Hopf modules, Frobenius functors and (one-sided) Hopf algebras

We investigate the property of being Frobenius for some functors strictly related with Hopf modules over a bialgebra and how this property reflects on the latter. In particular, we characterize one-sided Hopf algebras with anti-(co)multiplicative one-sided antipode as those for which the free Hopf module functor is Frobenius. As a by-product, this leads us to relate the property of being an FH-algebra (in the sense of Pareigis) for a given bialgebra with the property of being Frobenius for certain naturally associated functors.Comment: 20 pages. Major changes: considerably shortened. Comments are welcom

Globalization for geometric partial comodules

We discuss globalization for geometric partial comodules in a monoidal category with pushouts and we provide a concrete procedure to construct it, whenever it exists. The mild assumptions required by our approach make it possible to apply it in a number of contexts of interests, recovering and extending numerous ad hoc globalization constructions from the literature in some cases and providing obstruction for globalization in some other cases.Comment: 18 pages. Major revision. Results and global presentation improved. Comments are welcome

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

A Catalogue of Galaxies in the HDF-South: Photometry and Structural Parameters

We describe the construction of a catalogue of galaxies in the optical field of the Hubble Deep Field South. The HDF-S observations produced WFPC2 images in U, B, V, and I, the version 1 data have been made public on 23 November 1999. The effective field of view is 4.38 arcmin$^2$, and the 5$\sigma$ limiting magnitudes (in a FWHM aperture) are 28.87, 29.71, 30.19, 29.58 in the U, B, V and I bands, respectively. We created a catalogue for each pass-band (I$_{814}$, V$_{606}$, B$_{450}$, U$_{300}$), using simulations to account for incompleteness and spurious sources contamination. Along with photometry in all bands, we determined on the I$_{814}$-selected catalogue (I$_{814}<26$) structural parameters, such as a metric apparent size, derived from the petrosian radius, an asymmetry index, light concentration indexes and the mean surface brightness within the petrosian radius.Comment: 10 pages, 11 figures. Accepted for publication in A&ASS. The catalog is available in the source and at http://www.merate.mi.astro.it/~saracco/science.htm

PBWPBW-deformations of graded rings

We prove in a very general framework several versions of the classical Poincar\'e-Birkhoff-Witt Theorem, which extend results from [BeGi, BrGa, CS, HvOZ, WW]. Applications and examples are discussed in the last part of the paper

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