37 results on '"Joost Vercruysse"'
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2. V-universal Hopf algebras (co)acting on Ω-algebras
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Joost Vercruysse, Ana Agore, A. S. Gordienko, Algebra and Analysis, Mathematics, and Algebra
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Pure mathematics ,Computer Science::Information Retrieval ,Applied Mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,Astrophysics::Instrumentation and Methods for Astrophysics ,Mathematics - Category Theory ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,Mathematics - Rings and Algebras ,Hopf algebra ,Bialgebra ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,Computer Science::General Literature ,Mathematics - Representation Theory ,Mathematics - Abstract
We develop a theory which unifies the universal (co)acting bi/Hopf algebras as studied by Sweedler, Manin and Tambara with the recently introduced \cite{AGV1} bi/Hopf-algebras that are universal among all support equivalent (co)acting bi/Hopf algebras. Our approach uses vector spaces endowed with a family of linear maps between tensor powers of $A$, called $\Omega$-algebras. This allows us to treat algebras, coalgebras, braided vector spaces and many other structures in a unified way. We study $V$-universal measuring coalgebras and $V$-universal comeasuring algebras between $\Omega$-algebras $A$ and $B$, relative to a fixed subspace $V$ of $\Vect(A,B)$. By considering the case $A=B$, we derive the notion of a $V$-universal (co)acting bialgebra (and Hopf algebra) for a given algebra $A$. In particular, this leads to a refinement of the existence conditions for the Manin--Tambara universal coacting bi/Hopf algebras. We establish an isomorphism between the $V$-universal acting bi/Hopf algebra and the finite dual of the $V$-universal coacting bi/Hopf algebra under certain conditions on $V$ in terms of the finite topology on $\End_F(A)$., Comment: To appear in Commun. Contemp. Math
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- 2023
3. A Torsion Theory in the Category of Cocommutative Hopf Algebras.
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Marino Gran, Gabriel Kadjo, and Joost Vercruysse
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- 2016
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4. Equivalences of (co)module algebra structures over Hopf algebras
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Ana Agore, Alexey Sergeevich Gordienko, Joost Vercruysse, Algebra and Analysis, Mathematics, Mathematics-TW, and Algebra
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Polynomial ,Algebraic structure ,Structure (category theory) ,Primary 16W50, Secondary 16T05, 16T25, 16W22, 16W25 ,Algèbre - théorie des anneaux - théorie des corps ,Equivalence class (music) ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Category Theory (math.CT) ,Equivalence (measure theory) ,Mathematical Physics ,Mathematics ,Algebra and Number Theory ,Group (mathematics) ,Mathematics::Rings and Algebras ,Dual number ,Mathematics - Category Theory ,Mathematics - Rings and Algebras ,Théorie des ensembles et catégories ,Hopf algebra ,Algebra ,Rings and Algebras (math.RA) ,Geometry and Topology ,Groupes algébriques - Abstract
We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) equivalence of gradings. We show that for each equivalence class of (co)module algebra structures on a given algebra A, there exists a unique universal Hopf algebra H together with an H-(co)module structure on A such that any other equivalent (co)module algebra structure on A factors through the action of H. We study support equivalence and the universal Hopf algebras mentioned above for group gradings, Hopf-Galois extensions, actions of algebraic groups and cocommutative Hopf algebras. We show how the notion of support equivalence can be used to reduce the classification problem of Hopf algebra (co)actions. In particular, we apply support equivalence to study the asymptotic behaviour of codimensions of H-identities of a certain class of H-module algebras. This result proves the analogue (formulated by Yu.A. Bahturin) of Amitsur's conjecture which was originally concerned with ordinary polynomial identities., preprint submitted for publication, info:eu-repo/semantics/inPress
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- 2021
5. Correspondence theorems for Hopf algebroids with applications to affine groupoids
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Laiachi El Kaoutit, Aryan Ghobadi, Paolo Saracco, and Joost Vercruysse
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Mathematics - Algebraic Geometry ,Rings and Algebras (math.RA) ,General Mathematics ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Mathematics - Rings and Algebras ,Algebraic Geometry (math.AG) - Abstract
We provide a correspondence between one-sided coideal subrings and one-sided ideal two-sided coideals in an arbitrary bialgebroid. We prove that, under some expected additional conditions, this correspondence becomes bijective for Hopf algebroids. As an application, we investigate normal Hopf ideals in commutative Hopf algebroids (affine groupoid schemes) in connection with the study of normal affine subgroupoids.
