Colloquium on Algebras and Representations

A gathering in Córdoba to discuss the most recent developments in Hopf algebras, tensor categories and related structures.

December 10 - 14, 2018
Casa Serrana - Huerta Grande
Córdoba, Argentina


Yuri Bahturin (Memorial University Newfoundland, Canada)

Julien Bichon (Université Clermont Auvergne, France)

Giovanna Carnovale (Universitá di Padova, Italia)

Juan Cuadra (Universidad de Almerí­a, España)

Stefaan Caenepeel (Vrije Universiteit Brussel, Belgium)

François Dumas (Université Clermont Auvergne, France)

Ben Elias (University of Oregon, USA)

Pavel Etingof (MIT, USA)

César Galindo (Universidad de los Andes, Colombia)

Istvan Heckenberger (Philipps-Universität Marburg, Germany)

Vladislav Kharchenko (Universidad Nacional Autónoma de México, México)

Nicolás Libedinsky (Universidad de Chile, Chile)

Michael Müger (Radboud Universiteit Nijmegen, Netherland)

Dmitry Nikshych (University of New Hampshire, USA)

Victor Ostrik (University of Oregon, USA)

Marí­a Ofelia Ronco (Universidad de Talca, Chile)

Eric Rowell (Texas A&M University, USA)

Hans-Jürgen Schneider (LMU München, Germany)

Vera Serganova (University of California, Berkeley, USA)

Andrea Solotar (Universidad de Buenos Aires, Argentina)

Yorck Sommerhäuser (Memorial University of Newfoundland, Canada)

Milen Yakimov (Louisiana State University, USA)

Hiroyuki Yamane (University of Toyama, Japan)


For titles and abstracts of talks and posters see below. There is also a direct link on the speaker's name

Time Monday 10 Tuesday 11 Wednesday 12 Thursday 13 Friday 14
9:00 - 9:50 Registration Elias Solotar Carnovale Yakimov
9:50 - 10:00 Break
10:00 - 10:50 Libedinsky Rowell Caenepeel Heckenberger
10:50 - 11:10 Coffee break
11:10 - 12:00 Etingof Ronco Sommerhäuser Yamane Schneider
12:00 - 14:30 Lunch
14:30 - 15:20 Nikshych Serganova Free afternoon
Poster session Return to Córdoba
15:20 - 15:40 Coffee break Coffee break
15:40 - 16:30 Ostrik Kharchenko Cuadra
16:30 - 16:40 Break Break
16:40 - 17:30 Müger Dumas Bichon
17:30 - 17:40 Break Break
17:40 - 18:30 Galindo Bahturin Andruskiewitsch


Yuri Bahturin
Graded algebras and Hopf algebras
This is a survey talk devoted to the role Hopf algebras play in the theory of graded algebras.

Julien Bichon
Monoidal invariance of the cohomological dimension of a Hopf algebra
I will discuss the question of the invariance of the cohomological dimension of a Hopf algebra under monoidal equivalences of comodule categories

Giovanna Carnovale
On parity sheaves for Kashiwara's flag manifold

Juan Cuadra
Non-existence of integral Hopf orders for several twists of group algebras
A theorem of Frobenius asserts that the degree of any complex irreducible representation of a finite group G divides the order of G. The proofs known use a specific property of the group algebra CG: namely, the group ring ZG is a Hopf order of CG.
Kaplansky conjectured that Frobenius' Theorem holds for complex semisimple Hopf algebras. As in the case of groups, Larson proved in 1972 that this is true if the Hopf algebra admits a Hopf order over a number ring (integral Hopf order). For a long time it was an open question whether a complex semisimple Hopf algebra always admits an integral Hopf order.
We settled this in the negative in [1] for a family of semisimple Hopf algebras constructed by Galindo and Natale as a Drinfeld twist of certain group algebras. This reveals an important arithmetic difference between group algebras and semisimple Hopf algebras: complex semisimple Hopf algebras may not admit integral Hopf orders. In this talk we will show that this phenomenon occurs for two other families of twists of group algebras introduced by Bichon and Nikshych. Curiously, all these examples are simple Hopf algebras.
The results that will be presented appear in [2] and are joint with Ehud Meir (University of Aberdeen, UK).

[1] J. Cuadra and E. Meir, On the existence of orders in semisimple Hopf algebras. Trans. Amer. Math. Soc. 368 (2016), 2547-2562. ArXiv:1307.3269.
[2] J. Cuadra and E. Meir, Non-existence of Hopf orders for a twist of the alternating and symmetric groups. Accepted in J. London Math. Soc. ArXiv:1804.01121.

