Algebraic identity for the Schouten tensor and bi-Hamiltonian systems

A (0,3)-tensor Tijk is introduced in an invariant form. Algebraic identities are derived that connect the Schouten (2,1)-tensor and tensor Tijk with the Nijenhuis tensor . Applications to the bi-Hamiltonian dynamical systems are presented.     

Algebraic identities for the Nijenhuis tensors

The general algebraic identities are discovered for the Nijenhuis and Haantjes tensors on an arbitrary manifold Mn. For n=3, the special algebraic identities involving the symmetric bilinear form H(u,v) are derived.     

Exact Gerstenhaber algebras and Lie bialgebroids

We show that to any Poisson manifold and, more generally, to any triangular Lie bialgebroid in the sense of Mackenzie and Xu, there correspond two differential Gerstenhaber algebras in duality, one of which is canonically equipped with an operator generating the graded Lie algebra bracket, i.e. with the structure of a Batalin-Vilkovisky algebra.     

On the connection theory of extensions of transitive Lie groupoids

Due to a result by Mackenzie, extensions of transitive Lie groupoids are equivalent to certain Lie groupoids which admit an action of a Lie group. This paper is a treatment of the equivariant connection theory and holonomy of such groupoids, and shows that such connections give rise to the transition data necessary for the classification of their respective Lie algebroids.     

Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces

In this paper we give global characterisations of Gevrey–Roumieu and Gevrey–Beurling spaces of ultradifferentiable functions on compact Lie groups in terms of the representation theory of the group and the spectrum of the Laplace–Beltrami operator. Furthermore, we characterise their duals, the spaces of corresponding ultradistributions. For the latter, the proof is based on first obtaining the characterisation of their α-duals in the sense of Köthe and the theory of sequence spaces. We also give the corresponding characterisations on compact homogeneous spaces.     

On the role of effective representations of Lie groupoids

In this paper, we undertake the study of the Tannaka duality construction for the ordinary representations of a proper Lie groupoid on vector bundles. We show that for each proper Lie groupoid G, the canonical homomorphism of G into the reconstructed groupoid T(G) is surjective, although — contrary to what happens in the case of groups — it may fail to be an isomorphism. We obtain necessary and sufficient conditions in order that G may be isomorphic to T(G) and, more generally, in order that T(G) may be a Lie groupoid. We show that if T(G) is a Lie groupoid, the canonical homomorphism G→T(G) is a submersion and the two groupoids have isomorphic categories of representations.     

Tannaka duality for proper Lie groupoids

By replacing the category of smooth vector bundles of finite rank over a manifold with the category of what we call smooth Euclidean fields, which is a proper enlargement of the former, and by considering smooth actions of Lie groupoids on smooth Euclidean fields, we are able to prove a Tannaka duality theorem for proper Lie groupoids. The notion of smooth Euclidean field we introduce here is the smooth, finite dimensional analogue of the usual notion of continuous Hilbert field.     

Orbifolds and groupoids

We define a 2-category structure (Pre-Orb) on the category of reduced complex orbifold atlases. We construct a 2-functor F from (Pre-Orb) to the 2-category (Grp) of proper étale effective groupoid objects over the complex manifolds. Both on (Pre-Orb) and (Grp) there are natural equivalence relations on objects: (a natural extension of) equivalence of orbifold atlases on (Pre-Orb) and Morita equivalence in (Grp). We prove that F induces a bijection between the equivalence classes of its source and target.     

The Koszul homomorphism for a pair of Lie algebras in the theory of exotic characteristic classes of Lie algebroids

We examine fundamental properties of the universal exotic characteristic homomorphism in the category of Lie algebroids, introduced by the authors in Balcerzak and Kubarski (2012) [4]. The properties under study include: (a) functorial properties with respect to arbitrary morphisms of Lie algebroids, (b) homotopy properties, (c) relationships with the Koszul homomorphism for a pair of isotropy Lie algebras, (d) conditions under which the universal exotic characteristic homomorphism is a monomorphism.     

On duality between étale groupoids and Hopf algebroids

For any étale Lie groupoid G over a smooth manifold M, the groupoid convolution algebra of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over we construct the associated spectral étale Lie groupoid over M such that is naturally isomorphic to G. Both these constructions are functorial, and is fully faithful left adjoint to . We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid of an étale Lie groupoid G.     

The geometry of the bundle of connections

A generalized symplectic structure on the bundle of connections of an arbitrary principal G-bundle is defined by means of a -valued differential 2-form on C(P), which is related to the generalized contact structure on . The Hamiltonian properties of are also analyzed. Received August 31, 1999; in final form January 4, 2000 / Published online February 5, 2001     

A k-tableau characterization of k-Schur functions

We study k-Schur functions characterized by k-tableaux, proving combinatorial properties such as a k-Pieri rule and a k-conjugation. This new approach relies on developing the theory of k-tableaux, and includes the introduction of a weight-permuting involution on these tableaux that generalizes the Bender-Knuth involution. This work lays the groundwork needed to prove that the set of k-Schur Littlewood-Richardson coefficients contains the 3-point Gromov-Witten invariants; structure constants for the quantum cohomology ring.     

Some remarks on the global structure of proper Lie groupoids in low codimensions

We observe that any connected proper Lie groupoid whose orbits have codimension at most two admits a globally effective representation, i.e. one whose kernel consists only of ineffective arrows, on a smooth vector bundle. As an application, we deduce that any such groupoid can up to Morita equivalence be presented as an extension, by some bundle of compact Lie groups, of some action groupoid G?X with G compact.