New trends in Hopf algebras and tensor categories

Transcription

1 New trends in Hopf algebras and tensor categories 2-5 JUNE 2015 Royal Flemish Academy of Belgium for Science and the Arts Brussels List of participants Program Book of abstracts background picture: Lin ChangChih

2 List of participants (Speakers are indicated with an asterisk) (1) Ana Agore* Vrije Universiteit Brussel, Belgium (2) Nicolás Andruskiewitsch* Universidad Nacional de Córdoba, Argentina (3) Ivan Angiono MPIM, Germany and Universidad Nacional de Córdoba, Argentina (4) Alessandro Ardizzoni* University of Turin, Italy (5) Eliezer Batista* Universidade Federal de Santa Catarina, Brazil (6) Gabriella Böhm* Wigner Research Centre for Physics, Hungary (7) Tomasz Brzezinski* Swansea University, UK (8) Hoan-Phung Bui Université 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 Université 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 Université Libre de Bruxelles, Belgium (27) Lars Kadison* niversity of Porto, Portugal (28) Gabriel Kadjo* Université 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 López Centella* University of Granada, Spain (33) Laura Martín-Valverde University of Almería, Spain (34) Ehud Meir* Center for Symmetry and Deformation, University of Copenhagen, Denmark

3 New trends in Hopf algebras and tensor categories 3 (35) Claudia Menini* Università di Ferrara, Italy (36) Gigel Militaru* University of Bucharest, Romania (37) Laura Nastasescu University of Bucharest and Simion Stoilow Institute of Mathematics, 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* Université de Bourgogne, France (42) Yorck Sommerhäuser* State University of New York at Buffalo, US (43) Tim Van der Linden* Université catholique de Louvain, Belgium (44) Fred Van Oystaeyen Universiteit Antwerpen, Belgium (45) Joost Vercruysse Université 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 4 New trends in Hopf algebras and tensor categories Program Tuesday June Coffee and registration Nicolás Andruskiewitsch (Universidad Nacional de Córdoba, Argentina) Pointed Hopf algebras with finite Gelfand-Kirillov dimension Ehud Meir (University of Copenhagen, Denmark) Finite dimensional Hopf algebras and invariant theory Coffee Giovanna Carnovale (University of Padova, Italy) On Finite-dimensional pointed Hopf algebras over finite simple groups Gaston Andres Garcia (Universidad Nacional de La Plata, Argentina) On collapsing simple racks Lunch Alessandro Ardizzoni (University of Turin, Italy) Cohomology and Coquasi-bialgebras in Yetter-Drinfeld Modules - Part Claudia Menini (University of Ferrara, Italy) Cohomology and Coquasi-bialgebras in Yetter-Drinfeld Modules - Part Coffee Paolo Saracco (University of Turin, Italy) On the Structure Theorem for Quasi-Hopf Bimodules Theo Raedschelders (Vrije Universiteit Brussel, Belgium) Representation theory of universal coacting Hopf algebras Welcome reception

5 New trends in Hopf algebras and tensor categories 5 Wednessday June George Janelidze (University of Cape Town, South Africa) Commutative Hopf algebras in the classical, differential, and difference Galois theories Lars Kadison (University of Porto, Portugal) Invariants of Morita equivalent ring extensions Coffee Sarah Westreich (Bar Ilan University, Israel) Commutators, counting functions and probabilistically nilpotent Hopf algebras Gigel Militaru (University of Bucharest, Romania) Jacobi and Poisson algebras Ana Agore (Vrije Universiteit Brussel, Belgium) On the classification of bicrossed products of Hopf algebras Lunch Daniel Bulacu (University of Bucharest, Romania) On Frobenius and separable algebra extensions in monoidal categories Huixiang Chen (Yangzhou University, China) Monoidal categories of finite rank Coffee Tim Van der Linden (Université catholique de Louvain, Belgium) Tensors via commutators Gabriel Kadjo (Université catholique de Louvain, Belgium) A torsion theory in the category of cocommutative Hopf algebras

