The localization of 1-cohomology of transitive Lie algebroids

For a transitive Lie algebroid A on a connected manifold M and its representation on a vector bundle F, we define a morphism of cohomology groups rk: Hk(A,F) → Hk(Lx,Fx), called the localization map, where Lx is the adjoint algebra at x ∈ M. The main result in this paper is that if M is simply connected, or H (LX,FX) is trivial, then T is injective. This means that the Lie algebroid 1-cohomology is totally determined by the 1-cohomology of its adjoint Lie algebra in the above two cases.     

Bott's vanishing theorem for regular Lie algebroids

Differential geometry has discovered many objects which determine Lie algebroids playing a role analogous to that of Lie algebras for Lie groups. For example:

--- differential groupoids,

--- principal bundles,

--- vector bundles,

--- actions of Lie groups on manifolds,

--- transversally complete foliations,

--- nonclosed Lie subgroups,

--- Poisson manifolds,

--- some complete closed pseudogroups.

We carry over the idea of Bott's Vanishing Theorem to regular Lie algebroids (using the Chern-Weil homomorphism of transitive Lie algebroids investigated by the author) and, next, apply it to new situations which are not described by the classical version, for example, to the theory of transversally complete foliations and nonclosed Lie subgroups in order to obtain some topological obstructions for the existence of involutive distributions and Lie subalgebras of some types (respectively).

     

Nondegenerate cohomology pairing for transitive Lie algebroids,characterization

The Evens-Lu-Weinstein representation (Q _A , D) for a Lie algebroid A on a manifold M is studied in the transitive case. To consider at the same time non-oriented manifolds as well, this representation is slightly modified to (Q _A ^or , D^or) by tensoring by orientation flat line bundle, Q _A ^or =Q_A⊗or (M) and D ^or=D⊗∂ _A ^or . It is shown that the induced cohomology pairing is nondegenerate and that the representation (Q _A ^or , D^or) is the unique (up to isomorphy) line representation for which the top group of compactly supported cohomology is nontrivial. In the case of trivial Lie algebroid A=TM the theorem reduce to the following: the orientation flat bundle (or (M), ∂ _A ^or ) is the unique (up to isomorphy) flat line bundle (ξ, ∇) for which the twisted de Rham complex of compactly supported differential forms on M with values in ξ possesses the nontrivial cohomology group in the top dimension. Finally it is obtained the characterization of transitive Lie algebroids for which the Lie algebroid cohomology with trivial coefficients (or with coefficients in the orientation flat line bundle) gives Poincaré duality. In proofs of these theorems for Lie algebroids it is used the Hochschild-Serre spectral sequence and it is shown the general fact concerning pairings between graded filtered differential ℝ-vector spaces: assuming that the second terms live in the finite rectangular, nondegeneration of the pairing for the second terms (which can be infinite dimensional) implies the same for cohomology spaces.     

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.     

Equivariant Novikov Inequalities

We establish an equivariant generalization of the Novikov inequalities which allows us to estimate the topology of the set of critical points of a closed basic invariant form by means of twisted equivariant cohomology of the manifold. We apply these inequalities to study cohomology of the fixed points set of a symplectic torus action. We show that in this case our inequalities are perfect, i.e. they are in fact equalities.     

Nonabelian cohomology with coefficients in Lie groups

In this paper we prove some properties of the nonabelian cohomology of a group with coefficients in a connected Lie group . When is finite, we show that for every -submodule of which is a maximal compact subgroup of , the canonical map is bijective. In this case we also show that is always finite. When and is compact, we show that for every maximal torus of the identity component of the group of invariants , is surjective if and only if the -action on is -semisimple, which is also equivalent to the fact that all fibers of are finite. When , we show that is always surjective, where is a maximal compact torus of the identity component of . When is cyclic, we also interpret some properties of in terms of twisted conjugate actions of .

     

Nijenhuis forms on Lie-infinity algebras associated to Lie algebroids

Introducing Nijenhuis forms on L_∞-algebras gives a general frame to understand deformations of the latter. We give here a Nijenhuis interpretation of a deformation of an arbitrary Lie algebroid into an L_∞-algebra. Then we show that Nijenhuis forms on L_∞-algebras also give a short and e?cient manner to understand Poisson-Nijenhuis structures and, more generally, the so-called exact Poisson quasi-Nijenhuis structures with background.     

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.     

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 of K. It is a classical question whether there exists a -principal bundle on M such that . Neeb (Commun. Algebra 34:991–1041, 2006) defines in this context a crossed module of topological Lie algebras whose cohomology class is an obstruction to the existence of . In the present article, we show that is up to torsion a full obstruction for this problem, and we clarify its relation to crossed modules of Lie algebroids and Lie groupoids, and finally to gerbes.      

