Seminar of March 22nd

22/03/2018, 14.30 hs, aula 27 FaMAF

Restrictions of representations of Lie groups to reductive subgroups

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Speaker: Jorge Vargas

Abstract: We will present an overview of the subject and open problems. We will announce some results from the speaker obtained in collaboration with Michael Duflo or Bent Orsted.

Seminario del 22 de marzo

22/03/2018, 14.30 hs, aula 27 FaMAF

Restricciones de representaciones de grupos de Lie a subgrupos reductivos

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Expositor: Jorge Vargas

Resumen: Se presentara un panorama sobre el tema y problemas abiertos. Se enunciaran algunos resultados
del expositor obtenidos conjuntamente con Michael Duflo o Bent Orsted.

Prerequisitos: cuarto año de la licenciatura, quizas!!!

Jornada de doble seminario del 23 de noviembre

La jornada constará de dos seminarios, con el siguiente cronograma:

* 14.30-15.30: Seminario de Giovana Carnovale,

* 15.30-16.00: Café en la Sala de Matemática,

* 16.00-17.00: Seminario de Iván Darío Gomez

Más información sobre cada charla a continuación.

23/11/2017, 14:30 hs, aula 27 FaMAF

The Jordan stratification in Lie algebras and algebraic groups

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Expositor: Giovana Carnovale

Resumen: Semisimple Lie algebras and algebraic groups can be stratified in terms of so-called Jordan classes or decomposition classes. In the Lie algebra case they were introduced in Borho and Kraft’s work on sheets whereas the group analogue appeared in Lusztig’s construction of generalised Springer’s correspondence. Roughly speaking, Jordan classes are unions of adjoint orbits (or conjugacy classes) that are isomorphic as homogeneous space. We are interested in their geometry and in the geometry of the induced stratifications on the geometric quotients of the Lie algebra and of the group. We will show how some of these problems can be interpreted in terms of hyperplane arrangements.
It is based on a joint project with Francesco Esposito.

 

23/11/2017, 16:00 hs, aula 27 FaMAF

Estructura del producto tensorial de los sl(2)⋉ V(m)​​ -módulos uniseriales

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Expositor: Iván Darío Gomez

Resumen:  Sea ℊ​​ un álgebra de Lie sobre ℂ​​, el sócalo de un ℊ-módulo V​​ es el único ℊ​​-submódulo maximal semisimple de V y se denota por soc​​(V). A V se le denomina uniserial si la serie del sócalo es una serie de composición, es decir,


soc0(V)⊂ soc1(V)⊂ \hdots ⊂ socn(V)=V​​


es una serie de composición donde soci(V)/soci-1(V)=soc(V/soci-1(V))  para 1≤ i≤ n.


En [C-S] se obtiene la clasificación de los ℊ​​-módulos uniseriales cuando la descomposición de Levi de ℊ​​ es sl(2)⋉ V(m) ​ para m≥ 1​​, donde V(m)​​ es un sl(2)​​-módulo irreducible de peso máximo m​​.
Tales módulos uniseriales son los Z(a,l)​​ (salvo algunos caso especiales) los cuales vistos como sl(2)​​-módulos son Z(a,l)=⨁i=0l V(a+im)​​  y sus respectivos duales Z(a,l)*​​ con a,l ∈ ℕ∪ {0}​​.


En la primera parte de esta charla se hablará sobre el sócalo del producto tensorial de sl(2)⋉ V(m)​​ -módulos uniseriales, el cual nos permite construir nuevos módulos y demostrar con m​​​​ impar que Z(0,1)⊗ Z(b,1)​​​​ es indescomponible si b≠ 0​​​​.


Recordamos que un ℊ​​​​-módulo V​​​​ es cíclico si V=U(ℊ)v​​ para algún v∈ V​​​​ y donde U(ℊ)v​​​ es la envolvente universal de ℊ​​​​. En la segunda parte de la charla, se mostrará que ciertos productos tensoriales de sl(2)⋉ V(m)​​​​-módulos uniseriales son módulos cíclicos.


