and tensor categories New trends in Hopf...

New trends in Hopf algebrasand tensor categories2-5 JUNE 2015

Royal Flemish Academy of Belgium for Science and the Arts – Brussels

List of participants

Program

Book of abstracts

background picture: Lin ChangChih

List of participants(Speakers are indicated with an asterisk)

(1) Ana Agore* Vrije Universiteit Brussel, Belgium(2) Nicolas Andruskiewitsch* Universidad Nacional de Cordoba, Argentina(3) Ivan Angiono MPIM, Germany and Universidad Nacional de Cordoba, Argentina(4) Alessandro Ardizzoni* University of Turin, Italy(5) Eliezer Batista* Universidade Federal de Santa Catarina, Brazil(6) Gabriella Bohm* Wigner Research Centre for Physics, Hungary(7) Tomasz Brzezinski* Swansea University, UK(8) Hoan-Phung Bui Universite Libre de Bruxelles, Belgium(9) Daniel Bulacu* University of Bucharest, Romania

(10) Sebastian Burciu* Institute of Mathematics of Romanian Academy, Romania(11) Frederik Caenepeel Universiteit Antwerpen, Belgium(12) Stefaan Caenepeel Vrije Universiteit Brussel, Belgium(13) Giovanna Carnovale* University of Padova, Italy(14) Huixiang Chen* Yangzhou University, China(15) Kenny De Commer* Vrije Universiteit Brussel, Belgium(16) Laiachi El Kaoutit* University of Granada, Spain(17) Fatima Zahra El Khamouri University Hassan II, Morocco(18) Timmy Fieremans* Vrije Universiteit Brussel, Belgium(19) Gaston Andres Garcia* Universidad Nacional de La Plata, Argentina(20) Alexey Gordienko* Vrije Universiteit Brussel, Belgium(21) Isar Goyvaerts* University of Turin, Italy(22) Marino Gran Universite catholique de Louvain, Belgium(23) Miodrag Iovanov* University of Iowa, US(24) George Janelidze* University of Cape Town, South Africa(25) Geoffrey Janssens Vrije Universiteit Brussel, Belgium(26) Jiawei Hu Universite Libre de Bruxelles, Belgium(27) Lars Kadison* niversity of Porto, Portugal(28) Gabriel Kadjo* Universite catholique de Louvain, Belgium(29) Ulrich Kraehmer* University of Glasgow, UK(30) Victoria Lebed* University of Nantes, France(31) Simon Lentner* University of Hamburg, Germany(32) Esperanza Lopez Centella* University of Granada, Spain(33) Laura Martın-Valverde University of Almerıa, Spain(34) Ehud Meir* Center for Symmetry and Deformation, University of Copenhagen, Den-

mark

New trends in Hopf algebras and tensor categories 3

(35) Claudia Menini* Universita di Ferrara, Italy(36) Gigel Militaru* University of Bucharest, Romania(37) Laura Nastasescu University of Bucharest and ”Simion Stoilow” Institute of Math-

ematics, Romania(38) Theo Raedschelders* Vrije Universiteit Brussel, Belgium(39) Ana Rovi* University of Glasgow, UK(40) Paolo Saracco* University of Turin, Italy(41) Peter Schauenburg* Universite de Bourgogne, France(42) Yorck Sommerhauser* State University of New York at Buffalo, US(43) Tim Van der Linden* Universite catholique de Louvain, Belgium(44) Fred Van Oystaeyen Universiteit Antwerpen, Belgium(45) Joost Vercruysse Universite libre de Bruxelles, Belgium(46) Sara Westreich* Bar Ilan University, Israel(47) Yuping Yang Universiteit Hasselt, Belgium(48) Zhankui Xiao Universiteit Hasselt, Belgium(49) Yinhuo Zhang Universiteit Hasselt, Belgium

4 New trends in Hopf algebras and tensor categories

Program

Tuesday June 2

09.00-10.00 Coffee and registration

10.00-10.30 Nicolas Andruskiewitsch (Universidad Nacional de Cordoba, Argentina)Pointed Hopf algebras with finite Gelfand-Kirillov dimension

10.35-11.05 Ehud Meir (University of Copenhagen, Denmark)Finite dimensional Hopf algebras and invariant theory