6 6 New trends in Hopf algebras and tensor categories Thursday June Gabriella Böhm (Wigner Research Centre for Physics, Hungary) Multiplier bialgebras and Hopf algebras in braided monoidal categories Yorck Sommerhäuser (University at Buffalo, US) Triviality in Yetter-Drinfel d Hopf Algebras Coffee Miodrag Iovanov (University of Iowa, US) Generators for coalgebras and universal constructions Sebastian Burciu (Institute of Mathematics of Romanian Academy, Romania) On the irreducible modules of semisimple Drinfeld doubles and their fusion rules Victoria Lebed (University of Nantes, France) On braidings, Yetter-Drinfel d modules, crossed modules, and selfdistributivity Lunch Tomasz Brzezinski (Swansea University, UK) Covariant bialgebras Esperanza López Centella (Universidad de Granada, Spain) Weak multiplier bialgebras Coffee Ulrich Kraehmer (University of Glasgow, UK) Cyclic homology arising from adjunctions Ana Rovi (University of Glasgow, UK) Lie-Rinehart algebras, Hopf algebroids with and without antipodes Conference dinner at La Chaloupe d Or

7 New trends in Hopf algebras and tensor categories 7 Friday June Peter Schauenburg (Institut de Mathématiques de Bourgogne, France) Finite Hopf algebroids, their module categories, and (self-)duality Isar Goyvaerts (University of Turin, Italy) Some invariants for finite isocategorical groups Coffee Eliezer Batista (Universidade Federal de Santa Catarina, Brazil) Dual constructions for partial Hopf actions Laiachi El Kaoutit (University of Granada, Spain) New characterizations of geometrically transitive Hopf algebroids Timmy Fieremans (Vrije Universiteit Brussel, Belgium) Modules over pseudomonoids Lunch Kenny De Commer (Vrije Universiteit Brussel, Belgium) Partial compact quantum groups Simon Lentner (University of Hamburg, Germany) Quantum groups and logarithmic conformal field theories Coffee Alexey Gordienko (Vrije Universiteit Brussel, Belgium) Hopf algebra actions and related polynomial identities

8 8 New trends in Hopf algebras and tensor categories 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 isomorphism that stabilizes A all Hopf algebras E that factorize through A and H by a cohomological type object H 2 (A, H). Equivalently, we classify up to a left A-linear Hopf algebra isomorphism, the set of all bicrossed products A H associated to all possible matched pairs of Hopf algebras (A, H,, ) that can be defined between A and H. In the construction of H 2 (A, H) the key role is played by special elements of CoZ 1 (H, A) Aut 1 CoAlg (H), where CoZ 1 (H, A) is the group of unitary cocentral maps and Aut 1 CoAlg (H) is the group of unitary automorphisms of the coalgebra H. Among several applications and examples, all bicrossed products H 4 k[c n ] are described by generators and relations and classified: they are quantum groups at roots of unity H 4n, ω which are classified by pure arithmetic properties of the ring Z n. The Dirichlet s theorem on primes is used to count the number of types of isomorphisms of this family of 4n-dimensional quantum groups. As a consequence of our approach the group Aut Hopf (H 4n, ω ) of Hopf algebra automorphisms is fully described. Nicolás Andruskiewitsch (Universidad Nacional de Córdoba, Argentina) Pointed Hopf algebras with finite Gelfand-Kirillov dimension Abstract. I will report on work in progress on the classification of pointed Hopf algebras with 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 category YD of Yetter-Drinfeld modules over a Hopf algebra H. Using Hochschild cohomology in YD, we will investigate whether Q is gauge equivalent to a braided bialgebra in YD in the case 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 partial actions 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.)

9 New trends in Hopf algebras and tensor categories 9 Gabriella Böhm (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 category can 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 the so-called fusion morphism A A id A A A id µ A A. This treatment has the advantage of applicability also in the absence of a unit and a proper comultiplication; as Van Daeles approach to multiplier Hopf algebras shows. Based on the use of counital (but no longer unital) fusion morphisms, we work out the theory of multiplier bialgebras and multiplier Hopf algebras in arbitrary braided monoidal categories. The talk is based on a joint work with Stephen Lack (Macquarie University, Sydney). Tomasz Brzeziński (Swansea University, UK) Covariant bialgebras Abstract Infinitesimal bialgebras were introduced by Marcelo Aguiar at the turn of the centuries; an infinitesimal bialgebra is defined as an associative algebra with coassociative comultiplication that is a derivation. Quastiriangular infinitesimal bialgebars arise from solutions of classical associative Yang-Baxter equations and thus are related to Rota-Baxter algebras. In this talk we discuss covariant bialgebras, defined as associative algebras with coproduct that is a covariant derivation (with respect to a pair of derivations). These algebras are related to Yang-Baxter pairs and Rota-Baxter systems. We also describe some elementary facts about covariant modules the representation category of covariant bialgebras. 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 the extension is Frobenius, respectively separable, if and only if S is a Frobenius, respectively separable, algebra in the category of bimodules over R. In the case when R is separable we show that the extension is separable if and only if S is a separable algebra. Similarly, in the case when R is Frobenius and separable in a sovereign monoidal category we show that the extension is Frobenius if and only if S is a Frobenius algebra and the restriction at R of its Nakayama 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 of a Drinfeld double of a semisimple Hopf algebra, as induced modules from some special Hopf subalgebras. We also give a formula for the decomposition in direct sum of simple modules