Stefaan Caenepeel
Galois theory for Hopf categories
(Joint work with T. Fieremans)
We introduce Hopf categories enriched over braided monoidal categories. The notion is linked to several recently developed notions in Hopf algebra theory, such as Hopf group (co)algebras, weak Hopf algebras and duoidal categories. We generalize the fundamental theorem for Hopf modules and some of its applications to Hopf categories. We then focus on Hopf-Galois theory; there are two versions of the theory. The second one can be used to give an alternative approach to the Galois theory of algebras with an action by a groupoid, as introduced and studied by Bagio, Paques and Tamusiunas.

François Dumas
Homogeneous localizations of classical and quantum enveloping algebras of orthosymplectic Lie superalgebras
The enveloping algebra U of the orthosymplectic Lie superalgebra osp(1,2n) is a superalgebra and a noetherian domain. Its superskewfield of fractions is the localized superalgebra obtained making invertible all nonzero homogeneous elements of U. In this joint work with Jacques Alev in view of a formulation of Gelfand-Kirillov problem in the super context, we describe the superskewfield of fractions of the enveloping algebra of the nilpotent positive part and of the Borel subsuperalgebra of osp(1,2n) in terms of Z2-graded versions of polynomial algebras and Weyl algebras. We give some exploratory results for corresponding quantum analogues in low dimension.

Benjamin Elias
Generalized KLR algebras and mutation: the first steps towards categorification of Nichols algebras

Pavel Etingof
Semisimplification of tensor categories
We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. By definition, the semisimplification of a tensor category is its quotient by the tensor ideal of negligible morphisms, i.e., morphisms fsuch that Tr(fg)=0 for any morphism g in the opposite direction. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic p in terms of representations of the normalizer of its Sylow p-subgroup. This allows us to compute the semisimplification of the representation category of the symmetric group Sn+p in characteristic p, where n=0,...,p-1, and of the abelian envelope of the Deligne category, Repab St. We also compute the semisimplification of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of sl2. Finally, we study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplification construction, and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplification). This is joint work with Victor Ostrik.

César Galindo
The five-term exact sequence for Kac cohomology
Kac described the group of equivalence classes of abelian extensions of Hopf algebras associated to a matched pair of finite groups in the 60's as the first cohomology group of a double complex, whose total cohomology is known as the Kac cohomology. Masuoka generalized this result and used it to construct semisimple Hopf algebra extensions. In this talk, we will use relative group cohomology to describe this cohomology and the five-term exact sequence associated to Kac's double complex. Through several examples, we will show the usefulness of the five-term exact sequence in computing groups of abelian extensions. The talk is based on joint work with Yiby Morales.

Istvan Heckenberger
PBW deformations of Nichols algebras
Typically, a finite-dimensional Nichols algebra is a local algebra with a rather complicated structure. In this talk a relation to other well-known algebras like matrix algebras is discussed in the framework of deformation theory.

Vladislav Khartchenko
Braided Lie operations for nonassociative product
The question raised by K.H. Hofmann and K. Strambach of whether the commutator and associator are the only primitive operations in a nonassociative bialgebra was answered negatively by I.P. Shestakov and U.U. Umirbaev in 2002, when they discovered infinitely many independent operations. In fact the Shestakov-Umirbaev operations together with the commutator form a complete set of nonassociative Lie operations. In the talk, we consider operations when the (nonassociative) variables span a braided space (V, c). The braiding c has a unique extension on the free nonassociative algebra k{V} so that k{V} is a braided algebra. Moreover, the free braided algebra k{V} has a natural structure of a braided nonassociative Hopf algebra (H-bialgebra in sense of Pérez-Izquierdo) such that every element of the space of generators V is primitive. In the case of involutive braidings, c^2=id braided analogues of Shestakov-Umirbaev operations appear, and these operations are primitive. The set of all primitive elements of a nonassociative c-algebra is a Sabinin c-algebra.

Nicolás Libedinsky
On the p-canonical basis
There has been a lot of movement in modular representation theory lately. The concept of p-canonical basis has now become central (it encodes the full information of modular representations of the symmetric group, reductive algebraic groups, etc). We will tell a bit of its recent history and some approaches towards their calculation.

Michael Müger

Dmitri Nikshych
Braided module categories and braided extensions

Victor Ostrik
Dimensions modulo p of tilting modules over GL(n)
This is a joint work with Pavel Etingof. We give a conjectural description of tilting modules over algebraic group GL(n) in characteristic p with dimension not divisible by p. We can prove our conjecture for p=2 and p=3.