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- 2022
6. Partial corepresentations of Hopf algebras
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Joost Vercruysse, Felipe Nalon Castro, Marcelo Muniz S. Alves, Glauber Quadros, and Eliezer Batista
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Pure mathematics ,Partial comodules ,Algèbre linéaire et matricielle ,Existential quantification ,Coalgebra ,Structure (category theory) ,Partial corepresentation ,01 natural sciences ,Algèbre - théorie des anneaux - théorie des corps ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,Hopf coalgebroid ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Partial cosmash coproducts ,Quantum Algebra (math.QA) ,Category Theory (math.CT) ,Bicoalgebroid ,Representation Theory (math.RT) ,0101 mathematics ,16T05, 16S40 ,Mathematics ,Algebra and Number Theory ,Mathematics::Rings and Algebras ,010102 general mathematics ,Mathematics - Category Theory ,Mathematics - Rings and Algebras ,Hopf algebra ,Partial modules ,Partial representation ,Rings and Algebras (math.RA) ,010307 mathematical physics ,Groupes algébriques ,Géométrie non commutative ,Mathematics - Representation Theory - Abstract
We introduce the notion of a partial corepresentation of a given Hopf algebra H over a coalgebra C and the closely related concept of a partial H-comodule. We prove that there exists a universal coalgebra Hpar, associated to the original Hopf algebra H, such that the category of regular partial H-comodules is isomorphic to the category of Hpar-comodules. We introduce the notion of a Hopf coalgebroid and show that the universal coalgebra Hpar has the structure of a Hopf coalgebroid over a suitable coalgebra., SCOPUS: ar.j, info:eu-repo/semantics/published
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- 2021
7. The Eilenberg-Moore Category and a Beck-type Theorem for a Morita Context.
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Tomasz Brzezinski, Adrian Vazquez Marquez, and Joost Vercruysse
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- 2011
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8. Constructing Infinite Comatrix Corings from Colimits.
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Stefaan Caenepeel, Erwin de Groot, and Joost Vercruysse
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- 2006
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9. Partial and global representations of finite groups
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Michele D’Adderio, William Hautekiet, Paolo Saracco, and Joost Vercruysse
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partial action ,20C05, 20C30, 20M30 ,General Mathematics ,Group Theory (math.GR) ,Partial representation ,symmetric group ,groupoid algebra ,Partial representation, partial action, groupoid algebra, globalization, symmetric group ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Representation Theory (math.RT) ,Mathematics - Group Theory ,globalization ,Mathematics - Representation Theory - Abstract
Given a subgroup H of a finite group G, we begin a systematic study of the partial representations of G that restrict to global representations of H. After adapting several results from [DEP00] (which correspond to the case where H is trivial), we develop further an effective theory that allows explicit computations. As a case study, we apply our theory to the symmetric group and its subgroup of permutations fixing 1: this provides a natural extension of the classical representation theory of the symmetric group., 31 pages
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- 2020
10. BiHom Hopf algebras viewed as Hopf monoids
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Gabriella Böhm and Joost Vercruysse
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Pure mathematics ,Unital ,Mathematics::General Topology ,Symmetric monoidal category ,Mathematics - Category Theory ,Hopf algebra ,Mathématiques ,Mathematics::Logic ,Mathematics::Probability ,Mathematics::Category Theory ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Mathematics::Metric Geometry ,Category Theory (math.CT) ,Mathematics - Abstract
We introduce monoidal categories whose monoidal products of any positive number of factors are lax coherent and whose nullary products are oplax coherent. We call them $\mathsf{Lax}^+\mathsf{Oplax}^0$-monoidal. Dually, we consider $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal categories which are oplax coherent for positive numbers of factors and lax coherent for nullary monoidal products. We define $\mathsf{Lax}^+_0\mathsf{Oplax}^0_+$-duoidal categories with compatible $\mathsf{Lax}^+\mathsf{Oplax}^0$- and $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal structures. We introduce comonoids in $\mathsf{Lax}^+\mathsf{Oplax}^0$-monoidal categories, monoids in $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal categories and bimonoids in $\mathsf{Lax}^+_0\mathsf{Oplax}^0_+$- duoidal categories. Motivation for these notions comes from a generalization of a construction due to Caenepeel and Goyvaerts. This assigns a $\mathsf{Lax}^+_0\mathsf{Oplax}^0_+$-duoidal category $\mathsf D$ to any symmetric monoidal category $\mathsf V$. The unital $\mathsf{BiHom}$-monoids, counital $\mathsf{BiHom}$-comonoids, and unital and counital $\mathsf{BiHom}$-bimonoids in $\mathsf V$ are identified with the monoids, comonoids and bimonoids in $\mathsf D$, respectively., 40 pages, a few figures of commutative diagrams