11.05-11.35 Coffee

11.35-12.05 Giovanna Carnovale (University of Padova, Italy)On Finite-dimensional pointed Hopf algebras over finite simple groups

12.10-12.40 Gaston Andres Garcia (Universidad Nacional de La Plata, Argentina)On collapsing simple racks

12.40-14.30 Lunch

14.30-15.00 Alessandro Ardizzoni (University of Turin, Italy)Cohomology and Coquasi-bialgebras in Yetter-Drinfeld Modules - Part 1

15.05-15.35 Claudia Menini (University of Ferrara, Italy)Cohomology and Coquasi-bialgebras in Yetter-Drinfeld Modules - Part 2

15.35-16.15 Coffee

16.15-16.45 Paolo Saracco (University of Turin, Italy)On the Structure Theorem for Quasi-Hopf Bimodules

16.50-17.20 Theo Raedschelders (Vrije Universiteit Brussel, Belgium)Representation theory of universal coacting Hopf algebras

17.30-19.30 Welcome reception


Wednessday June 3

09.15-09.45 George Janelidze (University of Cape Town, South Africa)Commutative Hopf algebras in the classical, differential, and difference Galoistheories

09.50-10.20 Lars Kadison (University of Porto, Portugal)Invariants of Morita equivalent ring extensions

10.20-10.50 Coffee

10.50-11.20 Sarah Westreich (Bar Ilan University, Israel)Commutators, counting functions and probabilistically nilpotent Hopf alge-bras

11.25-11.55 Gigel Militaru (University of Bucharest, Romania)Jacobi and Poisson algebras

12.00-12.30 Ana Agore (Vrije Universiteit Brussel, Belgium)On the classification of bicrossed products of Hopf algebras

12.30-14.30 Lunch

14.30-15.00 Daniel Bulacu (University of Bucharest, Romania)On Frobenius and separable algebra extensions in monoidal categories

15.05-15.35 Huixiang Chen (Yangzhou University, China)Monoidal categories of finite rank

15.35-16.05 Coffee

16.05-16.35 Tim Van der Linden (Universite catholique de Louvain, Belgium)Tensors via commutators

16.40-17.10 Gabriel Kadjo (Universite catholique de Louvain, Belgium)A torsion theory in the category of cocommutative Hopf algebras


Thursday June 4

09.15-09.45 Gabriella Bohm (Wigner Research Centre for Physics, Hungary)Multiplier bialgebras and Hopf algebras in braided monoidal categories

09.50-10.20 Yorck Sommerhauser (University at Buffalo, US)Triviality in Yetter-Drinfel’d Hopf Algebras

10.20-10.50 Coffee

10.50-11.20 Miodrag Iovanov (University of Iowa, US)Generators for coalgebras and universal constructions

11.25-11.55 Sebastian Burciu (Institute of Mathematics of Romanian Academy, Romania)On the irreducible modules of semisimple Drinfeld doubles and their fusionrules

12.00-12.30 Victoria Lebed (University of Nantes, France)On braidings, Yetter-Drinfel’d modules, crossed modules, and self-distributivity

12.30-14.30 Lunch

14.30-15.00 Tomasz Brzezinski (Swansea University, UK)Covariant bialgebras

15.05-15.35 Esperanza Lopez Centella (Universidad de Granada, Spain)Weak multiplier bialgebras

15.35-16.05 Coffee

16.05-16.35 Ulrich Kraehmer (University of Glasgow, UK)Cyclic homology arising from adjunctions

16.40-17.10 Ana Rovi (University of Glasgow, UK)Lie-Rinehart algebras, Hopf algebroids with and without antipodes

19.30 Conference dinner at “La Chaloupe d’Or”


Friday June 5

09.15-09.45 Peter Schauenburg (Institut de Mathematiques de Bourgogne, France)Finite Hopf algebroids, their module categories, and (self-)duality

09.50-10.20 Isar Goyvaerts (University of Turin, Italy)Some invariants for finite isocategorical groups

10.20-10.50 Coffee

10.50-11.20 Eliezer Batista (Universidade Federal de Santa Catarina, Brazil)Dual constructions for partial Hopf actions

11.25-11.55 Laiachi El Kaoutit (University of Granada, Spain)New characterizations of geometrically transitive Hopf algebroids

12.00-12.30 Timmy Fieremans (Vrije Universiteit Brussel, Belgium)Modules over pseudomonoids

