Seminar of September 27th

 27/9/2018, Room 27, 14:30 hs FAMAF

How to lift

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Speaker: Agustín García

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 27, 14:30 hs FAMAF

How to lift!f4

Expositor: Agustín García

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

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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.

 

 

 

 

Seminario del 13/9

 13/9/2018, Aula 27, 14:30 hs FAMAF

Álgebras de Hopf con coradical de Hopf básicof4

Expositor: Iván Angiono

Resumen: 

El Método del Levante introducido por Andruskiewitsch y Schneider a fines de los ’90 ha sido muy fructífero para el problema de clasificación de álgebras de Hopf (de dimensión finita). En efecto, ha permitido la clasificación de todas las álgebras de Hopf punteadas con corradical un álgebra de grupo abeliano, y diversos casos de grupos no abelianos. Dicho método comienza con el estudio de un álgebra de Hopf graduada, y la existencia de dicha álgebra de Hopf graduada se basa en que el corradical es una subálgebra, lo cual no ocurre en todos los casos.
Cuando el corradical no es una subálgebra existe una versión generalizada del Método que introdujeron Andruskiewitsch y Cuadra: se basa en reemplazar el corradical por el corradical de Hopf, que siempre es una subálgebra, aunque no es semisimple. En la presente charla mostraremos cómo se puede proceder en el caso en que el corradical de Hopf es un álgebra básica. La respuesta se basa en resultados de clasificación de álgebras de Hopf punteadas

 

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Seminar of August 30th

 30/8/2018, Room 27, 14:30 hs FAMAF

On the SO(n+3) to SO(n) branching multiplicity space

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Speaker: Fiorella Rossi Bertone

Abstract:

Let \(\pi\) be an irreducible representation of a compact Lie group G. Consider the restriction of  \(\pi\) to a closed subgroup L of G. In general, this restriction is not irreducible and its decomposition as irreducible L-modules is called branching law.

In this talk, we will focus on the case when the subgroup L of G is KxH with K and H are closed subgroups of G. Since K and H commute to each other, the multiplicity spaces of the restriction of G to K inherit an H-module structure.
We will present some results obtained in a joint work with E. Lauret when G=SO(n+3) and KxH= SO(n) x SO(3).

 

 

 

 

Seminario del jueves 30/8

 30/8/2018, Aula 27, 14:30 hs FAMAF

Reglas de restricción de SO(n+3) a SO(n)f4

Expositor: Fiorela Rossi Bertone

Resumen: 

 Dada una representación irreducible \(\pi\) de un grupo de Lie compacto G, consideramos la restricción de \(\pi\) a un subgrupo cerrado L de G. En general, dicha restricción no es irreducible y su descomposición como L-módulos irreducibles es lo que se conoce como regla de restricción o branching law.

En esta charla consideraremos el caso en que el subgrupo L de G es de la forma KxH con K y H subgrupos cerrados de G. Como K y H conmutan entre sí, los espacios de multiplicidades de la restricción de G a K heredan una estructura de H-módulo.
Presentaremos algunos resultados obtenidos en un trabajo conjunto con E. Lauret cuando G=SO(n+3) y KxH= SO(n)xSO(3).

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Seminar of August 16th

 16/8/2018, Room 23, 14:30 hs FAMAF

Matrix spherical analysis on nilmanifoldsf4

Speaker: Rocío Díaz

Abstract: 

Let N be a connected, simply connected nilpotent Lie group, which is provided with a left invariant riemannian metric.  The isometry group of N is the semi-direct product $ K\ltimes N $, where K is the group of orthogonal automorphisms of N. We consider all the  Gelfand pairs $(K \ltimes N , K)$ determined by Jorge Lauret in [“Gelfand pairs attached to representations of compact Lie groups” Transformation Groups, 5(4):307-324 (2000)]. In these cases, the real-valued functions over N that are integrable and K-invariant form a commutative algebra with convolution as a product. For each irreducible representation W of K, we will determine whether the  \(End(W)\)-valued functions over N that are integrable and bi-W-equivariant form a commutative algebra.

 

 

 

Seminar of July 5th

 4/7/2018, Room 27, 14:30 hs FaMAF

Nichols algebras over dihedral groupsf4

Speaker: Sergio Beltran Cubillos

Abstract:

 The aim of this talk is to give an overview of what is known about the dimensions of the Nichols algebras over dihedral groups. We will first set up the problem and show with examples the methods that are used to find such dimensions. Then we will present some partial results in low dimensions that as far as we known are not yet available in the literature on the subject.

 

 

 

 

Seminar of June 22nd

 22/6/2018, Room 11, 11:00 hs FaMAF

Perverse sheaves in the language of diagrammatic categories.

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Speaker: Cristian Vay

Abstract:

Important results in the theory of representation of Lie groups and Lie algebras have been proved initially using geometry. The geometric interpretation is given via the category of perverse sheaves on a flag variety. Algebraic proofs were available later with the developing of the theory of Soergel Bimodules, and more generally, with the theory of Diagrammatic Categories of Elias-Williamson.
The geometric context is very rich in tools and intuition, and one hopes to have something similar in the context of Diagrammatic Categories. In a recent paper with P. Achar and S. Riche [https://arxiv.org/abs/1802.07651], we explain how to reintroduce perverse sheaves in the diagrammatic context.
In this seminar, we review the definition of Diagrammatic Categories, we define perverse sheaves in this context and we shall prove that they satisfy lots of the properties of their counterpart in geometry., which are properties of the category of representations $\mathcal{O}$ of a semisimple Lie algebra.

 

 

 

 

Seminario del Jueves 16/8

 16/8/2018, Aula 23, 14:30 hs FaMAF

Análisis esférico matricial en nilvariedadesf4

Expositor: Rocío Díaz

Resumen: 

 Sea N un grupo de Lie nilpotente, conexo, simplemente conexo dotado de una métrica Riemanniana invariante a izquierda. El grupo de isometrías de N está dado por el producto semidirecto $ K\ltimes N $, donde K es el grupo de automorfismos ortogonales de N. Consideraremos todos los pares de Gelfand $(K \ltimes N , K)$ determinados por Jorge Lauret en [“Gelfand pairs attached to representations of compact Lie groups” Tansformation Groups, 5(4):307-324 (2000)]. En estos casos las funciones sobre \(N\) a valores escalares que son integrables y K-invariantes forman un álgebra conmutativa con el producto de convolución. Para cada representación irreducible W de K determinaremos si las funciones de N en \(End(W)\) que son integrables y bi-W-equivariantes forman un álgebra conmutativa.