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- 2020
11. A semi-abelian extension of a theorem by Takeuchi
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Marino Gran, Joost Vercruysse, Florence Sterck, and UCL - SST/IRMP - Institut de recherche en mathématique et physique
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Crossed modules ,Pure mathematics ,Internal groupoids ,Abelian extension ,Semi-abelian categories ,Context (language use) ,Field (mathematics) ,01 natural sciences ,Algèbre - théorie des anneaux - théorie des corps ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Cocommutative Hopf algebrasSemi-abelian categoriesCrossed modulesInternal groupoidsCategorical commutator ,18E10, 18D35, 18G50, 16T05, 16B50, 18B40 ,Cocommutative Hopf algebras ,Categorical commutator ,Category Theory (math.CT) ,0101 mathematics ,Abelian group ,Commutative property ,Mathematics ,Commutator ,Algebra and Number Theory ,010102 general mathematics ,Mathematics::Rings and Algebras ,Zero (complex analysis) ,Mathematics - Category Theory ,Mathematics - Rings and Algebras ,Théorie des ensembles et catégories ,Hopf algebra ,Rings and Algebras (math.RA) ,010307 mathematical physics - Abstract
We prove that the category of cocommutative Hopf algebras over a field is a semi-abelian category. This result extends a previous special case of it, based on the Milnor–Moore theorem, where the field was assumed to have zero characteristic. Takeuchi's theorem asserting that the category of commutative and cocommutative Hopf algebras over a field is abelian immediately follows from this new observation. We also prove that the category of cocommutative Hopf algebras over a field is action representable. We make some new observations concerning the categorical commutator of normal Hopf subalgebras, and this leads to the proof that two definitions of crossed modules of cocommutative Hopf algebras are equivalent in this context., SCOPUS: ar.j, info:eu-repo/semantics/published
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- 2019
12. Rationality Properties for Morita Contexts associated to Corings
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Stefaan Caenepeel, Joost Vercruysse, Shuanhong Wang, Mathematics-TW, and Vrije Universiteit Brussel
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Algèbre - théorie des anneaux - théorie des corps - Abstract
info:eu-repo/semantics/published
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- 2019
13. Dilations of partial representations of Hopf algebras
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Joost Vercruysse, Marcelo Muniz S. Alves, and Eliezer Batista
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Pure mathematics ,General Mathematics ,Structure (category theory) ,01 natural sciences ,Representation theory ,Projection (linear algebra) ,Dilation (operator theory) ,Algèbre - théorie des anneaux - théorie des corps ,18D10 (secondary) ,Mathematics::Category Theory ,0103 physical sciences ,FOS: Mathematics ,Representation Theory (math.RT) ,0101 mathematics ,16S40 ,Mathematics ,Smash product ,010102 general mathematics ,Mathematics - Rings and Algebras ,Hopf algebra ,Action (physics) ,Rings and Algebras (math.RA) ,Category of modules ,16T05 (primary) ,010307 mathematical physics ,Groupes algébriques ,Mathematics - Representation Theory - Abstract
We introduce the notion of a dilation for a partial representation (that is, a partial module) of a Hopf algebra, which in case the partial representation origins from a partial action (that is, a partial module algebra) coincides with the enveloping action (or globalization). This construction leads to categorical equivalences between the category of partial H-modules, a category of (global) H-modules endowed with a projection satisfying a suitable commutation relation and the category of modules over a (global) smash product constructed upon H, from which we deduce the structure of a Hopfish algebra on this smash product. These equivalences are used to study the interactions between partial and global representation theory., SCOPUS: ar.j, info:eu-repo/semantics/published
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- 2019
14. Split extension classifiers in the category of cocommutative Hopf algebras
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Marino Gran, Gabriel Kadjo, Joost Vercruysse, and UCL - SST/IRMP - Institut de recherche en mathématique et physique