10 10 New trends in Hopf algebras and tensor categories for a tensor product of two such modules. The talk is based on New examples of the Green functors arising from representation theory of semisimple Hopf algebras J. Lond. Math. Soc. 89, (1), (2014), 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 the understanding of related Nichols algebras. If the group G of grouplikes is fixed, the latter can be carried over in terms of conjugacy classes in G and representations of the corresponding centralizer. There exist criteria on a conjugacy class ensuring that the attached Nichols algebras are infinite dimensional for every representation of the centralizer. In this talk we will 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 is a very important invariant (under monoidal equivalence) describing how tensor products decompose. 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 order to determine the monoidal category. In this talk, we introduce a new invariant of a monoidal category of finite rank, which gives us the necessary information about morphisms between the objects. This invariant is the Auslander algebra of a monoidal category. We show that a monoidal category of finite rank is uniquely determined, up to monoidal equivalence, by its 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 of Hayashi s definition to the case of an infinite object set. Partial compact quantum groups can for example be constructed from any rigid tensor C -category, and from any ergodic action of a compact quantum group. We give some details on a concrete example related to 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 with a 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,

11 New trends in Hopf algebras and tensor categories 11 implies that each extension of the base ring is Landweber exact. This in fact is the algebrogeometric counterpart of the well known characterization of transitive groupoids by means of weak equivalences. We also show that the characters groupoid of GT Hopf algebroid is transitive if and only if any two isotropy Hopf algebras are conjugated. If time allows, several others properties of these 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 monoidal bicagegory. Using this we generalize the definition of the Drinfeld center of a monoidal category 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 approached through 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 crucial to establish criteria on a rack to make sure that the attached Nichols algebra is infinite dimensional for every cocycle. In this talk we will focus on the translation of existing results into 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 algebras themselves. We will discuss asymptotic behaviour of polynomial H-identities in H-module algebras 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 algebra action. 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-dimensional complex representations of G. Two finite groups are called (C)-isocategorical if Rep-G is equivalent to Rep-H as a tensor category (without regard of the symmetric structure). In general it seems not always easy a question to determine whether two given finite groups are isocategorical or not. In this talk, I would like to sketch some invariants for isocategorical groups and illustrate how these invariants can be useful to study the above-mentioned question for some (rather small) groups. Amongst other things, I would like to report on a recent joint work with Ehud Meir.

12 12 New trends in Hopf algebras and tensor categories Miodrag Iovanov (University of Iowa, US) Generators for coalgebras and universal constructions Abstract. We investigate cofree coalgebras, and limits and colimits of coalgebras in some abelian monoidal categories of interest, such as bimodules over a ring, and modules and comodules over a bialgebra or Hopf algebra. We find concrete generators for the categories of coalgebras in these monoidal categories, and explicitly construct cofree coalgebras, products and limits of coalgebras in each case. In particular, this answers an open question of A.Agore on the existence of cofree corings and complements work of H.Porst; it also constructs the cofree (co)module coalgebra on a B-(co)module, for a bialgebra B. This is a joint work with Adnan 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. Grothendieck in 60s, continues with A. R. Magid in 70s, and arrives at its full generality with the author in 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 becomes a 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 Hopf algebra of idempotents in S, while in the differential and difference Galois theories it becomes the Hopf algebra of constants in S. There is, on the other hand, an important distinction: in the differential and difference cases only the very first step is made, and there is a number of 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 algebra or coalgebra stabilise, which we call its depth, possibly infinite. We observe that depth of a ring 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-dimensional Hopf algebra in a smash product or in a bigger Hopf algebra from taking place in a too large bimodule category down to a finite tensor category of Etingof-Ostrik. Based on a joint work with Alberto Hernandez. Gabriel Kadjo (Université 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 field K of characteristic zero, is a semi-abelian category. Moreover, we show that this category 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.

