Programa-HKolleg-quantum16

Humboldt Kolleg COLLOQUIUM ON ALGEBRAS AND REPRESENTATIONS - QUANTUM 16.

Córdoba, Argentina, February 29 - March 4 2016

Salón de Actos, Academia Nacional de Ciencias

Monday 29.02 | |

09:30 - 10:20 | Opening by local authorities *** Welcome by Dr. Thomas Hesse, Deputy Secretary General of the Alexander von Humboldt Foundation *** Presentation of the Programs of the Humboldt Foundation by Dr. Thomas Hesse |

10:20 - 10:30 | Pause |

10:30 - 11:20 | |

11:20 - 11:40 | Coffee break |

11:40 - 12:30 | |

12:30 - 15:00 | Reception at the Academia |

15:00 - 15:50 | Groupoids in Group Representation Theory |

15:50 - 16:10 | Coffee break |

16:10 - 17:00 |

Salón de Actos, Pabellón Argentina

18:30 - 20:00 | Honorary Doctorate of Harald Helfgott followed by Public Lecture |

Aula Magna, Facultad de Matemática, Astronomía, Física y Computación

Tuesday 01.03 | Wednesday 02.03 | Thursday 03.03 | Friday 04.03 | |

09:30 - 10:20 | Crecimiento y expansión en grupos lineales: un enfoque desde las álgebras de Lie / Growth in linear groups: an approach through Lie algebras | Decomposition of unipotents: coming of age | ||

10:20 - 10:30 | Pause | Pause | Pause | Pause |

10:30 - 11:20 | Hopf crossed products and their applications to super algebraic groups | |||

11:20 - 11:40 | Coffee break | Coffee break | Coffee break | Coffee break |

11:40 - 12:30 | Positivity of parabolic Kazhdan-Lusztig polynomials | Non-semisimple factorisable ribbon categories from symplectic fermions | Algebraic operads and bialgebras on spaces of m-Dyck paths | |

12:30 - 14:30 | Lunch | Lunch | Lunch | Lunch |

14:30 - 15:20 | Free Afternoon | Ribbon quasi-Hopf algebras for symplectic fermions and restricted quantum groups | ||

15:20 - 15:40 | Coffee break | Coffee break | Coffee break | |

15:40 - 16:30 | Lusztig's families and Cherednik algebras | On the classification of group theoretical categories up to weak Morita Equivalence | ||

16:30 - 16:40 | Pause | Pause | Pause | |

16:40 - 17:30 | Fusion for principal W-algebras | Multi-parameter quantum groups via quantum quasi-symmetric algebras | ||

17:30 - 17:40 | Pause | Coffee, tea, snacks | Pause | |

17:40 - 18:30 | Verma and simple modules for quantum groups at non-abelian groups |

Abstracts

Iván Angiono

Pointed Hopf algebras of finite Gelfand-Kirillov dimension

This talk is based in a joint work with N. Andruskiewitsch and I. Heckenberger. We recall the definition of Nichols algebras and present different examples of finite GK dimension. We recall the connection between these Nichols algebras and pointed Hopf algebras (of finite GK dimension), and present a family of new examples of pointed Hopf algebras. Finally we give an overview about the classification of Nichols algebras of finite GK dimension over abelian groups.

Camilo Arias Abad

Connections and loop spaces

The pull back of a flat bundle E→X along the evaluation map π: LX→X from the free loop space LX to X comes equipped with a canonical automorphism given by the holonomies of E.

This construction naturally generalizes to flat ℤ-graded connections on X. Our main result is that the restriction of this holonomy automorphism to the based loop space Ω∗X of X provides an A∞ quasi-equivalence between the dg category of flat ℤ-graded connections on X and the dg category of representations of C∙(Ω∗X), the dg algebra of singular chains on Ω∗X. This is joint work with F. Schaetz.

Guillermo Cortiñas

Linear categories vs non-unital rings

We will explain how many statements and results usually formulated in terms of linear categories can be translated in terms of non-unital rings.

Ben Elias

Diagonalization in categorical representation theory

The representation theory of the Hecke algebra in type A can be understood by examining the Young-Jucys-Murphy (YJM) operators, which form a large commutative subalgebra. One proves that these operators are (simultaneously) diagonalizable, and classifies their spectrum via standard tableaux. Alternatively, one can use full twists of parabolic subgroups to replace YJM operators, with the same results.

