26 research outputs found
Morita theory for Hopf algebroids, principal bibundles, and weak equivalences
We show that two flat commutative Hopf algebroids are Morita equivalent if
and only if they are weakly equivalent and if and only if there exists a
principal bibundle connecting them. This gives a positive answer to a
conjecture due to Hovey and Strickland. We also prove that principal (left)
bundles lead to a bicategory together with a 2-functor from flat Hopf
algebroids to trivial principal bundles. This turns out to be the universal
solution for 2-functors which send weak equivalences to invertible 1-cells. Our
approach can be seen as an algebraic counterpart to Lie groupoid Morita theory.Comment: 50 pages; v2: added a section in which we exhibit the categorical
group structure of monoidal symmetric autoequivalences. v3: added a section
which explains the abstract groupoid case as a guideline and for motivation.
To appear in Doc. Mat
Topological tensor product of bimodules, complete Hopf Algebroids and convolution algebras
Given a finitely generated and projective Lie-Rinehart algebra, we show that
there is a continuous homomorphism of complete commutative Hopf algebroids
between the completion of the finite dual of its universal enveloping Hopf
algebroid and the associated convolution algebra. The topological Hopf
algebroid structure of this convolution algebra is here clarified, by providing
an explicit description of its topological antipode as well as of its other
structure maps. Conditions under which that homomorphism becomes an
homeomorphism are also discussed. These results, in particular, apply to the
smooth global sections of any Lie algebroid over a smooth (connected) manifold
and they lead a new formal groupoid scheme to enter into the picture. In the
Appendix we develop the necessary machinery behind complete Hopf algebroid
constructions, which involves also the topological tensor product of filtered
bimodules over filtered rings.Comment: Minor changes, 33 pages. To appear in CC
Towards differentiation and integration between Hopf algebroids and Lie algebroids
In this paper we set up the foundations around the notions of formal
differentiation and formal integration in the context of commutative Hopf
algebroids and Lie-Rinehart algebras. Specifically, we construct a
contravariant functor from the category of commutative Hopf algebroids with a
fixed base algebra to that of Lie-Rinehart algebras over the same algebra, the
differentiation functor, which can be seen as an algebraic counterpart to the
differentiation process from Lie groupoids to Lie algebroids. The other way
around, we provide two interrelated contravariant functors form the category of
Lie-Rinehart algebras to that of commutative Hopf algebroids, the integration
functors. One of them yields a contravariant adjunction together with the
differentiation functor. Under mild conditions, essentially on the base
algebra, the other integration functor only induces an adjunction at the level
of Galois Hopf algebroids. By employing the differentiation functor, we also
analyse the geometric separability of a given morphism of Hopf algebroids.
Several examples and applications are presented along the exposition.Comment: Minor changes. Comments are very welcome
Functorial Constructions for Non-associative Algebras with Applications to Quasi-bialgebras
The aim of this paper is to establish a contravariant adjunction between the
category of quasi-bialgebras and a suitable full subcategory of dual
quasi-bialgebras, adapting the notion of finite dual to this framework. Various
functorial constructions involving non-associative algebras and
non-coassociative coalgebras are then carried out. Several examples
illustrating our methods are expounded as well