13 New trends in Hopf algebras and tensor categories 13 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 them from lifts of one through an adjunction for the other. In particular, this yields for any bialgebroid the Yetter-Drinfeld distributive law between the comonad given by a module coalgebra and the monad given by a comodule algebra. It is this self-dual setting that reproduces the cyclic homology of associative and of Hopf algebras in the monadic framework of Böhm and Ştefan. In fact, their approach generates two duplicial objects and morphisms between 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 mathematics. Its solutions are built, among others, from Yetter-Drinfel d modules over a Hopf algebra, from self-distributive structures, and from crossed modules of groups. This talk unifies these three sources of solutions inside the framework of Yetter-Drinfel d modules over a braided system. We also present a new source of solutions, originating from crossed modules of racks, and include it in the same framework. We discuss if these solutions stem from a braided monoidal category, discovering in the self-distributive case a non-strict tensor category 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 some more recent results about arbitrary roots of unity (e.g. even) and similar pointed Hopf algebras 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 in finite characteristic. Then I explain some open conjectures by Feigin, Gainutdinov, Semikhatov, Tipunin, which describe how to realize these categories from a uniformly constructed family of vertex algebras. Esperanza López Centella (University of Granada, Spain) Weak multiplier bialgebras Abstract. Weak Hopf algebras and multiplier Hopf algebras are generalizations of a Hopf algebra in different directions. In the first ones, the compatibility between the algebra and the coalgebra structure is weakened; in the second ones, the underlying algebra is not supposed to be unital and the comultiplication is multiplier-valued. Whereas (weak) Hopf algebras are classically defined as (weak) bialgebras admitting the further structure of an antipode, in Van Daele (and Wang)s approach, (weak) multiplier Hopf algebras are defined

14 14 New trends in Hopf algebras and tensor categories directly without considering the antipodeless situation of (weak) multiplier bialgebra. In this talk 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 regularity or fullness of the comultiplication), the most characteristic features of usual, unital, weak bialgebras extend to this generalization: There is a bijective correspondence between the weak bialgebra structures and the weak multiplier bialgebra structures on any unital algebra. The multiplier algebra of a weak multiplier bialgebra contains two canonical commuting anti-isomorphic firm Frobenius algebras; the so-called base algebras. (In the route, multiplier bialgebra is defined as the particular case when the base algebra is trivial; that is, it contains only multiples of the unit element.) Appropriately defined modules over a (nice enough) weak multiplier bialgebra constitute 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 and necessary conditions for a weak multiplier bialgebra to be a weak multiplier Hopf algebra in the sense of Van Daele and Wangs definition. We show a desired intermediate class between regular and arbitrary weak multiplier Hopf algebras, big enough to contain any unital weak Hopf 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. Abstract In 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 can be determined by a (very big) collection of invariant scalars, and will discuss the intuitive meaning 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 between these invariants and questions about fields of definition, some open problems in Hopf algebra theory (Kaplansky s sixth conjecture) and how one can use them in order to prove that every such algebra satisfies a certain finiteness condition (the existence of at most finitely many Hopf orders). If time permits, I will also explain the connection with invariants of 3-manifolds one receives in Topological Quantum Field Theory. Claudia Menini (Università 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 zero such that the coradical H of A is a sub-hopf algebra (i.e. A has the dual Chevalley Property). We will investigate conditions in order that A is quasi-isomorphic to the Radford-Majid bosonization of some connected bialgebra E in the category YD of Yetter-Drinfeld modules over H, by H. Gigel Militaru (University of Bucharest, Romania) Jacobi and Poisson algebras Abstract. Jacobi/Poisson algebras are algebraic counterparts of Jacobi/Poisson manifolds. If we look at Poisson algebras as the differential version of Hopf algebras, then,