Given a categorical representation of the Hecke algebra, one obtains a categorical representation of the braid group, and thus categorical versions of the YJM operators and the full twists. We describe what it means for these operators to be "categorically diagonalizable," and conjecture that the full twist of any finite Coxeter group is categorically diagonalizable. We have proven this conjecture in type A and for dihedral groups. (On the other hand, the categorical YJM operators are not diagonalizable.)

Given a diagonalizable operator F whose spectrum is known, linear algebra tells one how to construct operators which project to the eigenspaces of F. This construction lifts for categorically diagonalizable functors. For example, projection to eigenspaces of YJM operators gives any representation a canonical decomposition whose summands are isotypic. Upstairs, any categorical representation has a canonical and explicit filtration whose subquotients are isotypic categorifications.

This is joint work with Matt Hogancamp.

Vyacheslav Futorny

Representations of quantized gl(n)

We will discuss Gelfand-Tsetlin representations of Uq(gl(n)) based on a recent joint work with J.Hartwig and M.Rosso.

Agustín García Iglesias

Pointed Hopf algebras and deformations

After the classification of the finite-dimensional Nichols algebras of diagonal type by Heckenberger, the determination of its defining relations and the verification of the generation in degree one conjecture by Angiono, there is still one step missing in the classification of complex finite-dimensional Hopf algebras with abelian group, without restrictions on the order of the latter: the computation of all deformations or liftings. A technique towards solving this question, built on cocycle deformations, was developed previously by the author in collaboration with Andruskiewitsch, Angiono, Masuoka and Vay. In this talk, we shall discuss a recent article in collaboration with Andruskiewitsch and Angiono in which we elaborate further this technique and present an explicit algorithm to compute the liftings. In particular, we apply this algorithm to classify all liftings of finite-dimensional Nichols algebras of Cartan type A, over a cosemisimple Hopf algebra. As a by-product of our calculations, we present new infinite families of finite-dimensional pointed Hopf algebras.

Azat Gainutdinov

Ribbon quasi-Hopf algebras for symplectic fermions and restricted quantum groups

Having a finite ribbon category, like those coming from Vertex Operator Algebras, it is a natural question whether the category is equivalent to the representation category of a finite-dimensional ribbon Hopf algebra.

I will present a family of ribbon quasi-Hopf algebras with representation categories equivalent to the ribbon categories of the symplectic-fermions super VOAs (joint work with V. Farsad and I. Runkel). I will also discuss these quasi-Hopf algebras from perspectives of quantum groups at roots of unity.

Stephen Griffeth

Lusztig's families and Cherednik algebras

In 2007, Gordon and Martino noticed a mysterious connection between the representation theory of finite Hecke algebras, and that of restricted rational Cherednik algebras. Recent papers by several authors have added empirical weight to this mystery, but so far there seems to be no conceptual explanation for these coincidences. We will survey recent work on one instance of this phenomenon, involving Lusztig's families of representations of Weyl groups, and suggest a possible explanation in this instance.

Istvan Heckenberger

On Fomin-Kirillov algebras and related structures

Fomin-Kirillov algebras are a special class of quadratic non-commutative non-cocommutative braided Hopf algebras given by generators and relations. These algebras have strong connections to classical geometry, but a structural description is missing. Recently, Blasiak, Liu and Meszaros initiated the study of coideal subalgebras of Fomin-Kirillov algebras. In this talk, based on joint work with B. Roehrig, this approach is extended to related structures. I will indicate how this approach leads to a better understanding of known finite-dimensional Fomin-Kirillov algebras.

Harald Helfgott

Crecimiento y expansión en grupos lineales: un enfoque desde las álgebras de Lie/ Growth in linear groups: an approach through Lie algebras

TBA

Naihong Hu

Multi-parameter quantum groups via quantum quasi-symmetric algebras

It is proved that the entire multi-parameter (small-)quantum groups of symmetrizable Kac-Moody algebras can be realized as certain subquotients of the cotensor Hopf algebras. This is an axiomatic construction. Hopf 2-cocycle deformations variation for the construction machinery is described, moreover, the integrable irreducible modules can be constructed using this setting, even available for those in the fundamental alcove at root of unity case. This is a joint work with Yunnan Li and Marc Rosso.