12.30-14.30 Lunch

14.30-15.00 Kenny De Commer (Vrije Universiteit Brussel, Belgium)Partial compact quantum groups

15.05-15.35 Simon Lentner (University of Hamburg, Germany)Quantum groups and logarithmic conformal field theories

15.35-16.05 Coffee

16.05-16.35 Alexey Gordienko (Vrije Universiteit Brussel, Belgium)Hopf algebra actions and related polynomial identities


List of abstracts

Ana Agore (Vrije Universiteit Brussel, Belgium)On the classification of bicrossed products of Hopf algebras

Abstract. Let A and H be two Hopf algebras. We shall classify up to an isomorphismthat stabilizes A all Hopf algebras E that factorize through A and H by a cohomologicaltype object H2(A,H). Equivalently, we classify up to a left A-linear Hopf algebra isomor-phism, the set of all bicrossed products A ./ H associated to all possible matched pairs ofHopf algebras (A,H, /, .) that can be defined between A and H. In the construction ofH2(A,H) the key role is played by special elements of CoZ1(H,A)× Aut 1

CoAlg(H), where

CoZ1(H,A) is the group of unitary cocentral maps and Aut 1CoAlg(H) is the group of unitary

automorphisms of the coalgebra H. Among several applications and examples, all bicrossedproducts H4 ./ k[Cn] are described by generators and relations and classified: they arequantum groups at roots of unity H4n, ω which are classified by pure arithmetic propertiesof the ring Zn. The Dirichlet’s theorem on primes is used to count the number of types ofisomorphisms of this family of 4n-dimensional quantum groups. As a consequence of ourapproach the group Aut Hopf(H4n, ω) of Hopf algebra automorphisms is fully described.

Nicolas Andruskiewitsch (Universidad Nacional de Cordoba, Argentina)Pointed Hopf algebras with finite Gelfand-Kirillov dimension

Abstract. I will report on work in progress on the classification of pointed Hopf algebraswith finite Gelfand-Kirillov dimension and abelian group.This is joint work with I. Angiono and I. Heckenberger.

Alessandro Ardizzoni (University of Turin, Italy)Cohomology and Coquasi-bialgebras in Yetter-Drinfeld Modules - Part 1

Abstract. This talk deals with coquasi-bialgebras Q in the pre-braided monoidal categoryYD of Yetter-Drinfeld modules over a Hopf algebra H. Using Hochschild cohomology inYD, we will investigate whether Q is gauge equivalent to a braided bialgebra in YD in thecase when Q is connected and H is both semisimple and cosemisimple. A second talk,continuing this subject, will be delivered by Claudia Menini.

Eliezer Batista (Universidade Federal de Santa Catarina, Brazil)Dual constructions for partial Hopf actions

Abstract. In this talk, we introduce the construction of dual objects associated to partialactions of a Hopf algebra, such as partial module coalgebras and partial comodule coalgebras.We explore some dualities between these structures and their associated Hopf algebroids.(Joint work with Joost Vercruysse.)


Gabriella Bohm (Wigner Research Centre for Physics, Hungary)Multiplier bialgebras and Hopf algebras in braided monoidal categories

Abstract. A bialgebra A over a field or, more generally, in any braided monoidal categorycan equivalently be described without referring separately to the multiplication µ : A⊗A→A and the comultiplication ∆ : A → A ⊗ A; just in terms of the unit, the counit and theso-called fusion morphism

A⊗ A ∆⊗id // A⊗ A⊗ Aid⊗µ // A⊗ A .

This treatment has the advantage of applicability also in the absence of a unit and a propercomultiplication; as Van Daeles approach to multiplier Hopf algebras shows.Based on the use of counital (but no longer unital) fusion morphisms, we work out thetheory of multiplier bialgebras and multiplier Hopf algebras in arbitrary braided monoidalcategories.The talk is based on a joint work with Stephen Lack (Macquarie University, Sydney).

Tomasz Brzezinski (Swansea University, UK)Covariant bialgebras

Abstract Infinitesimal bialgebras were introduced by Marcelo Aguiar at the turn of thecenturies; an infinitesimal bialgebra is defined as an associative algebra with coassociativecomultiplication that is a derivation. Quastiriangular infinitesimal bialgebars arise from so-lutions of classical associative Yang-Baxter equations and thus are related to Rota-Baxteralgebras. In this talk we discuss covariant bialgebras, defined as associative algebras withcoproduct that is a covariant derivation (with respect to a pair of derivations). These al-gebras are related to Yang-Baxter pairs and Rota-Baxter systems. We also describe someelementary facts about covariant modules the representation category of covariant bialge-bras.