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General Mathematics ,20J99 ,Groups ,Semi-abelian categories ,Center (group theory) ,01 natural sciences ,18E10 ,16S40, 16T05, 16U70, 18E10, 20J99 ,16T05 ,center ,Lie algebras ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,0103 physical sciences ,FOS: Mathematics ,Category Theory (math.CT) ,0101 mathematics ,Algebraically closed field ,16S40 ,Categorical variable ,Mathematics ,Quantum group ,010102 general mathematics ,Mathematics::Rings and Algebras ,Zero (complex analysis) ,Mathematics - Category Theory ,Extension (predicate logic) ,Mathematics - Rings and Algebras ,centralizer ,Hopf algebra ,Centralizer and normalizer ,Algebra ,Rings and Algebras (math.RA) ,Hopf algebras ,universal object ,010307 mathematical physics ,16U70 ,cocommutative Hopf algebra ,split extension classifier - Abstract
We describe the split extension classifiers in the semi-abelian category of cocommutative Hopf algebras over an algebraically closed field of characteristic zero. The categorical notions of centralizer and of center in the category of cocommutative Hopf algebras is then explored. We show that the categorical notion of center coincides with the one that is considered in the theory of general Hopf algebras., 24 pages
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- 2018
15. Geometrically Partial actions
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Joost Vercruysse and Jiawei Hu
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Pure mathematics ,Applied Mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,Mathematics - Rings and Algebras ,Algebraic geometry ,Théorie des ensembles et catégories ,Hopf algebra ,Action (physics) ,Algèbre - théorie des anneaux - théorie des corps ,Mathématiques ,Rings and Algebras (math.RA) ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,FOS: Mathematics ,Groupes algébriques ,Mathematics - Abstract
We introduce "geometric" partial comodules over coalgebras in monoidal categories as an alternative notion to the notion of partial action and coaction of a Hopf algebra introduced by Caenepeel and Janssen. The name is motivated by the fact that our new notion suits better if one wants to describe phenomena of partial actions in algebraic geometry. Under mild conditions, the category of geometric partial comodules is shown to be complete and cocomplete and the category of partial comodules over a Hopf algebra is lax monoidal. We develop a Hopf-Galois theory for geometric partial coactions to illustrate that our new notion might be a useful additional tool in Hopf algebra theory., SCOPUS: ar.j, info:eu-repo/semantics/published
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- 2018
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16. Lie monads and dualities
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Joost Vercruysse, Isar Goyvaerts, Mathematics-TW, Mathematics, and Algebra
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Pure mathematics ,Monoidal category ,Algebra and Number Theory ,Lie algebra ,Mathematics::Rings and Algebras ,Duality (optimization) ,Mathematics - Category Theory ,Mathematics - Rings and Algebras ,Lie monad ,Théorie des ensembles et catégories ,Hopf algebra ,Algèbre - théorie des anneaux - théorie des corps ,Lie coalgebra ,Morphism ,Rings and Algebras (math.RA) ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,FOS: Mathematics ,Category Theory (math.CT) ,Monad ,Indecomposable module ,Mathematics - 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 indecomposable elements, respectively. © 2014 Elsevier Inc., SCOPUS: ar.j, info:eu-repo/semantics/published
- Published
- 2014
17. MULTIPLIER BI- AND HOPF ALGEBRAS
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Joost Vercruysse, K. Janssen, Mathematics-TW, and Algebra
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Algebra and Number Theory ,Multiplier algebra ,Quantum group ,Applied Mathematics ,monoidal category ,Mathematics::Rings and Algebras ,Monoidal category ,Representation theory of Hopf algebras ,Tensor algebra ,Quasitriangular Hopf algebra ,Hopf algebra ,Algèbre - théorie des anneaux - théorie des corps ,Algebra ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,Multiplier Hopf algebra ,Géométrie non commutative ,Monoidal functor ,Mathematics - 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 [Multiplier Hopf algebras, Trans. Amer. Math. Soc.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 nonunital, idempotent, nondegenerate, 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.
- Published
- 2010
18. The Eilenberg-Moore Category and a Beck-type Theorem for a Morita Context
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Tomasz Brzeziński, Adrian Vazquez Marquez, Joost Vercruysse, Mathematics-TW, and Algebra
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Pure mathematics ,Adjoint functor ,Algebra and Number Theory ,Functor ,General Computer Science ,Derived functor ,Functor category ,Mathematics - Category Theory ,Morita context ,Mathematics::Algebraic Topology ,Algèbre - théorie des anneaux - théorie des corps ,Theoretical Computer Science ,Algebra ,Eilenberg Moore category ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,18A40 ,Natural transformation ,FOS: Mathematics ,Tor functor ,Category Theory (math.CT) ,Morita equivalence ,Exact functor ,Adjoint functors ,Mathematics - Abstract