Bibliografía:
[Ca] P. Casati, The classification of the perfect cyclic sl(n+1)⋉ ℂn+1, Journal of Algebra 476 (2017) 311-343.


[C-S] L. Cagliero and F. Szechtman, The classification of uniserial sl(2)⋉ V(m)-modules and a new interpretation of the Racah-Wigner 6j-symbol, J. of Algebra, Volume 386 (2013), 142-175.


[Pi] A. Piard, Sur des représentations indécomposables de dimension finie de \matfrak{SL}(2).R2​​, Journal of Geometry and Physics, Volume 3, Issue 1, 1986, 1–53.

 

 

 

Seminario del 09 de noviembre

09/11/2017, 16.00 hs, aula 27 FaMAF

Hopf algebras having a dense big cell

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Expositor: Julien Bichon

Resumen: We will discuss some axioms that ensure that a quantum goup has its irreducible representations classified by means of an analogue of the Borel-Weil construction. The axioms are inspired by the work of Parshall-Wang for the usual q-deformation of GL(n).
We will examine in detail the example of the the free quantum group GL(2), for which the weight group is the free group on two generators. The talk will be based on joint work with Simon Riche.

Seminar of November 9th

09/11/2017, 16.00 hs, aula 27 FaMAF

Hopf algebras having a dense big cell

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Speaker: Julien Bichon

Abstract: We will discuss some axioms that ensure that a quantum goup has its irreducible representations classified by means of an analogue of the Borel-Weil construction. The axioms are inspired by the work of Parshall-Wang for the usual q-deformation of GL(n).
We will examine in detail the example of the the free quantum group GL(2), for which the weight group is the free group on two generators. The talk will be based on joint work with Simon Riche.

Seminario del 26 de Octubre

26/10/2017, 14.30 hs, aula 27 FaMAF

Rigidez de álgebras de Lie k pasos nilpotentes vía el teorema de Nash-Moser

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Expositor: Augusto Chavez

Resumen: Al conjunto de las álgebras de Lie de dimensión finita n, k-pasos nilpotentes, se les asocia un conjunto algebraico N_{n,k}. Nos dedicaremos a aplicar el teorema de Nash-Moser para secuencias exactas cotas de R. Hamilton en el contexto de rigidez en el conjunto N_{n,k}.

Dada un álgebra de Lie g de dimensión finita n k-pasos nilpotente, discutiremos algunos aspectos de cierto espacio vectorial H^{2}_{k−nil}(g, g), el cual nos provee información sobre la rigidez de g en N_{n,k}. Daremos algunos ejemplos de álgebras de Lie rígidas en N_{n,k}. Cuando k=2, presentaremos algunos criterios mas sobre rígidez en N_{n,2}.

Este es un trabajo en conjunto con Cagliero, Leandro y Brega, Oscar.

Seminar of October 26th

26/10/2017, 14.30 hs, aula 27 FaMAF

Rigidity of the k-steps nilpotent Lie algebras through the Nash-Moser theorem

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Speaker: Augusto Chavez

Abstract: Given the set of k-step nilpotent Lie algebras of dimension n, we associate an algebraic set N_{n, k} for them. We apply the Nash-Moser theorem to exact sequences of R. Hamilton in the context of rigidity in the set N_{n, k}.
Given a k-step nilpotent Lie algebra of dimension n, we will discuss some aspects of a certain vector space H^{2} _{k-nil} (g, g), which gives us information about the rigidity of g in N_{n, k}. We will give some examples of rigid Lie algebras in N_{n,k}. When k = 2, we will present some more criteria on rigidity in N_ {n, 2}.

Double seminar day of November 23th

The seminar will have two talks, with the following schedule:

* 14.30-15.30: Seminar of Giovanna Carnovale,

* 15.30-16.00: Coffee at Sala de Matemática,

* 16.00-17.00: Seminar of Iván Darío Gomez.

More information about each talk is available below.