Marí­a Ofelia Ronco

Rowell Eric
Algebraic Questions from Topological Quantum Computation

Hans-Jürgen Schneider
Yetter-Drinfeld modules in braided categories and reflections of Nichols systems
I will discuss a new approach to a braided equivalence which is fundamental for the construction of Cartan graphs of Nichols algebras and their root systems. These results will appear in a forthcoming book ''Hopf algebras and root systems'' by I. Heckenberger and myself.

Vera Serganova
Finite W algebras and Yangians of type Q
We will discuss finite W-algebras associated with nilpotent orbits in the coadjoint representations for the Lie superalgebras of type Q. In particular, we explain their relation to the type Q Yangian introduced by Nazarov and describe irreducible representations in the case of the principal orbit.

Andrea Solotar
On the Lie algebra structure of the first Hochschild cohomology of gentle algebras and Brauer graph algebras
We determine the first Hochschild cohomology of gentle algebras and we give a geometrical interpretation of these cohomologies using the ribbon graph of a gentle algebra as defined by Schroll. We give an explicit description of the Lie algebra structure of the first Hochschild cohomology of gentle and Brauer graph algebras (with multiplicity one) based on trivial extensions of gentle algebras and we show how the Hochschild cohomology is encoded in the Brauer graph. In particular, we show that except in one low-dimensional case, the resulting Lie algebras are all solvable.

Yorck Sommerhäuser
Hochschild Cohomology and Mapping Class Groups
It is known that the modular group acts on the center of a factorizable Hopf algebra. The center is isomorphic to the zeroth Hochschild cohomology group of the Hopf algebra. In a recently published article, we have generalized this action of the modular group to an arbitrary Hochschild cohomology group. As we stated already in that article, this result is a special case of a more general fact: The modular group is the mapping class group of the torus, and the action of the modular group on the center is a special instance of the action of the mapping class groups of surfaces on the Hom-spaces between certain representations of the Hopf algebra. As we now show, the action on the Hom-spaces can be generalized to an action on the Ext-spaces, which in the case of the torus reduces to the action of the modular group on the Hochschild cohomology groups. In the talk, we explain how techniques from mapping class groups, such as the Birman exact sequence, can be used to establish the generalization just mentioned. The talk is based on joint work with Simon Lentner, Svea Nora Mierach, and Christoph Schweigert.

Milen Yakimov
Quantum symmetric spaces for Drinfeld doubles of diagonal pre-Nichols algebras
We will describe a construction of quantum symmetric spaces (coideal subalgebras) for the Drinfeld doubles of all pre-Nichols algebras of diagonal type. We will also prove a characterization result about when they have the correct size so an Iwasawa decomposition holds. This includes the case of infinite GK dimension. For the doubles of all Nichols algebras we will describe an explicit construction of universal K-matrices. The previous constructions of K-matrices worked for quantum groups and generic q and were recursive, based on the Lusztig bar involution. Our construction uses *-products on graded connected algebras and gives an explicit formula for the K-matrices as sums of dual bases like R-matrices. This is a joint work with Stefan Kolb (Newcastle Univ).

Hiroyuki Yamane
Generalized quantum groups and their representations
I demonstrate that Weyl groupoids are useful for study of representation theory of generalized quantum groups.


Mauricio Angel
Generalized quantum groups and their representations

Dirceu Bagio
Representations of the quantum Borel Subalgebra of sl(2) at -1

Angelo Bianchi
Finite-dimensional representations of hyper multicurrent and multiloop algebras

Leonardo Duarte Silva
Nichols algebras associated to upper triangular solutions of the Yang-Baxter equation in rank 3

Marco Farinati
Quantum Determinants from Nichols Algebras

Javier Gutierrez
Operads and F-Algebroides

Patricia Jancsa
2-step nilpotent Lie bialgebras

Joao Matheus Jury Giraldi
Nichols algebras that are quantum planes

Marcelo Paez
An identity on Rota-Baxter algebras

Eddy Pariguan
Quantum product of symmetric functions

Barbara Pogorelsky
On the representation theory of a quantum group attached to the Fomin-Kirillov algebra FK3

Carolina Renz
Right Coideal Subalgebras of a quantum group attached to the Fomin-Kirillov algebra FK3

Alveri Sant'Ana
On Weak Hopf-Ore extensions

Paolo Saracco
Preantipodes vs Antipodes

Lukas Woike
Orbifold Construction for Topological Field Theories


Vyacheslav Futorny (Universidade de Sao Paulo)

Marí­a Ofelia Ronco (Universidad de Talca)

Hans-Jürgen Schneider (LMU München)

Andrea Solotar (Universidad de Buenos Aires)



You may find an album with pictures from the conference here.

Universidad Nacional de Córdoba
Medina Allende s/n
Ciudad Universitaria
5000 Córdoba
República Argentina

Email: cordobaquantum60@gmail.com