## From groupoid cardinality to Drinfeld doubles

Christoph Schweigert

Viernes 15 de Marzo, 16hs. Aula 13.

We explain how to “count” the number of G-covers on a compact manifold, where G is a finite group. This yields in particular a simple invariant of three-manifolds. We show how to extract from this invariant (and its associated three-dimensional topological field theory) many important notions of modern algebra, in particular the Drinfeld double of the group and module categories over the group ring of G.

## Seminar of November 22nd

22/11/2018, Room 27, 2:30 pm, FAMAF

Time-Band-Limiting for matrix-valued functions

Speaker: Ignacio Zurrián

Abstract:

## Seminario del 22 de noviembre

22/11/2018, Aula 27, 14:30 hs FAMAF

Time-Band-Limiting para funciones matriciales.

Expositor: Ignacio Zurrián

Resumen:

## Seminar of November 8th

8/11/2018, Room 27, 2:30 pm, FAMAF

The adjoint algebra for tensor categories.

Speaker: Noelia Belén Bortolussi

Abstract:

Representation theory and category theory are useful tools for the study of groups and Hopf algebras. In this talk, we shall introduce the background to understand the notion of characters in the context of tensor categories. This concept was introduced by K. Shimizu, who also extended many related notions and properties about classical characters to this categorical context, such as the orthogonality relations, the relation with conjugacy classes, etc.
Notice that a character of a finite group $$G$$ can be viewed as a morphism of G-modules $$\chi: kG \to k$$, where$$kG$$ is equipped with the adjoint action. So the first step in the definition of characters for tensor categories is the generalization of the notion of adjoint algebra for arbitrary (finite) tensor categories.  This generalization plays a fundamental roll in the theory of  Lyubashenko on the action of the modular group on no-semisimple tensor categories. In order to introduce the adjoint algebra, we will first introduce the concepts of (co)monads and of (co)ends in categories.

## Seminario del 8 de noviembre

8/11/2018, Aula 27, 14:30 hs FAMAF

El álgebra adjunta para categorías tensoriales

Expositor: Noelia Belén Bortolussi

Resumen:

La teoría de representaciones y la teoría de categorías son fuertes herramientas en el estudio de los grupos y las álgebras de Hopf. En esta charla explicaremos las herramientas necesarias para entender la noción del álgebra de caracteres en el contexto de categorías tensoriales. Esta noción fue introducida por K. Shimizu, quien demostró generalizaciones de la ortogonalidad de caracteres, clases de conjugación, entre otros resultados de la teoría de caracteres de grupos al ámbito de categorías de fusión.
Un carácter para un grupo finito $$G$$ puede verse como un morfismo de G-módulos $$\chi: kG \to k$$, donde a $$kG$$ se lo equipa con la acción adjunta. El primer paso para entender el álgebra de caracteres será explicar como generalizar la noción del álgebra adjunta en una categoría tensorial finita arbitraria. Esta generalización juega un rol fundamental en la teoría de Lyubashenko de la acción del grupo modular en categorías tensoriales no semisimples. Para introducir el álgebra adjunta, presentaremos las nociones de (co)mónadas y (co)ends en categorías

## Seminar of October 25th

25/10/2018, Room 27, 14:30 hs FAMAF

Matrix valued spherical functions and multivariable matrix valued orthogonal polynomials

Abstract:

For a specific case of a symmetric pair $(G,K)$ corresponding to the group case $SU(n+1)$, we discussmatrix-valued spherical functions as a generalization of the spherical functions, which arise as bi-K-invariant functions.
The main input is an irreducible representation of $K$, such that the induced representation splits multiplicity free. To such a fixed representation one can associate matrix valued spherical functions on $G$. The restriction to the corresponding abelian subgroup gives a column in the matrix-valued orthogonal polynomial upon choosing suitable coordinates depending on $n$ variables.
The polynomials satisfy orthogonality relations, and can also be described in terms of eigenfunctions of matrix-valued (partial) differential operators.
The case $n=1$ gives matrix valued analogs of Chebyshev polynomials. The case $n=2$ gives matrix valued $2$-variable orthogonal polynomials on Steiner’s hypocycloid, generalizing the scalar case by Koornwinder (1974).

Joint work with Maarten van Pruijssen (U. Paderborn, Germany) and Pablo Rom\’an (UN Córdoba).

## Seminar of October 11th

11/10/2018, Room 27, 14:30 hs FAMAF

Burchnall type identities for orthogonal polynomials

Abstract:

L.J. Burchnall (1892-1975) proved a converse to the linearization formula for Hermite polynomials in 1941. In doing so, he calculated powers of a differential operator in terms of Hermite polynomials and (standard) derivatives. We show that this approach generalizes to all families of orthogonal polynomials in the Askey-scheme and its $q$- analog. We discuss several applications of this approach. One application lies in expansion formulas, and a related application is related to the Lax pair formalism for the Toda lattice.
It is possible to extend this approach to some families of matrix-valued orthogonal polynomials and
a corresponding non-abelian Toda lattice. The lecture reports on joint work with Mourad Ismail (University of Central Florida, USA) and Pablo Román (UNC).

## Seminar of September 27th

27/9/2018, Room 23, 16:30 hs FAMAF

How to lift

Speaker: Agustín García Iglesias

Abstract:

In this talk, we will give a detailed introduction to a technique of deformation of graded Hopf algebras that we introduced, in joint work with various authors, and which was successfully employed to conclude the classification of finite-dimensional pointed Hopf algebras with abelian coradical.  After a  brief historical contextualization of the subject, we shall present the aforementioned classification result and show the ideas behind the proof. Finally, we shall present the deformation algorithm, which can be also applied for other types of Hopf algebras. We shall illustrate this with examples.

## Seminario del 27 de setiembre

27/9/2018, Aula 23, 16:30 hs FAMAF

How to lift

Expositor: Agustín García Iglesias

Resumen:

En esta charla daremos una introducción detallada a una técnica de deformación de álgebras de Hopf graduadas que desarrollamos en conjunto con varios autores y que pudimos aplicar con éxito para concluir la clasificación de las álgebras de Hopf punteadas de dimensión finita, con coradical abeliano. Luego de una breve contextualización histórica, presentaremos este resultado de clasificación y mostraremos las ideas detrás de la prueba. Finalmente estudiaremos el algoritmo de deformación, que puede aplicarse para otros tipos de álgebras de Hopf, y que vamos a ilustrar con algunos ejemplos.

## Seminar of September 13th

13/9/2018, Room 27, 14:30 hs FAMAF

Hopf algebra with basic Hopf coradical

Speaker: Iván Angiono

Abstract:

The Lifting Method, which was introduced by  Andruskiewitsch and Schneider in the late 1990s, has been very fruitful for the problem of classification of Hopf algebras (of finite dimension). In fact, it has led to the classification of all pointed Hopf algebras whose coradical is the group-algebra of an abelian group and several cases where the groups are not abelian.  The Method starts by associating to a Hopf algebra a graded Hopf algebra, whose existence assume that the coradical of the original Hopf algebra is a subalgebra, which is not always the case.
When the coradical is not a subalgebra, Andruskiewitsch and Cuadra introduced a generalized version of the Method: it is based on replacing the coradical by the Hopf coradical, which is always a subalgebra, though it is not semi-simple. In the present talk, we shall show how to proceed in the case that the Hopf coradical is a basic algebra. The answer rely on results of classification of pointed Hopf algebras.