The Eilenberg-Moore constructions and a Beck-type theorem for pairs of monads are described. More specifically, a notion of a {\em Morita context} comprising of two monads, two bialgebra functors and two connecting maps is introduced. It is shown that in many cases equivalences between categories of algebras are induced by such Morita contexts. The Eilenberg-Moore category of representations of a Morita context is constructed. This construction allows one to associate two pairs of adjoint functors with right adjoint functors having a common domain or a {\em double adjunction} to a Morita context. It is shown that, conversely, every Morita context arises from a double adjunction. The comparison functor between the domain of right adjoint functors in a double adjunction and the Eilenberg-Moore category of the associated Morita context is defined. The sufficient and necessary conditions for this comparison functor to be an equivalence (or for the {\em moritability} of a pair of functors with a common domain) are derived., 33 pages, some corrections, new section 5.3
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- 2009
19. Corrigendum to 'Morita theory for coring extensions and cleft bicomodules' [Adv. Math. 209 (2) (2007) 611–648]
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Joost Vercruysse and Gabriella Böhm
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Pure mathematics ,Mathematics(all) ,Mathematics::Quantum Algebra ,General Mathematics ,Morita therapy ,Order (group theory) ,Coring ,Structured program theorem ,Mathematics - Abstract
The results in our paper heavily rely on the journal version of [T. Brzezinski, A note on coring extensions, Ann. Univ. Ferrara Sez. VII (N.S.) 51 (2005) 15–27; a corrected version is available at http://arxiv.org/abs/math/0410020v3 , Theorem 2.6]. Since it turned out recently that in the proof of the quoted theorem there are some assumptions missing, our derived results are not expected to hold at the stated level of generality either. Here we supplement the constructions in our article with the missing assumptions and show that they hold in most of our examples. In order to handle also the non-fitting case of cleft extensions by arbitrary Hopf algebroids, Morita contexts are constructed that do not necessarily correspond to coring extensions. They are used to prove a Strong Structure Theorem for cleft extensions by arbitrary Hopf algebroids. In this way we obtain in particular a corrected form of the journal version of [G. Bohm, Integral theory for Hopf algebroids, Algebr. Represent. Theory 8 (4) (2005) 563–599; Corrigendum, to be published; see also http://arxiv.org/abs/math/0403195v4 , Theorem 4.2], whose original proof contains a similar error.
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- 2009
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20. Morita theory for coring extensions and cleft bicomodules
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Joost Vercruysse, Gabriella Böhm, Mathematics-TW, and Vrije Universiteit Brussel
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Mathematics(all) ,Pure mathematics ,General Mathematics ,Coalgebra ,Context (language use) ,Algèbre - théorie des anneaux - théorie des corps ,Coring extension ,Comodule ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,16D90 ,FOS: Mathematics ,Mathematics ,Subcategory ,Ring (mathematics) ,Functor ,Mathematics::Rings and Algebras ,Mathematics - Rings and Algebras ,Morita theory ,Hopf algebra ,Normal basis ,Rings and Algebras (math.RA) ,16W30 ,Géométrie non commutative ,Weak and strong structure theorems ,Cleft bicomodule - 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., Comment: 34 pages LaTeX. v2:A missing purity assumption is added throughout Sections 3, 4 and 5
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- 2007
21. A torsion theory in the category of cocommutative Hopf algebras
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Joost Vercruysse, Marino Gran, Gabriel Kadjo, and UCL - SST/IRMP - Institut de recherche en mathématique et physique
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Pure mathematics ,General Computer Science ,Semi-abelian category ,Category of groups ,01 natural sciences ,Theoretical Computer Science ,QA150 ,Mathematics::K-Theory and Homology ,Torsion theory ,Mathematics::Category Theory ,0103 physical sciences ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Category Theory (math.CT) ,0101 mathematics ,Mathematics ,Algebra and Number Theory ,010102 general mathematics ,Mathematics - Category Theory ,18E40, 18E10, 20J99, 16T05, 16S40 ,torsion theory ,Mathematics - Rings and Algebras ,Hopf algebra ,Rings and Algebras (math.RA) ,Torsion (algebra) ,010307 mathematical physics ,cocommutative Hopf algebra - Abstract
The purpose of this article is to prove that the category of cocommutative Hopf $K$-algebras, over a field $K$ of characteristic zero, is a semi-abelian category. Moreover, we show that this category is action representable, and that it contains a torsion theory whose torsion-free and torsion parts are given by the category of groups and by the category of Lie $K$-algebras, respectively.