15 New trends in Hopf algebras and tensor categories 15 mutatis-mutandis, Jacobi algebras can be seen as generalizations of Poisson algebras in the same way as weak Hopf algebras generalize Hopf algebras. We introduce representations of 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 on Jacobi algebras. For a vector space V a non-abelian cohomological type object J H 2 (V, A) is constructed: it classifies all Jacobi algebras containing A as a subalgebra of codimension equal 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 Hopf algebra coacting on A, as considered by Manin. We provide a construction inspired by Tannaka-Krein theory and prove that this Hopf algebra is quasi-hereditary as a coalgebra, so its 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 class of 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 Hopf algebroids 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 to characterize Hopf algebras: a bialgebra H is a Hopf algebra (i.e. it is endowed with a so-called antipode) if and only if every Hopf module M can be decomposed in the form M coh H, where M coh denotes the space of coinvariant elements in M. The main aim of this talk is to show how this characterization could be extended to the framework of quasibialgebras by introducing the notion of preantipode and by proving a Structure Theorem for quasi-hopf bimodules. As a consequence, some previous results as the Structure Theorem for Hopf modules and the Hausser-Nill theorem for quasi-hopf algebras, can be deduced from our Structure Theorem. This talk is based on the paper: P. Saracco, On the Structure Theorem for Quasi-Hopf Bimodules (arxiv: ). Peter Schauenburg (Université de Bourgogne, France) Finite Hopf algebroids, their module categories, and (self-)duality Abstract. We give an intrinsic characterization of the module categories over finite Hopf algebroids (apologizing for the use of this term for Takeuchi s R -bialgebras satisfying a

16 16 New trends in Hopf algebras and tensor categories Hopf-like property). Here a Hopf algebroid is called finite if it is finitely generated projective with respect to a certain one among its four module structures over the base algebra. For a finite-dimensional bialgebra over a base field it is well-known that its being Hopf is equivalent to 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 finitely generated projective bialgebras over a base ring). We will discuss the appropriate weakening of the notion of rigidity that characterizes the module categories of finite Hopf algebroids among those of finite bialgebroids. Moreover, every abelian category satisfying our condition allows the reconstruction of a finite Hopf algebroid having it as its module category. In many cases the Hopf algebroid can be chosen to be self-dual. Obviously the dual bialgebroid of a self-dual Hopf algebroid is itself Hopf. More generally we can show that the dual bialgebroid of any finite Hopf algebroid is itself Hopf. Yorck Sommerhäuser (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 d Hopf 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 modules accidentally coincides with the ordinary flip of tensor factors on the second tensor power of the Yetter-Drinfel d Hopf algebra. In certain situations, every Yetter-Drinfel d Hopf algebra is trivial. In the talk, we consider a semisimple Yetter-Drinfel d Hopf algebra A over the group ring K[G] of a finite abelian group G, where K is an algebraically closed field of characteristic zero, and first briefly discuss 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 at least 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 (Université catholique de Louvain, Belgium) Tensors via commutators Abstract. While the co-smash product of objects in a semi-abelian category may be used as a formal commutator [3, 2], after Carboni and Janelidze s paper [1] it has also been clear that certain tensor products appear as co-smash products. This leads to an approach of tensor products via commutators. In my talk I will explain how the co-smash product of objects in the two-nilpotent core Nil 2 (X) of a semi-abelian category X determines a so-called bilinear product on the abelian core Ab(X) of X. In certain homological applications, this may 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. Algebra 183 (2003), [2] M. Hartl and B. Loiseau, On actions and strict actions in homological categories, Theory Appl. Categ. 27 (2013), no. 15, [3] S. Mantovani and G. Metere, Normalities and commutators, J. Algebra 324 (2010), no. 9,

17 New trends in Hopf algebras and tensor categories 17 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 of characteristic 0. We define Hopf algebraic analogues of commutators and their generalizations and show how they are related to H, the Hopf algebraic analogue of the commutator subgroup. We introduce a family of central elements of H which on one hand generate H and on the other hand give rise to a family of functionals on H: When H = kg; G a finite group, these functionals are counting functions on G. We define nilpotency via a descending chain of commutators and give a criterion for nilpotency via a family of central invertible elements. These elements can be obtained from a commutator matrix A which depends only on the Grothendieck ring of H. When H is almost cocommutative we introduce a probabilistic method. We prove that every semisimple quasitriangular Hopf algebra is probabilistically nilpotent.