Nicolás Libedinsky

Positivity of parabolic Kazhdan-Lusztig polynomials

TBA

Akira Masuoka

Hopf crossed products and their applications to super algebraic groups

I will start with reminding you of (now old) fundamental results on Hopf crossed products, due to Blatter-Montgomery and Doi-Takeuchi. After discussing equivariant smoothness of Hopf algebras, I will then show you some applications to super algebraic groups; they will include the super-analogue of the Kempf Vanishing Theorem which was recently proved by Taiki Shibata in his PhD Thesis.

Victor Petrogradsky

Growth in Lie algebras and groups

The notion of growth appeared in group theory and geometry. One of outstanding results is Gromov's theorem on groups of polynomial growth. Also, Gelfand and Kirillov used it to distinguish fields of fractions of enveloping algebras. Free finitely generated Lie algebras have exponential growth. But there are natural examples of Lie algebras that have growth between polynomial and exponent. For example, Lie algebras of vector fields have such an intermediate growth. We suggest an hierarchy of types of intermediate growth, which consists of a countable series of functions. In terms of these functions, we describe the growth of finitely generated solvable Lie algebras that have a fixed solubility length q and which are free under this condition. These algebras belong to the level q of the hierarchy, where the level q=1 corresponds to finite dimensional Lie algebras. The level q=2 corresponds to the polynomial growth. As an application, we obtain an asymptotic for ranks of the lower central series factors for free solvable groups of finite rank. For free groups one has well-known Schreier's formula. We use analogues of this formula for free Lie algebras in terms of generating functions.

María Ofelia Ronco

Algebraic operads and bialgebras on spaces of m-Dyck paths

TBA

Ingo Runkel

Non-semisimple factorisable ribbon categories from symplectic fermions

Modular tensor categories are semi-simple ribbon categories with a finite number of simple objects and a non-degenerate braiding. When dropping the semi-simplicity requirement, one arrives at the notion of factorisable ribbon categories. Such categories enter the construction of three-dimensional topological field theories and hence have applications to knot invariants and representations of mapping class groups of surfaces.

In this talk I would like to consider a series of examples arising from symplectic fermion conformal field theories. I will discuss non-degeneracy conditions on the braiding and the notion of Lagrangian algebras. The latter enter the definition of the so-called Witt group of modular tensor categories, and are related to holomorphic extensions of vertex operator algebras.

Hans-Jürgen Schneider

Nichols algebras with finite Weyl groupoid

This is joint work with I. Heckenberger. It was shown by Lusztig that the plus part Uq+(g) of the quantum group Uq(g), g a semisimple Lie algebra, q not a root of one, is a Nichols algebra with a PBW-basis generated by the positive root vectors. I will explain the Nichols algebra B(V) of a Yetter-Drinfeld module V over some Hopf algebra with bijective antipode and then concentrate on the structure of the Nichols algebra of semisimple YD-modules V. Based on a general result on braided monomial categories we extend Lusztig’s results to Nichols algebras with finite Weyl groupoid. The main technique is a stepwise construction of right coideal subalgebras. In particular, we obtain a new interpretation of the Lusztig automorphisms as braided Hopf algebra isomorphisms and new proofs in the quantum group case. Our categorical construction allows to introduce the useful notion of a Nichols system, to show that a given braided Hopf algebra is in fact the Nichols algebra.

Christoph Schweigert

TV theories with defects and representation theory

Surface defects in (extended) three-dimensional topological field theories have important applications, ranging from solid state physics to representation categories. The intersection of such defects is the location of a generalized Wilson line. Categories in which labels for these Wilson lines are constructed, for topological field theories of Turaev-Viro type, by a 2-functor that assigns to a bimodule category over a finite tensor category a k-linear category, which can be seen as a catetgorified trace. This 2-functor also enters crucially into the construction of conformal blocks, which give a realization of the description of the Brauer-Picard group of Etingof-Nikshych-Ostrik.