- Published
- 2015
22. Galois theory for comatrix corings: Descent theory, Morita theory, Frobenius and separability properties
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Stefaan Caenepeel, Joost Vercruysse, E. De Groot, Mathematics-TW, and Vrije Universiteit Brussel
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Pure mathematics ,Applied Mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,Galois theory ,Context (language use) ,Mathematics - Rings and Algebras ,Coring ,Algèbre - théorie des anneaux - théorie des corps ,Algebra ,Comodule ,Rings and Algebras (math.RA) ,Mathematics::Category Theory ,Mathematics - Quantum Algebra ,Morita therapy ,FOS: Mathematics ,Quantum Algebra (math.QA) ,16W30 ,Géométrie non commutative ,Descent (mathematics) ,Structured program theorem ,Mathematics - Abstract
El Kaoutit and G\'omez Torrecillas introduced comatrix corings, generalizing Sweedler's canonical coring, and proved a new version of the Faithfully Flat Descent Theorem. They also introduced Galois corings, as corings isomorphic to a comatrix coring. In this paper, we further investigate this theory. We prove a new version of the Joyal-Tierney Descent Theorem, and generalize the Galois Coring Structure Theorem. We associate a Morita context to a coring with a fixed comodule, and relate it to Galois-type properties of the coring. An affineness criterion is proved in the situation where the coring is coseparable. Further properties of the Morita context are studied in the situation where the coring is (co)Frobenius., Comment: 42 pages
- Published
- 2006
23. Dual Constructions for Partial Actions of Hopf Algebras
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Eliezer Batista and Joost Vercruysse
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Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,Mathematics::Rings and Algebras ,Duality (optimization) ,010103 numerical & computational mathematics ,Mathematics - Rings and Algebras ,Hopf algebra ,01 natural sciences ,Dual (category theory) ,Rings and Algebras (math.RA) ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,16T05, 16S40, 16S35, 16W50 ,0101 mathematics ,Connection (algebraic framework) ,Commutative property ,Mathematics - Abstract
The duality between partial actions (partial $H$-module algebras) and co-actions (partial $H$-comodule algebras) of a Hopf algebra $H$ is fully explored in this work. A connection between partial (co)actions and Hopf algebroids is established under certain commutativity conditions. Moreover, we continue this duality study, introducing also partial $H$-module coalgebras and their associated $C$-rings, partial $H$-comodule coalgebras and their associated cosmash coproducts, as well as the mutual interrelations between these structures., v3: strongly revised version
- Published
- 2014
24. On the duality of generalized Lie and Hopf algebras
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Isar Goyvaerts, Joost Vercruysse, Mathematics-TW, and Algebra
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Pure mathematics ,General Mathematics ,Hopf group-coalgebra ,Universal enveloping algebra ,Representation theory of Hopf algebras ,Quasitriangular Hopf algebra ,Hopf algebra ,Graded Lie algebra ,Algèbre - théorie des anneaux - théorie des corps ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Category Theory (math.CT) ,Monoidal categories ,Lie Algebra ,Mathematics ,Discrete mathematics ,Quantum group ,Mathematics::Rings and Algebras ,Mathematics - Category Theory ,Mathematics - Rings and Algebras ,Théorie des ensembles et catégories ,Lie conformal algebra ,Adjoint representation of a Lie algebra ,Rings and Algebras (math.RA) ,Hopf algebras ,duality ,Lie coalgebra ,Géométrie non commutative - Abstract
We show how, under certain conditions, an adjoint pair of braided monoidal functors can be lifted to an adjoint pair between categories of Hopf algebras. This leads us to an abstract version of Michaelis' theorem, stating that given a Hopf algebra H, there is a natural isomorphism of Lie algebras Q(H) * ≅ P(H {ring operator}), where Q(H) * is the dual Lie algebra of the Lie coalgebra of indecomposables of H, and P(H {ring operator}) is the Lie algebra of primitive elements of the Sweedler dual of H. We apply our theory to Turaev's Hopf group-(co)algebras. © 2014 Elsevier Inc., SCOPUS: ar.j, info:eu-repo/semantics/published
- Published
- 2014
25. Partial Representations of Hopf Algebras
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Marcelo Muniz S. Alves, Joost Vercruysse, and Eliezer Batista
- Subjects
Algebra and Number Theory ,Quantum group ,Mathematics::Rings and Algebras ,Monoidal category ,Representation theory of Hopf algebras ,Mathematics - Rings and Algebras ,Quasitriangular Hopf algebra ,Hopf algebra ,Algebra ,Morphism ,Rings and Algebras (math.RA) ,Category of modules ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Universal property ,16T05, 16S40, 16S35, 16W50 ,Mathematics - Abstract