23/11/2017, 14:30 hs, aula 27 FaMAF

The Jordan stratification in Lie algebras and algebraic groups

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Expositor: Giovanna Carnovale 

Resumen: Semisimple Lie algebras and algebraic groups can be stratified in terms of so-called Jordan classes or decomposition classes. In the Lie algebra case they were introduced in Borho and Kraft’s work on sheets whereas the group analogue appeared in Lusztig’s construction of generalised Springer’s correspondence. Roughly speaking, Jordan classes are unions of adjoint orbits (or conjugacy classes) that are isomorphic as homogeneous space. We are interested in their geometry and in the geometry of the induced stratifications on the geometric quotients of the Lie algebra and of the group. We will show how some of these problems can be interpreted in terms of hyperplane arrangements.
It is based on a joint project with Francesco Esposito.

 

23/11/2017, 16:00 hs, aula 27 FaMAF

Extensiones modulares de categorías de fusión super-Tannakianas

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Expositor: Iván Darío Gomez

Resumen: Let ℊ​​ be Lie algebra over ℂ​​, the socle of the ℊ​​-module V​​ is the unique maximal semi-simple ℊ​​-submodule of V and it is denote soc​​(V). A V​​ it is called uniserial if the socle series is a composition series, i.e,

 
soc0(V)⊂ soc1(V)⊂ \hdots ⊂ socn(V)=V​​

 
is a composition series where soci(V)/soci-1(V)=soc(V/soci-1(V)) for ​​1≤ i≤ n.


In [C-S] it is obtenied the classification of the uniserial ℊ​​-modules when the Levi descomposition of ℊ​​ is sl(2)⋉ V(m) for ​​m≥ 1, where ​​V(m) is a irreducible ​​sl(2)-module of highest weight m​​.
These uniserial modules are the Z(a,l)​​ (except for some special case) which as sl(2)​​-modules are

Z(a,l)=⨁i=0l V(a+im)​​ and its respective dual Z(a,l)*​​ with a,l ∈ ℕ∪ {0}​​.


In the first part of this talk will be discussed over the socle of the tensorial product of the uniserial sl(2)⋉ V(m)-modules, which we allow construct new modules and proof with ​​m odd that Z(0,1)⊗ Z(b,1)​​ is indecomposable if b≠ 0​​.


Remember that is V is a cyclic ​​ℊ-module if V=U(ℊ)v​​ for some v∈ V​​ y where U(ℊ)v​​ is the universal envelope of ℊ​​. In the second part of the talk, will be shown that certain tensorial products of uniserial sl(2)⋉ V(m)-modules are cyclics modules.


Bibliography:
[Ca] P. Casati, The classification of the perfect cyclic sl(n+1)⋉ ℂn+1, Journal of Algebra 476 (2017) 311-343.


[C-S] L. Cagliero and F. Szechtman, The classification of uniserial sl(2)⋉ V(m)-modules and a new interpretation of the Racah-Wigner 6j-symbol, J. of Algebra, Volume 386 (2013), 142-175.


[Pi] A. Piard, Sur des représentations indécomposables de dimension finie de \matfrak{SL}(2).R2​​, Journal of Geometry and Physics, Volume 3, Issue 1, 1986, 1–53.

 

 

 

 

Seminario del 12 de octubre

12/10/2017, 14.30 hs, aula 27 FaMAF

Álgebras de Hopf copunteadas sobre grupos diedrales

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Expositor: Fernando Fantino

Resumen: En esta charla se expondrán consideraciones generales sobre la clasificación de las álgebras de Hopf (co)punteadas de dimensión finita sobre un cuerpo algebraicamente cerrado de característica cero y se presentará la clasificación de las álgebras de Hopf copunteadas sobre los grupos diedrales de orden 8t, t>2.
Este es un trabajo en conjunto con G. A. Garcia y M. Mastnak.

Seminar of October 12th

12/10/2017, 14.30 hs, aula 27 FaMAF

Copointed Hopf algebras over dihedral groups

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Speaker: Fernando Fantino

Abstract: In this talk we will give general considerations on the classification of (co)pointed Hopf algebras of finite dimension over an algebraically closed filed of characteristic zero and we will present the classification of copointed Hopf algebras over dihedral groups of order 8t, t>2.
This is a joint work with G. A. Garcia y M. Mastna