Daniel Bulacu (University of Bucharest, Romania)On Frobenius and separable algebra extensions in monoidal categories

Abstract. We characterize Frobenius and separable monoidal algebra extensions i : R→S in terms given by R and S. For instance, under some conditions, we show that theextension is Frobenius, respectively separable, if and only if S is a Frobenius, respectivelyseparable, algebra in the category of bimodules over R. In the case when R is separable weshow that the extension is separable if and only if S is a separable algebra. Similarly, in thecase when R is Frobenius and separable in a sovereign monoidal category we show that theextension is Frobenius if and only if S is a Frobenius algebra and the restriction at R of itsNakayama automorphism is equal to the Nakayama automorphism of R.This talk is based on a joint work with Blas Torrecillas.

Sebastian Burciu (Institute of Mathematics of Romanian Academy, Romania)On the irreducible modules of semisimple Drinfeld doubles and their fusion rules

Abstract. In this talk we present a new description for the irreducible representations ofa Drinfeld double of a semisimple Hopf algebra, as induced modules from some special Hopfsubalgebras. We also give a formula for the decomposition in direct sum of simple modules


for a tensor product of two such modules. The talk is based on New examples of the Greenfunctors arising from representation theory of semisimple Hopf algebras J. Lond. Math. Soc.89, (1), (2014), 97-113 and another work in progress of the author.

Giovanna Carnovale (University of Padova, Italy)On Finite-dimensional pointed Hopf algebras over finite simple groups

Abstract. The classification of finite dimensional pointed Hopf algebras is based on theunderstanding of related Nichols algebras. If the group G of grouplikes is fixed, the latter canbe carried over in terms of conjugacy classes in G and representations of the correspondingcentralizer. There exist criteria on a conjugacy class ensuring that the attached Nicholsalgebras are infinite dimensional for every representation of the centralizer. In this talk wewill focus on how to apply those criteria to conjugacy classes in simple groups of Lie type.Based on a joint project with N. Andruskiewitsch and G. A. Garcia.

Huixiang Chen (Yangzhou University, China)Monoidal categories of finite rank

Abstract. In the study of monoidal categories, the representation (or Green) ring isa very important invariant (under monoidal equivalence) describing how tensor productsdecompose. However, this invariant is not enough to determine the monoidal category. So,the natural question is, next to the Green ring invariant, what else invariants we need in orderto determine the monoidal category. In this talk, we introduce a new invariant of a monoidalcategory of finite rank, which gives us the necessary information about morphisms betweenthe objects. This invariant is the Auslander algebra of a monoidal category. We show thata monoidal category of finite rank is uniquely determined, up to monoidal equivalence, byits two invariants: the Green ring and the Auslander algebra, together with the associator.(This is a joint work with Yinhuo Zhang.)

Kenny De Commer (Vrije Universiteit Brussel, Belgium)Partial compact quantum groups

Abstract. T. Hayashi introduced the notion of ’compact quantum group of face type’,which is to be seen as a compact quantum groupoid with a finite object set. In this talk,we introduce the notion of ’partial compact quantum group’, which is a generalization ofHayashi’s definition to the case of an infinite object set. Partial compact quantum groupscan for example be constructed from any rigid tensor C∗-category, and from any ergodicaction of a compact quantum group. We give some details on a concrete example relatedto the dynamical quantum SU(2) group. This is joint work with T. Timmermann.

Laiachi El Kaoutit (University of Granada, Spain)New characterizations of geometrically transitive Hopf algebroids

Abstract. The aim of this talk is to show that a commutative flat Hopf algebroid witha non trivial base ring and a non empty characters groupoid, is geometrically transitive(GT) if and only if each base change morphism is a weak equivalence, which in particular,


implies that each extension of the base ring is Landweber exact. This in fact is the algebro-geometric counterpart of the well known characterization of transitive groupoids by meansof weak equivalences.We also show that the characters groupoid of GT Hopf algebroid is transitive if and only ifany two isotropy Hopf algebras are conjugated. If time allows, several others properties ofthese Hopf algebroids, in relation with transitive groupoids, will be also displayed.