Jorge Soto Andrade

Groupoids in Group Representation Theory

In (ordinary) representation theory groupoids play a key role in the construction of Gelfand Models for finite groups G (i.e. representations containing all irreps of G exactly once). Indeed, geometric induction to G from characters of the motion groupoid associated to a suitable G-set affords Gelfand Models for a vast class of finite groups, that appear then as "twisted natural representations", and encompass all known cases of construction of Gelfand Models. We have conjectured that any finite group admits a Gelfand Model so constructed. This has been proved when G is a dihedral group, the symmetric group, the general linear group and the projective general linear group of rank 2. We discuss the scope of validity of this conjecture and its variants, counter examples included, as well as related conjectures, including the realisability of a Gelfand Model for a finite group G as a top cohomology space of a G-set naturally associated to G.

This is joint work with Anne-Marie Aubert and Antonio Behn.

Bernardo Uribe

On the classification of group theoretical categories up to weak Morita Equivalence

A pointed fusion category is a rigid tensor category with finitely many isomorphism classes of simple objects which moreover are invertible. Two tensor categories C and D are weakly Morita equivalent if there exists an indecomposable right module category M over C such that FunC(M,M) and D are tensor equivalent. We use the Lyndon-Hochschild-Serre spectral sequence associated to abelian group extensions to give necessary and sufficient conditions in terms of cohomology classes for two pointed fusion categories to be weakly Morita equivalent. This result may permit to classify the equivalence classes of pointed fusion categories of any given global dimension.

Christian Valqui

Twisted tensor products of Kn with Km

Joint work with Jack Arce, Jorge Guccione and Juan José Guccione. We give a detailed description of different families of twisted tensor products of Kn with Km. These families include the case Kn x K2 considered by Cibils in [C] and yield a simple description in the case of reduced rank one considered in [JLNS]. This family corresponds to truncated quiver algebras with square zero radical, whereas another family corresponds to (formal) deformations of some these algebras. We also construct all twisting maps of K3 x K3 and describe a method to deal with the case K3 x Kn.

[C] C. Cibils, Non-commutative duplicates of finite sets, J. Algebra Appl. 5 (2006), 361-377.

[JLNS] P. Jara, J. López Peña, G. Navarro, D. Stefan, On the classification of twisting maps between Kn and Km, Algebr. Represent. Theory 14 (2011), 869-895.

Fusion for principal W-algebras

I describe joint work with T. Arakawa in which we determine modular properties of characters of affine W-algebras. We apply these results to establish the fusion rules of these algebras as originally computed by Frenkel, Kac and Wakimoto.

Nicolai Vavilov

Decompostion of unipotents: coming of age.

The talk is devoted to the recent progress on and the new applications of a powerful geometric/representation theoretic approach towards the structure theory of algebraic groups over commutative rings, decomposition of unipotents.

Let be a root system, R be a commutative ring, and let be a Chevalley group of type over R. Further, denote by , where

and , the corresponding elementary root element.

For the group this method was initially proposed in 1987 by Alexei Stepanov, to give a simplified proof of Suslin's normality theorem. Soon thereafter I generalised it to split classical groups, and then together with Eugene Plotkin we generalised it to exceptional Chevalley groups.

In the simplest form, at the level of K1, this method gives explicit polynomial factorisations of root type elements , where , in terms of factors sitting in proper parabolic subgroups, and eventually in terms of elementary generators. Among other things, this allows to give extremely short and straightforward proofs of the main structure theorems for such groups.

However, for exceptional groups the early versions of the method relied on the presence of huge classical embeddings, such as A5<E6, A7<E7 and D8<E8. Also, even for some classical groups, the method would not give explicit bounds on the length of elementary decompositions of root elements.

Inspired by the A2-proof of structure theorems for Chevalley groups, proposed by myself, Mikhail Gavrilovich, Sergei Nikolenko, and Alexander Luzgarev, recently we succeeded in developing new powerful versions of the method, that only depend on the presence of small rank classical

embeddings. In 2014-2015 the following amazing progress occured.