In this work, the notion of partial representation of a Hopf algebra is introduced and its relationship with partial actions of Hopf algebras is explored. Given a Hopf algebra $H$, one can associate it to a Hopf algebroid $H_{par}$ which has the universal property that each partial representation of $H$ can be factorized by an algebra morphism from $H_{par}$. We define also the category of partial modules over a Hopf algebra $H$, which is the category of modules over its associated Hopf algebroid $H_{par}$. The Hopf algebroid structure of $H_{par}$ enables us to enhance the category of partial $H$ modules with a monoidal structure and such that the algebra objects in this category are the usual partial actions. Some examples of categories of partial $H$ modules are explored. In particular we can describe fully the category of partially $\mathbb{Z}_2$-graded modules., 39 pages, version 2: some additions in the section on partial gradings and minor corrections elsewhere
- Published
- 2013
26. Hopf Algebras—Variant Notions and Reconstruction Theorems
- Author
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Joost Vercruysse
- Published
- 2013
27. A Note on the Categorification of Lie Algebras
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Isar Goyvaerts, Joost Vercruysse, Dobrev, Vladimir, Mathematics-TW, and Algebra
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Pure mathematics ,Non-associative algebra ,18D10, 17B60 ,Mathematics - Category Theory ,Mathematics - Rings and Algebras ,Théorie des ensembles et catégories ,Killing form ,Affine Lie algebra ,Algèbre - théorie des anneaux - théorie des corps ,Graded Lie algebra ,Lie conformal algebra ,Yang Baxter equation ,Algebra ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,Rings and Algebras (math.RA) ,Mathematics::Category Theory ,FOS: Mathematics ,Freudenthal magic square ,Category Theory (math.CT) ,Lie Algebra ,Mathematics - Abstract
In this short note we study Lie algebras in the framework of symmetric monoidal categories. After a brief review of the existing work in this field and a presentation of earlier studied and new examples, we examine which functors preserve the structure of a Lie algebra., 8 pages, to appear in Proceedings of International Workshop "Lie Theory-IX", Varna, 2011
- Published
- 2013
28. Cohomology for bicomodules: separable and Maschke functors
- Author
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Joost Vercruysse, L. El Kaoutit, Mathematics-TW, and Vrije Universiteit Brussel
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Fiber functor ,Algebra and Number Theory ,Functor ,Grothendieck category ,Mathematics::Rings and Algebras ,Mathematics - Rings and Algebras ,Cone (category theory) ,Théorie des ensembles et catégories ,Cohomology ,Separable space ,Algebra ,Rings and Algebras (math.RA) ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,FOS: Mathematics ,16W30, 16D20 ,Geometry and Topology ,Exact functor ,Forgetful functor ,Algèbre commutative et algèbre homologique ,Mathematics - Abstract
We introduce the category of bicomodules for a comonad on a Grothendieck category whose underlying functor is right exact and preserves direct sums. We characterize comonads with a separable forgetful functor by means of cohomology groups using cointegrations into bicomodules. We present two applications: the characterization of coseparable corings stated in [14], and the characterization of coseparable coalgebra coextensions stated in [19].
- Published
- 2009
29. Quasi Frobenius corings as Galois comodules
- Author
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Joost Vercruysse and Mathematics-TW
- Subjects
Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,Mathematics::Rings and Algebras ,Galois comodule ,Morita context ,Quasi Frobenius Coring - 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.
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- 2008
30. Morita theory of comodules over corings
- Author
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Joost Vercruysse and Gabriella Böhm
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Pure mathematics ,Algebra and Number Theory ,Endomorphism ,Mathematics::Rings and Algebras ,Sigma ,Mathematics - Rings and Algebras ,Algèbre - théorie des anneaux - théorie des corps ,16D90, 16W30 ,Comodule ,Rings and Algebras (math.RA) ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Morita therapy ,FOS: Mathematics ,Canonical map ,Finitely-generated abelian group ,Equivalence (formal languages) ,Structured program theorem ,Mathematics - 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., Comment: LaTeX, 35 pages. v2: Minor changes including the title, examples added in Section 2
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- 2007
31. Comatrix corings and Galois comodules over firm rings
- Author
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José Gómez-Torrecillas, Joost Vercruysse, Mathematics-TW, and Vrije Universiteit Brussel
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Pure mathematics ,Mathematics::Operator Algebras ,General Mathematics ,Mathematics::Rings and Algebras ,Mathematics - Rings and Algebras ,Noncommutative geometry ,Algèbre - théorie des anneaux - théorie des corps ,Algebra ,General theory ,Rings and Algebras (math.RA) ,Mathematics::Category Theory ,FOS: Mathematics ,Mathematics ,Descent (mathematics) - Abstract
We construct comatrix corings on bimodules without finiteness conditions by using firm rings. This leads to the formulion of a notion of Galois coring which plays a key role in the statement of a Noncommutative Faithfully Flat Descent for comodules which generalizes previous versions. In particular, infinite comatrix corings fit in our general theory.
