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Search Results: 1 - 10 of 144369 matches for " B;Vercruysse "
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Morita theory of comodules over corings
Gabriella B?hm,Joost Vercruysse
Mathematics , 2007, DOI: 10.1080/00927870902747993
Abstract: By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule $\Sigma$ of an $A$-coring $\cC$. This allows to extend (weak and strong) structure theorems in the literature, in particular beyond the cases when any of the coring $\cC$ or the comodule $\Sigma$ is finitely generated and projective as an $A$-module. That is, we obtain relations between the category of $\cC$-comodules and the category of firm modules for a firm ring $R$, which is an ideal of the endomorphism algebra $^\cC(\Sigma)$. For a firmly projective comodule of a coseparable coring we prove a strong structure theorem assuming only surjectivity of the canonical map.
Morita theory for coring extensions and cleft bicomodules
Gabriella B?hm,Joost Vercruysse
Mathematics , 2006,
Abstract: A Morita context is constructed for any comodule of a coring and, more generally, for an $L$-$\cC$ bicomodule $\Sigma$ for a pure coring extension $(\cD:L)$ of $(\cC:A)$. It is related to a 2-object subcategory of the category of $k$-linear functors $\Mm^\Cc\to\Mm^\Dd$. Strictness of the Morita context is shown to imply the Galois property of $\Sigma$ as a $\cC$-comodule and a Weak Structure Theorem. Sufficient conditions are found also for a Strong Structure Theorem to hold. Cleft property of an $L$-$\cC$ bicomodule $\Sigma$ -- implying strictness of the associated Morita context -- is introduced. It is shown to be equivalent to being a Galois $\cC$-comodule and isomorphic to $\End^\cC(\Sigma)\otimes_{L} \cD$, in the category of left modules for the ring $\End^\cC(\Sigma)$ and right comodules for the coring $\cD$, i.e. satisfying the normal basis property. Algebra extensions, that are cleft extensions by a Hopf algebra, a coalgebra or a pure Hopf algebroid, as well as cleft entwining structures (over commutative or non-commutative base rings) and cleft weak entwining structures, are shown to provide examples of cleft bicomodules. Cleft extensions by arbitrary Hopf algebroids are described in terms of Morita contexts that do not necessarily correspond to coring extensions.
Equivalences between categories of modules and categories of comodules
Joost Vercruysse
Mathematics , 2006,
Abstract: We show the close connection between appearingly different Galois theories for comodules introduced recently in [J. G\'omez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, arXiv:math.RA/0509106.] and [R. Wisbauer, On Galois comodules (2004), to appear in Comm. Algebra.]. Furthermore we study equivalences between categories of comodules over a coring and modules over a firm ring. We show that these equivalences are related to Galois theory for comodules.
Quasi-co-Frobenius corings as Galois comodules
Joost Vercruysse
Mathematics , 2008,
Abstract: We compare several quasi-Frobenius-type properties for corings that appeared recently in literature and provide several new characterizations for each of these properties. By applying the theory of Galois comodules with a firm coinvariant ring, we can characterize a locally quasi-Frobenius (quasi-co-Frobenius) coring as a locally projective generator in its category of comodules.
Hopf algebras---Variant notions and reconstruction theorems
Joost Vercruysse
Mathematics , 2012,
Abstract: Hopf algebras are closely related to monoidal categories. More precise, $k$-Hopf algebras can be characterized as those algebras whose category of finite dimensional representations is an autonomous monoidal category such that the forgetful functor to $k$-vectorspaces is a strict monoidal functor. This result is known as the Tannaka reconstruction theorem (for Hopf algebras). Because of the importance of both Hopf algebras in various fields, over the last last few decades, many generalizations have been defined. We will survey these different generalizations from the point of view of the Tannaka reconstruction theorem.
Local units versus local projectivity. Dualisations: Corings with local structure maps
Joost Vercruysse
Mathematics , 2004,
Abstract: We unify and generalize different notions of local units and local projectivity. We investigate the connection between these properties by constructing elementary algebras from locally projective modules. Dual versions of these constructions are discussed, leading to corings with local comultiplications, corings with local counits and rings with local multiplications.