Fieremans (Vrije Universiteit Brussel, Belgium)Modules over pseudomonoids

Abstract. We introduce the notion of (bi)modules of pseudomonoids in a monoidalbicagegory. Using this we generalize the definition of the Drinfeld center of a monoidalcategory to pseudomonoids.Based on a joint work with Stef Caenepeel.

Gaston Andres Garcia (Universidad Nacional de La Plata, Argentina)On collapsing simple racks

Abstract. The study of finite dimensional pointed Hopf algebras can be approachedthrough the analysis of the Nichols algebras associated with simple Yetter-Drinfeld modules.The latter can be described in terms of racks and rack cocycles. In this direction it is crucialto establish criteria on a rack to make sure that the attached Nichols algebra is infinitedimensional for every cocycle. In this talk we will focus on the translation of existing resultsinto criteria which allow inductive reasoning.This talk is based on a joint project with N. Andruskiewitsch and G. Carnovale.

Alexey Gordienko (Vrije Universiteit Brussel, Belgium)Hopf algebra actions and related polynomial identities

Abstract. Study of polynomial identities is an important aspect of study of algebrasthemselves. We will discuss asymptotic behaviour of polynomial H-identities in H-modulealgebras and related problems in the structure theory of H-module algebras. In particular,we will classify finite dimensional associative algebras simple with respect to a Taft algebraaction.

Isar Goyvaerts (University of Turin, Italy)Some invariants for finite isocategorical groups

Abstract. Let G be a finite group and let Rep-G denote the category of finite-dimensionalcomplex representations of G. Two finite groups are called (C)-isocategorical if Rep-G isequivalent to Rep-H as a tensor category (without regard of the symmetric structure). Ingeneral it seems not always easy a question to determine whether two given finite groupsare isocategorical or not. In this talk, I would like to sketch some invariants for isocategor-ical groups and illustrate how these invariants can be useful to study the above-mentionedquestion for some (rather small) groups. Amongst other things, I would like to report on arecent joint work with Ehud Meir.


Miodrag Iovanov (University of Iowa, US)Generators for coalgebras and universal constructions

Abstract. We investigate cofree coalgebras, and limits and colimits of coalgebras in someabelian monoidal categories of interest, such as bimodules over a ring, and modules and co-modules over a bialgebra or Hopf algebra. We find concrete generators for the categories ofcoalgebras in these monoidal categories, and explicitly construct cofree coalgebras, productsand limits of coalgebras in each case. In particular, this answers an open question of A.Agoreon the existence of cofree corings and complements work of H.Porst; it also constructs thecofree (co)module coalgebra on a B-(co)module, for a bialgebra B. This is a joint work withAdnan Abdulwahid, PhD student, University of Iowa.

George Janelidze (University of Cape Town, South Africa)Commutative Hopf algebras in the classical, differential, and difference Galois theories

Abstract. This is a direction of abstract Galois theory that begins with A. Grothendieckin 60s, continues with A. R. Magid in 70s, and arrives at its full generality with the authorin 80s. The talk is devoted particularly to the evolution of the notion of Galois group,which eventually becomes an internal group in an abstract category, and therefore becomesa commutative Hopf algebra in important special cases. Specifically, for a normal (=Galois)extension E/K with the tensor square S, in the classical Galois theory it becomes the Hopfalgebra of idempotents in S, while in the differential and difference Galois theories it becomesthe Hopf algebra of constants in S. There is, on the other hand, an important distinction: inthe differential and difference cases only the very first step is made, and there is a numberof categorical conditions to be analysed further, including the definition of normality.

Lars Kadison (University of Porto, Portugal)Invariants of Morita equivalent ring extensions

Abstract. In a tensor category, we are interested in when and if tensor powers of an algebraor coalgebra stabilise, which we call its depth, possibly infinite. We observe that depth of aring extension is a Morita invariant among many other properties like QF, separable, Galois,centraliser isoclass. We reduce certain depth problems such as that of a finite-dimensionalHopf algebra in a smash product or in a bigger Hopf algebra from taking place in a ”toolarge” bimodule category down to a finite tensor category of Etingof-Ostrik.Based on a joint work with Alberto Hernandez.

