arXiv subjects

Joost Vercruysse

Publications and source records attributed to Joost Vercruysse.

At least 37 records · Page 2Linked to original sources

arXiv2020

Equivalences of (co)module algebra structures over Hopf algebras

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. We apply support equivalence in the study of the asymptotic behaviour of codimensions of H-identities and, in particular, to the analogue (formulated by Yu. A. Bahturin) of Amitsur's conjecture, which was originally concerned with ordinary polynomial identities. As an example we prove this analogue for all unital H-module structures on the algebra $F[x]/(x^2)$ of dual numbers.

math.RA↗

arXiv2020

Oplax Hopf Algebras

We introduce the notion of an oplax Hopf monoid in any braided monoidal bicategory, generalizing that of a Hopf monoid in a braided monoidal category in an appropriate way. We show that Hopf V-categories introduced in [BCV16] are a particular type of oplax Hopf monoids in the monoidal bicategory Span|V described in [Böh17]. Finally, we introduce Frobenius V-categories as the Frobenius objects in the same monoidal bicategory.

math.CT↗

arXiv2020

Partial and global representations of finite groups

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.

math.RT↗

arXiv2020

BiHom Hopf algebras viewed as Hopf monoids

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.

math.QA↗

arXiv2020

On the duality of generalized Lie and Hopf algebras

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 $Q(H)^*\cong P(H^\circ)$, where $Q(H)^*$ is the dual Lie algebra of the Lie coalgebra of indecomposables of $H$, and $P(H^\circ)$ is the Lie algebra of primitive elements of the Sweedler dual of $H$. We apply our theory to Turaev's Hopf group-(co)algebras.

math.RA↗

arXiv2019

Geometrically Partial actions

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.

math.RA↗

arXiv2019

Dilations of partial representations of Hopf algebras

We introduce the notion of a dilation for a partial representation (i.e. a partial module) of a Hopf algebra, which in case the partial representation origins from a partial action (i.e.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.

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arXiv2018

A semi-abelian extension of a theorem by Takeuchi

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.

math.CT↗

arXiv2017

Split extension classifiers in the category of cocommutative Hopf algebras

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.

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arXiv2015

A torsion theory in the category of cocommutative Hopf algebras

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.

math.CT↗

arXiv2015

Dual Constructions for Partial Actions of Hopf Algebras

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.

math.RA↗

arXiv2013

Lie monads and dualities

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.

math.RA↗

arXiv2013

Partial Representations of Hopf Algebras

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.

math.RA↗

arXiv2012

A Note on the categorification of Lie algebras

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.

math.RA↗

arXiv2012

Hopf algebras---Variant notions and reconstruction theorems

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.

math.RA↗

arXiv2009

The Eilenberg-Moore category and a Beck-type theorem for a Morita context

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.

math.CT↗

arXiv2008

Morita theory for coring extensions and cleft bicomodules

A Morita context is constructed for any comodule of a coring and, more generally, for an $L$-$\cC$ bicomodule $Σ$ 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 $Σ$ 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 $Σ$ -- implying strictness of the associated Morita context -- is introduced. It is shown to be equivalent to being a Galois $\cC$-comodule and isomorphic to $\End^\cC(Σ)\otimes_{L} \cD$, in the category of left modules for the ring $\End^\cC(Σ)$ 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.

math.RA↗

arXiv2008

Morita theory of comodules over corings

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 $Σ$ 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 $Σ$ 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 $^\cC(Σ)$. For a firmly projective comodule of a coseparable coring we prove a strong structure theorem assuming only surjectivity of the canonical map.

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