The coupling of the tangent bundle $TM$ with the Lie algebra bundle $L$ (K.Mackenzie,2005, Definition 7.2.2) plays the crucial role in the classification of the transitive Lie algebroids for Lie algebra bundle $L$ with fixed finite dimensional Lie algebra $\rg$ as a fiber of $L$. Here we give a necessary and sufficient condition for existence such a coupling. Namely we define a new topology on the group $\Aut(\rg)$ of all automorphisms of Lie algebra $\rg$ and show that tangent bundle $TM$ can be coupled with the Lie algebra bundle $L$ if and only if the Lie algebra bundle L admits a local trivial structure with structural group endowed with such new topology. The second version of the article is different from the first because of the improvement of the text by invaluable efforts of professor J.Stasheff, to whom we submit our sincere gratitude.
The talk was done at the International Conference "Analysis, Topology and Applications", Harbin, China, 23.08.2011. Transitive Lie algebroids have specific properties that allow to look at the transitive Lie algebroid as an element of the object of a homotopy functor. Roughly speaking each transitive Lie algebroids can be described as a vector bundle over the tangent bundle of the manifold which is endowed with additional structures. Therefore transitive Lie algebroids admits a construction of inverse image generated by a smooth mapping of smooth manifolds. The intention of my talk is to use a…
Transitive Lie algebroids have specific properties that allow to look at the transitive Lie algebroid as an element of the object of a homotopy functor. Roughly speaking each transitive Lie algebroids can be described as a vector bundle over the tangent bundle of the manifold which is endowed with additional structures. Therefore transitive Lie algebroids admits a construction of inverse image generated by a smooth mapping of smooth manifolds. Due to to K.Mackenzie (2005) the construction can be managed as a homotopy functor $\mathcal{TLA}_{\rg}$ from category of smooth manifolds to the transi…
The intention of this article is to make an attempt of classification of transitive Lie algebroids and on this basis to construct a classifying space. The realization of the intention allows to describe characteristic classes of transitive Lie algebroids form the point of view a natural transformation of functors similar to the classical abstract characteristic classes for vector bundles.
The technique developed in Hochschild's works "Lie algebra kernels and cohomology " and "Cohomology classes of finite dimensional kernels for Lie algebras " [2],[3] can be applied to the special case of transitive Lie algebroids. Our task is formulated to transfer the Hochschild construction to the case of transitive Lie algebroids and prove a similar theorem.
D.Sullivan (1977) (see also the book by H.Whitney "Geometric Integration Theory",1957) considered a new model for underlying cochain complex for classical cohomologies with rational coefficients for arbitrary simplicial spaces that gives the isomorphism with classical rational cohomologies. We apply the key ideas developed by K.MacKenzie (2005) and J.Kubarski (1991) to generalization of the D.Sullivan model for transitive Lie algebroids. The generalization is based on the existence of the inverse image of the transitive Lie algebroids and on the property of transitive Lie algebroids being triv…
The preprint is prepared as description of results that were obtained during joint scientific project No: 71NC /2015/VNCCCT on the VIASM (Vietnam Institute for Advanced Study in Mathematics) from 08.12.2015 to 06.02.2016. The problem was formulated how to calculate so called the Mackenzie obstruction for existing of transitive Lie algebroid for given coupling between a finite dimensional Lie algebra and the tangent bundle of a smooth manifold. It is proved that the Mackenzie obstruction for transitive Lie algebroids is trivial for the finite dimensional Lie algebra which is the direct sum of t…
In our previous paper (arXiv:1306.5449) we have given a sufficient and necessary condition when the coupling between Lie algebra bundle (LAB) and the tangent bundle exists in the sense of Mackenzie (\cite{Mck-2005}, Definition 7.2.2) for the theory of transitive Lie algebroids. Namely we have defined a new topology on the group $\Aut(\rg)$ of all automorphisms of the Lie algebra $\rg$, say $\Aut(\rg)^δ$, and show that tangent bundle $TM$ can be coupled with the Lie algebra bundle $L$ if and only if the Lie algebra bundle L admits a local trivial structure with structural group endowed with suc…
Let $M$ be a smooth manifold, smoothly triangulated by a simplicial complex $K$, and $\cA$ a transitive Lie algebroid on $M$. The Lie algebroid restriction of $\cA$ to a simplex $Δ$ of $K$ is denoted by $\cA^{!!}_Δ$. A piecewise smooth form of degree $p$ on $\cA$ is a family $ω=(ω_Δ)_{Δ\in K}$ such that $ω_Δ\in Ω^{p}(\cA^{!!}_Δ;Δ)$ for each $Δ\in K$, satisfying the compatibility condition concerning the restrictions of $ω_Δ$ to the faces of $Δ$, that is, if $Δ'$ is a face of $Δ$, the restriction of the form $ω_Δ$ to the simplex $Δ'$ coincides with the form $ω_{Δ'}$. The set $Ω^{\ast}(\cA;K)$ o…
Mishchenko's theorem states that piecewise smooth and Lie algebroid cohomology of a transitive Lie algebroid defined over a combinatorial manifold are isomorphic. In this paper, we describe two applications of that result. The first application consists in the relationship between piecewise de Rham cohomology of a Lie groupoid and piecewise smooth cohomology of its Lie algebroid. For the second application, we combine the classical result dealing with invariant cohomology in Lie algebroids with the Mishchenko's theorem.