Gabriel Kadjo (Universite catholique de Louvain, Belgium)A torsion theory in the category of cocommutative Hopf algebras.

Abstract. We prove that the category of cocommutative Hopf K-algebras, over a fieldK of characteristic zero, is a semi-abelian category. Moreover, we show that this categorycontains a torsion theory whose torsion-free and torsion parts are given by the category ofgroups and by the category of Lie K-algebras, respectively.


This is based on joint work with M. Gran and J. Vercruysse.

Ulrich Kraehmer (University of Glasgow, UK)Cyclic homology arising from adjunctions

Abstract. Given a monad and a comonad, one obtains a distributive law between themfrom lifts of one through an adjunction for the other. In particular, this yields for anybialgebroid the Yetter-Drinfeld distributive law between the comonad given by a modulecoalgebra and the monad given by a comodule algebra. It is this self-dual setting thatreproduces the cyclic homology of associative and of Hopf algebras in the monadic frameworkof Bohm and Stefan. In fact, their approach generates two duplicial objects and morphismsbetween them which are mutual inverses if and only if the duplicial objects are cyclic.Joint work with Niels Kowalzig and Paul Slevin.

Victoria Lebed (University of Nantes, France)On braidings, Yetter-Drinfel’d modules, crossed modules, and self-distributivity

Abstract. The Yang-Baxter equation plays a fundamental role in various areas of math-ematics. Its solutions are built, among others, from Yetter-Drinfel’d modules over a Hopfalgebra, from self-distributive structures, and from crossed modules of groups. This talkunifies these three sources of solutions inside the framework of Yetter-Drinfel’d modulesover a braided system. We also present a new source of solutions, originating from crossedmodules of racks, and include it in the same framework. We discuss if these solutions stemfrom a braided monoidal category, discovering in the self-distributive case a non-strict tensorcategory with an interesting associator.This is a joint work with F. Wagemann.

Simon Lentner (University of Hamburg, Germany)Quantum groups and logarithmic conformal field theories

Abstract. I will review Lusztig’s quantum group of divided powers and also include somemore recent results about arbitrary roots of unity (e.g. even) and similar pointed Hopf alge-bras involving other Nichols algebras. This leads to a family of non-semisimple ”modular”categories with deep connection to affine Lie algebras at negative level and Lie groups infinite characteristic. Then I explain some open conjectures by Feigin, Gainutdinov, Semikha-tov, Tipunin, which describe how to realize these categories from a uniformly constructedfamily of vertex algebras.

Esperanza Lopez Centella (University of Granada, Spain)Weak multiplier bialgebras

Abstract. Weak Hopf algebras and multiplier Hopf algebras are generalizations of aHopf algebra in different directions. In the first ones, the compatibility between the algebraand the coalgebra structure is weakened; in the second ones, the underlying algebra is notsupposed to be unital and the comultiplication is multiplier-valued. Whereas (weak) Hopfalgebras are classically defined as (weak) bialgebras admitting the further structure of anantipode, in Van Daele (and Wang)s approach, (weak) multiplier Hopf algebras are defined


directly without considering the antipodeless situation of (weak) multiplier bialgebra. In thistalk we present a suitable notion of weak multiplier bialgebras, filling this conceptual gap.Our definition is supported by the fact that (assuming some further properties like regularityor fullness of the comultiplication), the most characteristic features of usual, unital, weakbialgebras extend to this generalization:

• There is a bijective correspondence between the weak bialgebra structures and theweak multiplier bialgebra structures on any unital algebra.• The multiplier algebra of a weak multiplier bialgebra contains two canonical com-

muting anti-isomorphic firm Frobenius algebras; the so-called base algebras. (In theroute, multiplier bialgebra is defined as the particular case when the base algebra istrivial; that is, it contains only multiples of the unit element.)• Appropriately defined modules over a (nice enough) weak multiplier bialgebra con-

stitute a monoidal category via the module tensor product over the base algebra.

We provide a notion of antipode for regular weak multiplier bialgebras, and sufficient andnecessary conditions for a weak multiplier bialgebra to be a weak multiplier Hopf algebra inthe sense of Van Daele and Wangs definition. We show a desired intermediate class betweenregular and arbitrary weak multiplier Hopf algebras, big enough to contain any unital weakHopf algebra and answering the questions left open by the aforementioned authors.