Jointly creating digital abstracts: dealing with synonymy and polysemy
Steven Vercruysse, Martin Kuiper
BMC Research Notes , 2012, DOI: 10.1186/1756-0500-5-601
Abstract: We implemented a ‘controlled language’ as a more expressive representation method. We studied how usable this format was for wet-lab-biologists that volunteered as curators. We assessed some issues that arise with the usability of ontologies and other controlled vocabularies, for the encoding of structured information by ‘untrained’ curators. We take a user-oriented viewpoint, and make recommendations that may prove useful for creating a better curation environment: one that can engage a large community of volunteer curators.Entering information in a biocuration environment could improve in expressiveness and user-friendliness, if curators would be enabled to use synonymous and polysemous terms literally, whereby each term stays linked to an identifier.The Life Sciences are producing vast amounts of information. Each year, over half a million new publications is indexed by PubMed/Medline, and this volume keeps growing increasingly faster. All this information can only be processed effectively with computer assistance. However, most knowledge is only reported through natural language, a format that remains fairly opaque to computers despite flourishing text-mining research [1,2]. Comprehensive and accurate digital formalisation of the published information still needs human intervention, a process called manual curation. But it was shown that curators working in small, focused groups (like institutes) don’t have the capacity to keep up with the enormous growth of new findings [3]. This calls for a crowdsourced setup: a large, distributed community of scientists that collectively curates on a part-time, volunteer basis [4-7].As has been argued, every publication should best become accompanied by a manually created, or at least validated, structured digital abstract (SDA) [7-9]. A few years ago the journal FEBS Letters launched an initiative to let authors create digital abstracts when they submitted a paper [7]. An Excel-sheet was provided with a number of mandatory a
Bimodule herds
Tomasz Brzezinski,Joost Vercruysse
Mathematics , 2008,
Abstract: The notion of a bimodule herd is introduced and studied. A bimodule herd consists of a $B$-$A$ bimodule, its formal dual, called a pen, and a map, called a shepherd, which satisfies untiality and coassociativity conditions. It is shown that every bimodule herd gives rise to a pair of corings and coactions. If, in addition, a bimodule herd is tame i.e. it is faithfully flat and a progenerator, then these corings are associated to entwining structures; the bimodule herd is a Galois comodule of these corings. The notion of a bicomodule coherd is introduced as a formal dualisation of the definition of a bimodule herd. Every bicomodule coherd defines a pair of (non-unital) rings. It is shown that a tame $B$-$A$ bimodule herd defines a bicomodule coherd, and sufficient conditions for the derived rings to be isomorphic to $A$ and $B$ are discussed. The composition of bimodule herds via the tensor product is outlined. The notion of a bimodule herd is illustrated by the example of Galois co-objects of a commutative, faithfully flat Hopf algebra.
Multiplier Hopf and bi-algebras
K. Janssen,J. Vercruysse
Mathematics , 2009,
Abstract: We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele in [A. Van Daele, Multiplier Hopf algebras, {\em Trans. Amer. Math. Soc.}, {\bf 342}(2), (1994) 917--932.] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly non-unital, idempotent, non-degenerate, $k$-projective) algebra over a commutative ring $k$ is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of $k$-modules, into a diagram of strict monoidal forgetful functors.
Lie monads and dualities
Isar Goyvaerts,Joost Vercruysse
Mathematics , 2013,
Abstract: We study dualities between Lie algebras and Lie coalgebras, and their respective (co)representations. To allow a study of dualities in an infinite-dimensional setting, we introduce the notions of Lie monads and Lie comonads, as special cases of YB-Lie algebras and YB-Lie coalgebras in additive monoidal categories. We show that (strong) dualities between Lie algebras and Lie coalgebras are closely related to (iso)morphisms between associated Lie monads and Lie comonads. In the case of a duality between two Hopf algebras -in the sense of Takeuchi- we recover a duality between a Lie algebra and a Lie coalgebra -in the sense defined in this note- by computing the primitive and the indecomposables elements, respectively.
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