Given a good homology theory E and a topological space X, the E-homology of X is not just an E_{*}-module but also a comodule over the Hopf algebroid (E_{*}, E_{*}E). We establish a framework for studying the homological algebra of comodules over a well-behaved Hopf algebroid (A, Gamma). That is, we construct the derived category Stable(Gamma) of (A, Gamma) as the homotopy category of a Quillen model structure on the category of unbounded chain complexes of Gamma-comodules. This derived category is obtained by inverting the homotopy isomorphisms, NOT the homology isomorphisms. We establish the…
This paper studies the role of dg-Lie algebroids in derived deformation theory. More precisely, we provide an equivalence between the homotopy theories of formal moduli problems and dg-Lie algebroids over a commutative dg-algebra of characteristic zero. At the level of linear objects, we show that the category of representations of a dg-Lie algebroid is an extension of the category of quasi-coherent sheaves on the corresponding formal moduli problem. We describe this extension geometrically in terms of pro-coherent sheaves.
In this Note, for the future purposes of relative formal derived deformation theory and of derived coisotropic structures, we prove the existence of a model structure on the category of dg-Lie algebroids over a cochain differential non-positively graded commutative algebra over a commutative base ring k of characteristic 0.
We investigate Hopf algebroids in the category of $L$-complete modules over a commutative Noetherian regular complete local ring. The main examples are provided by the Hopf algebroids associated to Lubin-Tate spectra in the K(n)-local stable homotopy category and we show that these have Landweber filtrations for all finitely generated discrete modules. Along the way we investigate the canonical Hopf algebras associated to Hopf algebroids over fields and introduce a notion of unipotent Hopf algebroid generalising that for Hopf algebras. In two appendices we continue the discussion of the connec…
We show that there is an equivalence of $\infty$-categories between Lie algebroids and certain kinds of curved Lie algebras. For this we develop a method to study the $\infty$-category of curved Lie algebras using the homotopy theory of algebras over a complete operad.
Comodules over Hopf algebroids are of central importance in algebraic topology. It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff, the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of presheaves of groupoids with respect to a Grothendieck topology on Aff give rise to equivalences of categories of sheaves in that topology. We then show using faithfully flat descent that an internal equivalenc…
The stable Picard group of finite Adams Hopf algebroids with an application to the $\mathbb{R}$-motivic Steenrod subalgebra $\mathcal{A}(1)^{\mathbb{R}}$
We define the orbit category for transitive topological groupoids and their equivariant CW-complexes. By using these constructions we define equivariant Bredon homology and cohomology for actions of transitive topological groupoids. We show how these theories can be obtained by looking at the action of a single isotropy group on a fiber of the anchor map, extending equivariant results for compact group actions. We also show how this extension from a single isotropy group to the entire groupoid action can be applied to the structure of principal bundles and classifying spaces.
We generalize the van Est map and isomorphism theorem in three ways, and we discuss conjectured connections with homotopy theory, including a proposal of a category which unifies differentiable stacks, Lie algebroids and homotopy theory. In Part 2 of this thesis we generalize the van Est map from a comparison map between Lie groupoid cohomology and Lie algebroid cohomology to a (more conceptual) comparison map between the cohomology of a stack $\mathcal{G}$ and the cohomology of a simple foliation $\mathcal{H}\to\mathcal{G}$. In Part 1 we generalize the functions that we can take cohomology of…
The quotient $L/A[-1]$ of a pair $A\hookrightarrow L$ of Lie algebroids is a Lie algebra object in the derived category $D^b(\mathscr{A})$ of the category $\mathscr{A}$ of left $\mathcal{U}(A)$-modules, the Atiyah class $α_{L/A}$ being its Lie bracket. In this note, we describe the universal enveloping algebra of the Lie algebra object $L/A[-1]$ and we prove that it is a Hopf algebra object in $D^b(\mathscr{A})$.
