Lie Algebroids, Holonomy and Characteristic Classes

We extend the notion of connection in order to study singular geometric structures, namely, we consider a notion of connection on a Lie algebroid which is a natural extension of the usual concept of a covariant connection. It allows us to define holonomy of the orbit foliation of a Lie algebroid and prove a Stability Theorem. We also introduce secondary or exotic characteristic classes, thus providing invariants which generalize the modular class of a Lie algebroid.     

Equivariant cohomology and localization for Lie algebroids

Let M be a manifold carrying the action of a Lie group G, and let A be a Lie algebroid on M equipped with a compatible infinitesimal G-action. Using these data, we construct an equivariant cohomology of A and prove a related localization formula for the case of compact G. By way of application, we prove an analog of the Bott formula.     

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.     

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.     

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.     

Distinguished Connections on Finsler Algebroids

Considering the prolongation of a Lie algebroid,the authors introduce Finsler algebroids and present important geometric objects on these spaces.Important endomorphisms like conservative and Barthel,Cartan tensor and some distinguished connections like Berwald,Cartan,Chern-Rund and Hashiguchi are introduced and studied.     

Recurrence relations and convergence theory of the generalized polar decomposition on Lie groups

The subject matter of this paper is the analysis of some issues related to generalized polar decompositions on Lie groups. This decomposition, depending on an involutive automorphism , is equivalent to a factorization of , being a Lie group, as with and , and was recently discussed by Munthe-Kaas, Quispel and Zanna together with its many applications to numerical analysis. It turns out that, contrary to , an analysis of is a very complicated task. In this paper we derive the series expansion for , obtaining an explicit recurrence relation that completely defines the function in terms of projections on a Lie triple system and a subalgebra of the Lie algebra , and obtain bounds on its region of analyticity. The results presented in this paper have direct application, among others, to linear algebra, integration of differential equations and approximation of the exponential.

     

Vertex operator algebras associated to modified regular representations of affine Lie algebras

Let G be a simply-connected complex Lie group with simple Lie algebra g and let be its affine Lie algebra. We use intertwining operators and Knizhnik-Zamolodchikov equations to construct a family of N-graded vertex operator algebras (VOAs) associated to g. These vertex operator algebras contain the algebra of regular functions on G as the conformal weight 0 subspaces and are -modules of dual levels in the sense that , where h^∨ is the dual Coxeter number of g. This family of VOAs was previously studied by Arkhipov-Gaitsgory and Gorbounov-Malikov-Schechtman from different points of view. We show that when k is irrational, the vertex envelope of the vertex algebroid associated to G and the level k is isomorphic to the vertex operator algebra we constructed above. The case of rational levels is also discussed.     

Manifold of nondegenerate affinor fields

We consider the quotient set of the set of nondegenerate affinor fields with respect to the action of the group of nowhere vanishing functions. This set is endowed with a structure of infinite-dimensional Lie group. On this Lie group, we construct an object of linear connection with respect to which all left-invariant vector fields are covariantly constant (the Cartan connection).     

Primeness of the enveloping algebra of a Cartan type Lie superalgebra

We show that a primeness criterion for enveloping algebras of Lie superalgebras discovered by Bell is applicable to the Cartan type Lie superalgebras , even. Other algebras are considered but there are no definitive answers in these cases.

     

On the nonrationality of rigid Lie algebras

In his thesis, Carles made the following conjecture: Every rigid Lie algebra is defined on the field . This was quite an interesting question because a positive answer would give a nice explanation of the fact that simple Lie algebras are defined over . The goal of this note is to provide a large number of examples of rigid but nonrational and nonreal Lie algebras.

     

Left‐symmetric algebroids

In this paper, we introduce the notion of a left‐symmetric algebroid, which is a generalization of a left‐symmetric algebra from a vector space to a vector bundle. The left multiplication gives rise to a representation of the corresponding sub‐adjacent Lie algebroid. We construct left‐symmetric algebroids from ‐operators on Lie algebroids. We study phase spaces of Lie algebroids in terms of left‐symmetric algebroids. Representations of left‐symmetric algebroids are studied in detail. At last, we study deformations of left‐symmetric algebroids, which could be controlled by the second cohomology class in the deformation cohomology.     

Extensions of Modules over Hopf Algebras Arising from Lie Algebras of Cartan Type

In this paper we prove that there are no self-extensions of simple modules over restricted Lie algebras of Cartan type. The proof given by Andersen for classical Lie algebras not only uses the representation theory of the Lie algebra, but also representations of the corresponding reductive algebraic group. The proof presented in the paper follows in the same spirit by using the construction of a infinite-dimensional Hopf algebra D(G) u( ) containing u( ) as a normal Hopf subalgebra, and the representation theory of this algebra developed in our previous work. Finite-dimensional hyperalgebra analogs D(G _r ) u( ) have also been constructed, and the results are stated in this setting.     

On the length of the spectral sequence of a Lie algebra extension

The length of the spectral sequence of a Lie algebra extension is at most the dimension of the quotient algebra. We show that this bound can be attained for arbitrarily large quotient algebras even when the algebra is nilpotent and the extension splits.