Ehud Meir (Center for Symmetry and Deformation, University of Copenhagen, Denmark)Finite dimensional Hopf algebras and invariant theory.

AbstractIn this talk I will present an approach to study finite dimensional semisimple Hopf algebras,based on invariant theory. I will explain why the isomorphism type of such an algebra canbe determined by a (very big) collection of invariant scalars, and will discuss the intuitivemeaning of (some of) these scalars. While in some cases they can be very easily interpreted,in some other cases their meaning is much less clear. I will explain the connection betweenthese invariants and questions about fields of definition, some open problems in Hopf alge-bra theory (Kaplansky’s sixth conjecture) and how one can use them in order to prove thatevery such algebra satisfies a certain finiteness condition (the existence of at most finitelymany Hopf orders). If time permits, I will also explain the connection with invariants of3-manifolds one receives in Topological Quantum Field Theory.

Claudia Menini (Universita di Ferrara, Italy)Cohomology and Coquasi-bialgebras in Yetter-Drinfeld Modules - Part 2

Abstract. Let A be a finite-dimensional Hopf algebra over a field of characteristic zerosuch that the coradical H of A is a sub-Hopf algebra (i.e. A has the dual Chevalley Prop-erty). We will investigate conditions in order that A is quasi-isomorphic to the Radford-Majidbosonization of some connected bialgebra E in the category YD of Yetter-Drinfeld modulesover H, by H.

Gigel Militaru (University of Bucharest, Romania)Jacobi and Poisson algebras

Abstract. Jacobi/Poisson algebras are algebraic counterparts of Jacobi/Poisson mani-folds. If we look at Poisson algebras as the ’differential’ version of Hopf algebras, then,


mutatis-mutandis, Jacobi algebras can be seen as generalizations of Poisson algebras in thesame way as weak Hopf algebras generalize Hopf algebras. We introduce representationsof a Jacobi algebra A and Frobenius Jacobi algebras as symmetric objects in the category.A characterization theorem for Frobenius Jacobi algebras is given in terms of integrals onJacobi algebras. For a vector space V a non-abelian cohomological type object JH2 (V, A)is constructed: it classifies all Jacobi algebras containing A as a subalgebra of codimensionequal to dim(V ). Several examples and applications are given.Based on a joint work with Ana Agore.

Theo Raedschelders (Vrije Universiteit Brussel, Belgium)Representation theory of universal coacting Hopf algebras

Abstract. For any Koszul, Artin-Schelter regular algebra A, we study the universal Hopfalgebra coacting on A, as considered by Manin. We provide a construction inspired byTannaka-Krein theory and prove that this Hopf algebra is quasi-hereditary as a coalgebra, soits representation theory is ”nice”. In this lecture we will focus on the example A = k[x, y],where many of the non-trivial features can be explicitly exhibited.Joint work with Michel Van den Bergh.

Ana Rovi (University of Glasgow, UK)Lie-Rinehart algebras, Hopf algebroids with and without antipodes

Abstract. We discuss the enveloping algebra of Lie-Rinehart algebras as a very rich classof examples of Hopf algebroids. Some of these Hopf algebroids do not admit an antipode,while others do. Our examples clarify the discrepancy between different definitions of Hopfalgebroids in the literature.

Paolo Saracco (University of Turin, Italy)On the Structure Theorem for Quasi-Hopf Bimodules

Abstract. It is known that the Structure Theorem for Hopf modules can be used tocharacterize Hopf algebras: a bialgebra H is a Hopf algebra (i.e. it is endowed with aso-called antipode) if and only if every Hopf module M can be decomposed in the formM coH ⊗H, where M coH denotes the space of coinvariant elements in M . The main aim ofthis talk is to show how this characterization could be extended to the framework of quasi-bialgebras by introducing the notion of preantipode and by proving a Structure Theorem forquasi-Hopf bimodules. As a consequence, some previous results as the Structure Theoremfor Hopf modules and the Hausser-Nill theorem for quasi-Hopf algebras, can be deducedfrom our Structure Theorem.This talk is based on the paper: P. Saracco, On the Structure Theorem for Quasi-HopfBimodules (arXiv:1501.06061).