- Published
- 2007
32. Equivalences between categories of modules and categories of comodules
- Author
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Joost Vercruysse, Mathematics-TW, and Vrije Universiteit Brussel
- Subjects
Ring (mathematics) ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Mathematics::Number Theory ,Galois theory ,Mathematics::Rings and Algebras ,Galois group ,Mathematics - Rings and Algebras ,Algèbre - théorie des anneaux - théorie des corps ,Algebra ,Rings and Algebras (math.RA) ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Algebra over a field ,Connection (algebraic framework) ,16W30 ,Mathematics - 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., Comment: 19 pages, some corrections and additions to version 1
- Published
- 2006
33. Quasi-Frobenius functors. Applications
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Joost Vercruysse, F. Castano Iglesias, C. Nǎstǎsescu, Mathematics-TW, and Algebra
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Algebra and Number Theory ,Calculus of functors ,Derived functor ,Quasi Frobenius ring ,Mathematics::Rings and Algebras ,Functor category ,Mathematics - Rings and Algebras ,Mathematics::Algebraic Topology ,Algèbre - théorie des anneaux - théorie des corps ,Algebra ,Quasi Frobenius Functor ,Mathematics::K-Theory and Homology ,Rings and Algebras (math.RA) ,Mathematics::Category Theory ,Natural transformation ,Ext functor ,Bimodule ,Tor functor ,FOS: Mathematics ,Adjoint functors ,16W30, 16L60 ,Mathematics - Abstract
We investigate functors between abelian categories having a left adjoint and a right adjoint that are \emph{similar} (these functors are called \emph{quasi-Frobenius functors}). We introduce the notion of a \emph{quasi-Frobenius bimodule} and give a characterization of these bimodules in terms of quasi-Frobenius functors. Some applications to corings and graded rings are presented. In particular, the concept of quasi-Frobenius homomorphism of corings is introduced. Finally, a version of the endomorphism ring Theorem for quasi-Frobenius extensions in terms of corings is obtained., Comment: 20 pages, major changes in v2
- Published
- 2006
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34. Constructing infinite comatrix corings from colimits
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E. De Groot, Joost Vercruysse, Stefaan Caenepeel, Mathematics-TW, and Vrije Universiteit Brussel
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Class (set theory) ,Pure mathematics ,Algebra and Number Theory ,General Computer Science ,Mathematics - Rings and Algebras ,Théorie des ensembles et catégories ,Algèbre - théorie des anneaux - théorie des corps ,Theoretical Computer Science ,Rings and Algebras (math.RA) ,Theory of computation ,Calculus ,FOS: Mathematics ,Special case ,16W30 ,Mathematics - Abstract
We propose a class of infinite comatrix corings, and describe them as colimits of systems of usual comatrix corings. The infinite comatrix corings of El Kaoutit and G\'omez Torrecillas are special cases of our construction, which in turn can be considered as a special case of the comatrix corings introduced recently by G\'omez Torrecillas an the third author., Comment: 23 pages, some corrections and additions to the previous version, and a renumbering of theorems in section 4
- Published
- 2006
35. Local units versus local projectivity. Dualisations: Corings with local structure maps
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Joost Vercruysse, Mathematics-TW, and Vrije Universiteit Brussel
- Subjects
Algebra ,Algebra and Number Theory ,Rings and Algebras (math.RA) ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Mathematics - Rings and Algebras ,16W30 ,Local structure ,Mathematics ,Connection (mathematics) ,Dual (category theory) - 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., 22 pages, including a correction to Proposition 1.1
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- 2004
36. Morita Theory for corings and cleft entwining structures
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Joost Vercruysse, Stefaan Caenepeel, Shuanhong Wang, Algebra, Mathematics-TW, and Vrije Universiteit Brussel
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Pure mathematics ,Entwining structures ,Algebra and Number Theory ,Galois theory ,Mathematics::Rings and Algebras ,Structure (category theory) ,Mathematics - Rings and Algebras ,Hopf algebra ,Algebra ,Rings and Algebras (math.RA) ,Morita contexts ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,Morita therapy ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Corings ,Algebra over a field ,16W30 ,Mathematics - Abstract
Using the theory of corings, we generalize and unify Morita contexts introduced by Chase and Sweedler, Doi, and Cohen, Fischman and Montgomery. We discuss when the contexts are strict. We apply our theory corings arising from entwining structures, and this leads us to the notion of cleft entwining structure., 27 pages
- Published
- 2002
37. Co-Frobenius corings and adjoint functors
- Author
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Joost Vercruysse and Miodrag Cristian Iovanov
- Subjects
Algebra ,Pure mathematics ,Algebra and Number Theory ,Mathematics::K-Theory and Homology ,Coalgebra ,Mathematics::Category Theory ,Base (topology) ,Adjoint functors ,Mathematics - Abstract
We study co-Frobenius and more generally quasi-co-Frobenius corings over arbitrary base rings and over PF base rings in particular. We generalize some results about co-Frobenius and quasi-co-Frobenius coalgebras to the case of non-commutative base rings and give several new characterizations for co-Frobenius and more generally quasi-co-Frobenius corings, some of them are new even in the coalgebra situation. We construct Morita contexts to study Frobenius properties of corings and a second kind of Morita contexts to study adjoint pairs. Comparing both Morita contexts, we obtain our main result that characterizes quasi-co-Frobenius corings in terms of a pair of adjoint functors (F,G) such that (G,F) is locally quasi-adjoint in a sense defined in this note.
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