie groupoids and the van Est map

We define the “localized index” of longitudinal elliptic operators on Lie groupoids associated with Lie algebroid cohomology classes. We derive a topological expression for these numbers using the algebraic index theorem for Poisson manifolds on the dual of the Lie algebroid. Underlying the definition and computation of the localized index, is an action of the Hopf algebroid of jets around the unit space, and the characteristic map it induces on Lie algebroid cohomology. This map can be globalized to differentiable groupoid cohomology, giving a definition of the “global index”, that can be computed by localization. This correspondence between the “global” and “localized” index is given by the van Est map for Lie groupoids.     

Lie algebroid fibrations

A degree 1 non-negative graded super manifold equipped with a degree 1 vector field Q satisfying [Q,Q]=1, namely a so-called NQ-1 manifold is, in plain differential geometry language, a Lie algebroid. We introduce a notion of fibration for such super manifolds, that essentially involves a complete Ehresmann connection. As it is the case for Lie algebras, such fibrations turn out not to be just locally trivial products. We also define homotopy groups and prove the expected long exact sequence associated to a fibration. In particular, Crainic and Fernandes's obstruction to the integrability of Lie algebroids is interpreted as the image of a transgression map in this long exact sequence.