Peter Schauenburg (Universite de Bourgogne, France)Finite Hopf algebroids, their module categories, and (self-)duality

Abstract. We give an intrinsic characterization of the module categories over finite Hopfalgebroids (apologizing for the use of this term for Takeuchi’s ×R-bialgebras satisfying a


Hopf-like property). Here a Hopf algebroid is called finite if it is finitely generated projectivewith respect to a certain one among its four module structures over the base algebra. For afinite-dimensional bialgebra over a base field it is well-known that its being Hopf is equivalentto the category of its finite-dimensional modules being rigid (i. e. admitting dual objects).For bialgebroids this requirement is too strong (in fact this is already the case for finitelygenerated projective bialgebras over a base ring). We will discuss the appropriate weakeningof the notion of rigidity that characterizes the module categories of finite Hopf algebroidsamong those of finite bialgebroids. Moreover, every abelian category satisfying our conditionallows the reconstruction of a finite Hopf algebroid having it as its module category. In manycases the Hopf algebroid can be chosen to be self-dual. Obviously the dual bialgebroid of aself-dual Hopf algebroid is itself Hopf. More generally we can show that the dual bialgebroidof any finite Hopf algebroid is itself Hopf.

Yorck Sommerhauser (State University of New York at Buffalo, US)Triviality in Yetter-Drinfel’d Hopf Algebras

Abstract. Usually, a Yetter-Drinfel’d Hopf algebra is not a Hopf algebra. Yetter-Drinfel’dHopf algebras that are ordinary Hopf algebras are called trivial; by a result of P. Schauenburg,this happens if and only if the quasisymmetry in the category of Yetter-Drinfel’d modulesaccidentally coincides with the ordinary flip of tensor factors on the second tensor power ofthe Yetter-Drinfel’d Hopf algebra.In certain situations, every Yetter-Drinfel’d Hopf algebra is trivial. In the talk, we considera semisimple Yetter-Drinfel’d Hopf algebra A over the group ring K[G] of a finite abeliangroup G, where K is an algebraically closed field of characteristic zero, and first brieflydiscuss the following triviality theorem:If A is commutative and its dimension is relatively prime to the order of G, then A is trivial.Even without relatively primeness assumptions, certain Yetter-Drinfel’d Hopf algebras are atleast partially trivial:If A is cocommutative and its dimension is greater than 1, then A contains a trivial Yetter-Drinfel’d Hopf subalgebra of dimension greater than 1.The main focus of the talk will be an outline of the proof of this partial triviality theorem.

Tim Van der Linden (Universite catholique de Louvain, Belgium)Tensors via commutators

Abstract. While the co-smash product of objects in a semi-abelian category may be usedas a formal commutator [3, 2], after Carboni and Janelidze’s paper [1] it has also been clearthat certain tensor products appear as co-smash products. This leads to an approach oftensor products via commutators. In my talk I will explain how the co-smash product ofobjects in the two-nilpotent core Nil2(X) of a semi-abelian category X determines a so-calledbilinear product on the abelian core Ab(X) of X. In certain homological applications, thismay then play the role of an intrinsic tensor product on X. (Joint work with Manfred Hartl.)

References

[1] A. Carboni and G. Janelidze, Smash product of pointed objects in lextensive categories, J. Pure Appl. Al-gebra 183 (2003), 27–43.

[2] M. Hartl and B. Loiseau, On actions and strict actions in homological categories, Theory Appl. Categ.27 (2013), no. 15, 347–392.

[3] S. Mantovani and G. Metere, Normalities and commutators, J. Algebra 324 (2010), no. 9, 2568–2588.


Sara Westreich (Bar Ilan University, Israel)Commutators, counting functions and probalistically nilpotent Hopf algebras

Abstract. Let H be a semisimple Hopf algebra over an algebraically closed field k ofcharacteristic 0. We define Hopf algebraic analogues of commutators and their generaliza-tions and show how they are related to H ′, the Hopf algebraic analogue of the commutatorsubgroup. We introduce a family of central elements of H ′ which on one hand generateH ′ and on the other hand give rise to a family of functionals on H: When H = kG; Ga finite group, these functionals are counting functions on G. We define nilpotency via adescending chain of commutators and give a criterion for nilpotency via a family of centralinvertible elements. These elements can be obtained from a commutator matrix A whichdepends only on the Grothendieck ring of H.When H is almost cocommutative we introduce a probabilistic method. We prove that everysemisimple quasitriangular Hopf algebra is probabilistically nilpotent.

and tensor categories New trends in Hopf...

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