Twisted Weyl groups of Lie groups and nonabelian cohomology

For a cyclic group A and a connected Lie group G with an A-module structure (with the additional assumptions that G is compact and the A-module structure on G is 1-semisimple if ), we define the twisted Weyl group W = W(G,A,T), which acts on T and H ¹(A,T), where T is a maximal compact torus of , the identity component of the group of invariants G ^A . We then prove that the natural map is a bijection, reducing the calculation of H ¹(A,G) to the calculation of the action of W on T. We also prove some properties of the twisted Weyl group W, one of which is that W is a finite group. A new proof of a known result concerning the ranks of groups of invariants with respect to automorphisms of a compact Lie group is also given.      

Equivariant crystalline cohomology and base change

Given a perfect field of characteristic , a smooth proper -scheme , a crystal on relative to and a finite group acting on and , we show that, viewed as a virtual -module, the reduction modulo of the crystalline cohomology of is the de Rham cohomology of modulo . On the way we prove a base change theorem for the virtual -representations associated with -equivariant objects in the derived category of -modules.

     

Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology

We associate to each infinite primitive Lie pseudogroup a Hopf algebra of ‘transverse symmetries,’ by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a ‘quantum group’ counterpart of the infinite-dimensional primitive Lie algebra of the pseudogroup. It is first constructed via its action on the étale groupoid associated to the pseudogroup, and then realized as a bicrossed product of a universal enveloping algebra by a Hopf algebra of regular functions on a formal group. The bicrossed product structure allows to express its Hopf cyclic cohomology in terms of a bicocyclic bicomplex analogous to the Chevalley-Eilenberg complex. As an application, we compute the relative Hopf cyclic cohomology modulo the linear isotropy for the Hopf algebra of the general pseudogroup, and find explicit cocycle representatives for the universal Chern classes in Hopf cyclic cohomology. As another application, we determine all Hopf cyclic cohomology groups for the Hopf algebra associated to the pseudogroup of local diffeomorphisms of the line.     

Equivariant cohomology distinguishes toric manifolds

The equivariant cohomology of a space with a group action is not only a ring but also an algebra over the cohomology ring of the classifying space of the acting group. We prove that toric manifolds (i.e. compact smooth toric varieties) are isomorphic as varieties if and only if their equivariant cohomology algebras are weakly isomorphic. We also prove that quasitoric manifolds, which can be thought of as a topological counterpart to toric manifolds, are equivariantly homeomorphic if and only if their equivariant cohomology algebras are isomorphic.     

Deformation cohomology of Lie algebroids and Morita equivalence

Let $A?M$ be a Lie algebroid. In this short note, we prove that a pull-back of A along a fibration with homologically m-connected fibers shares the same deformation cohomology of A up to degree m.     

Formality of equivariant intersection cohomology of algebraic varieties

We present a proof that the equivariant intersection cohomology of any complete algebraic variety acted by a connected algebraic group is a free module over .

     

Equivariant entire cyclic cohomology: I. Finite groups

We extend the framework of entire cyclic cohomology to the equivariant context.Supported in part by the Department of Energy under Grant DE-FG02-88ER25065.     

Detecting cohomology for Lie superalgebras

In this paper we use invariant theory to develop the notion of cohomological detection for Type I classical Lie superalgebras. In particular we show that the cohomology with coefficients in an arbitrary module can be detected on smaller subalgebras. These results are used later to affirmatively answer questions, which were originally posed in Boe et al. (2010) [5] and Bagci et al. (2008) [2], about realizing support varieties for Lie superalgebras via rank varieties constructed for the smaller detecting subalgebras.     

Weights in the cohomology of toric varieties

We describe the weight filtration in the cohomology of toric varieties. We present a role of the Frobenius automorphism in an elementary way. We prove that equivariant intersection homology of an arbitrary toric variety is pure. We obtain results concerning Koszul duality: nonequivariant intersection cohomology is equal to the cohomology of the Koszul complexIH _T ^* (X)⊗H*(T). We also describe the weight filtration inIH ^*(X). Supported by KBN 2P03A 00218 grant. I thank, Institute of Mathematics, Polish Academy of Science for hospitality.     

On the second cohomology of nilpotent orbits in exceptional Lie algebras

In [1], the second de Rham cohomology groups of nilpotent orbits in all the complex simple Lie algebras are described. In this paper we consider non-compact non-complex exceptional Lie algebras, and compute the dimensions of the second cohomology groups for most of the nilpotent orbits. For the rest of cases of nilpotent orbits, which are not covered in the above computations, we obtain upper bounds for the dimensions of the second cohomology groups.     

On the Lie algebroid of a derived self-intersection

Let i:X?Y$i:X?Y$ be a closed embedding of smooth algebraic varieties. Denote by N the normal bundle of X in Y  . The present paper contains two constructions of certain Lie structure on the shifted normal bundle N[−1]$N[-1]$ encoding the information of the formal neighborhood of X in Y. We also present a few applications of these Lie theoretic constructions in understanding the algebraic geometry of embeddings.