- Combining decomposition of unipotents with localisation and amalgam methods, Andrei Lavrenov and Sergei Sinchuk succeded in obtaining generalisations at the level of K2, which led to a solution of a 40++ years old open problem, centrality of K2 modeled on , and the Chevalley groups of types E6, E7 and E8.
- For the microweight case, Victor Petrov discovered a completely new approach towards the construction of small unipotents stabilising a vector from the highest weight orbit. Unlike original proofs, his construction does not rely on difficult explicit calculations, and uses only some elementary algebro-geometric facts such as motivic filtrations.
- Using generic constructions, Alexei Stepanov gave a new proof of the normal structure theorem for Chevalley groups, that does not invoke the full force of the method, and only relies on the presence of one non-trivial unipotent in the stabiliser of a column. Combined, these advances made the whole machinery much more manageable and flexible, and paved way for new applications. For instance, it turned out that decompositions of unipotents do not have to be parabolic, they can work in terms of proper reductive subgroups, which allows to dramatically reduce length bounds. Here are some typical recent applications.
- An explicit essentially linear length factorisation of transvections for .
- Applications to description of overgroups of subsystem subgroups, in particular, for the case A7<E7 (joint with Alexander Shchegolev).
- Description of subnormal subgroups of Chevalley groups of types E6 and E7 (joint with Zuhong Zhang).

Finally, I plan to mention possible generalisations to isotropic (but non necessarily split) reductive groups. In this generality, Victor Petrov and Anastasia Stavrova initiated the use of localisation methods, further developed by Stavrova, Luzgarev, Stepanov, and others. Decomposition of unipotents would constitute a viable alternative to and in some cases enhancement of localisation methods, and might have serious impact on the

structure theory of such groups. Actually, in many important cases it gives much better length bounds in terms of elementary generators. Another tantalising challenge would be to extend these methods to infinite-dimensional algebraic-like groups such as Kac-Moody groups, etc.

Cristian Vay

Verma and simple modules for quantum groups at non-abelian groups

The Drinfeld double D of the bosonization of a finite-dimensional Nichols algebra B(V) over a finite non-abelian group G is called a quantum group at a non-abelian group. We introduce Verma modules over such a quantum group D and prove that a Verma module has simple head and simple socle. This provides two bijective correspondences between the set of simple modules over D and the set of simple modules over the Drinfeld double D(G). As an example, we describe the lattice of submodules of the Verma modules over the quantum group at the symmetric group S3 attached to the 12-dimensional Fomin-Kirillov algebra, computing all the simple modules and calculating their dimensions.

This talk is based on the joint work with Barbara Pogorelsky [http://arxiv.org/abs/1409.0438].

Leandro Vendramin

Problems and results on Nichols algebras

Nichols algebras appear in several branches of mathematics going from Hopf algebras and quantum groups, to Schubert calculus and conformal field theories. In this talk we review the main problems related to Nichols algebras and we discuss some classification.

Konrad Waldorf

Transgressive central extensions of loop groups

Some central extensions of the free loop group LG of a Lie group G can be obtained as the transgression of a gerbe over G. I will describe a loop group-theoretical characterization of such central extensions in terms of loop fusion and thin homotopy equivariance. This is part of the programme of exploring the geometry and representation theory of loop groups via finite-dimensional, higher-categorical structure over the underlying Lie group.

Geordie Williamson

An example of higher representation theory

I will discuss recent work with Simon Riche where we use ideas from higher representation theory to understand modular representations of quantum groups and algebraic groups. We formulate a very natural conjecture (existence of a categorical action), and show that it has remarkable consequences (character formulas, description of categories by generators and relations). The talk will be non-technical, and I will try to emphasise the patterns arising in higher representation theory.

Public Lecture by Harald Helfgott

La conjetura ternaria de Goldbach

La conjetura ternaria de Goldbach (1742) afirma que todo número impar mayor que 5 se puede escribir como la suma de tres números primos. Después de los trabajos pioneros de Hardy y Littlewood, Vinogradov probó (1937) que todo número impar mayor que una cierta constante C satisface la conjetura. En los tres cuartos de siglo que siguieron, hubo varios resultados que reducían C, pero sólo hasta niveles que seguían siendo demasiado grandes (C>101300) como para que fuera posible una verificación por la fuerza bruta.

(Hubo varias otras aproximaciones a la conjetura; por ejemplo, Ramare probó que todo número par es la suma de a lo más seis primos, y Tao mostró que todo número impar es la suma de a lo más cinco primos.)

Mis trabajos prueban la conjetura ternaria para todo impar mayor que 5. Haremos un recorrido por las principales ideas, detallando los lazos con el resto de la teoría de números y otras áreas.