X Encuentro Rioplatense, 28 de abril al 2 de mayo 2003

Este encuentro rioplatense de álgebra tendrá lugar en Buenos Aires, Dto. de matemática F.C.E.yN. - Universidad de Buenos Aires, pabellón I.

Se enmarca dentro del proyecto PICS 1514 de colaboración Francia - Argentina, y recibe también el apoyo financiero de la Cooperación Regional Francesa, Ministère d'Affaires étrangères (proyecto de colaboración Francia - Uruguay), la Agencia Nacional de Promoción Científica y Tecnológica (Argentina), y el CONICET (Argentina).

Instrucciones para llegar desde Retiro a Ciudad Universitaria.

Actividades:

Horarios

Resúmenes de los cursos

Iván Pan (Instituto de Matematica UFRGS - Porto Alegre)

Transformaciones de Cremona

El estudio de las transformaciones de Cremona fue uno de los temas de investigación predilectos de los geómetras de la llamada Escuela Italiana (Cremona, Comessati, Noether, Castelnuovo, etc...) dando origen a lo que hoy conocemos como Geometría Birracional.

El objetivo de este mini-curso es, primeramente, dar un panorama general sobre el área describiendo algunas de las técnicas de trabajo y algunos de los problemas abiertos. Luego nos concentraremos en el caso más simple (y más conocido) de las transformaciones del plano indicando los teoremas de estructura del Grupo de Cremona y sus subgrupos. Finalmente, y dependiendo del tiempo que nos reste, estudiaremos alguna de las aplicaciones de este tema a otras áreas de la Matemática, principalmente a estudio birracional de foliaciones analíticas.

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Alexander Zimmermann (Université de Picardie)

Triangulated categories and equivalences in representation theory.

In a first part we shall explain the main method, equivalences between derived categories, which became a main tool in represenation theory after the major discovery of Rickard and Keller.

In joint work with Jan Schroer we showed that a nice class of algebras of tame representation type so-called gentle algebras, is closed under derived equivalences. Actually more is true, involving stable categories of a bigger algebra.

We will present some applications, and show some link to Hochschild cohomology to this and other algebras. Hochschild cohomology is a very useful tool which was used by Linckelmann to define group cohomology of a non principal block. We outline this development and give some further applications and examples.

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Resúmenes de las charlas

Ibrahim Assem (Université Sherbrooke - Québec)

The left and the right parts of a module category.

This is a report on a joint work with F. U. Coelho and S. Trepode. We study, for an Artin algebra A, the class L_A (and R_A) which is a full subcategory of the category modA of finitely generated A-modulos, and which consists of all indecomposables A-modulos whose predecessors (and successors) have projective dimension (and injective dimension, respectively) at most one. We consider certain quotient algebras of A, which contain the information on these classes, then define and characterize those algebras for which the class L_A is contravariantly finite (and R_A is covariantly finite, respectively) in modA.

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Claudia Chaio (Universidad Nacional de Mar del Plata)

On the composition of irreducible morphisms

The notion of irreducible morphism has played an important role in the study of the category modA of finitely generated modules over an Artin algebra A. The connection with the radical R of this category is well known and is given by the fact that a morphism between indecomposable modules is irreducible if and only if it lies in R and not in R². It is well known that the composition of n irreducible morphisms belongs to Rⁿ. An interesting question is when does the composition of n irreducible morphisms belongs to Rⁿ⁺¹.

We can associate to mod A an oriented graph, called the Auslander-Reiten quiver. The connected components of this graph are called the components of the Auslander-Reiten quiver. We show that in the directed components of the Auslander-Reiten quiver of a strongly simply connected algebra the composition of n irreducible morphisms, belongs to R^n+k with k>0 if and only if that composition is zero. In particular, this statement is true for the Auslander-Reiten quiver of a simply connected algebra of finite representation type. Now, we are interesting in answering the following question: when does the composition of two irreducible morphisms belongs to R^k\setminus R^k+1 with k >2. We analyze necessary and sufficient conditions for the existence of such two irreducible morphisms. Finally, we give a family of algebras of finite representation type having two irreducible morphisms f,g between indecomposable modules such that gf\in R^m\setminus R^m+1 with m >2.

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Diane Canstonguay

Derived-tame blowing of tree algebra

Let k be an algebraically closed field and A=kQ/I be a tree algebra. It is already shown by T. Br\"ustle and also by C. Geiss, with different methods, that A is derived-tame if and only if the Euler quadratic form \chi_A of A is non-negative. Moreover, in this case, A is derived equivalent to a hereditary algebra of type \E, \EE, or to a tubular algebra, or else to a special type of incidence algebras, called semichain algebra.

We consider here a class obtained by blowing a tree algebra A at a set of vertices D, such an algebra is denoted by A{D}. As will be seen, this procedure of blowing a tree is very natural in our context. For instance, semichain algebra are obtained in this way. The objective of this talk is to give the equivalence between derived-tameness and non-negativity of the Euler form for algebras of this form. We also show that, in this case, if D is a non-empty set that A{D} must be derived equivalent to a semichain algebra.

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Flávio Ulhoa Coelho (Universidade de São Paulo)

Two-sided gluings of tilted algebras

This talk is a following up of the talk by I. Assem and we shall discuss here our joint work contained in [A-C2].

Let A be an Artin algebra. We are interested in studying the representation theory of A, thus in characterizing A by properties of the category modA of finitely generated right A-modules. One method to achieve this goal is to start from a class of algebras whose representation theory is considered to be sufficiently well-understood, and then to generalize this class to another whose representation theory is close enough to that of the preceding class. Thus, tilted algebras were introduced in [H-R] as a generalization of hereditary algebras. The class of tilted algebras is now considered to be one of the most useful for the general theory. For instance, it is known that an indecomposable module over an arbitrary algebra which does not lie in an oriented cycle of non-zero non-isomorphisms, is a module over a tilted algebra [R]. It was therefore natural to consider various generalizations of this notion. Thus, over the years, the following classes of algebras were defined and investigated: the quasi-tilted (which generalize the tilted and the canonical algebras of [R]) [H-R-S], the shod algebras (which generalize the quasi-tilted) [C-L1], the weakly shod algebras (which generalize the shod and the representation-directed algebras) [C-L2,C-L3] and the left and the right glued algebras (which generalize the tilted and the representation-finite algebras) [A-C1]. The purpose of our talk is to introduce a new class of algebras which generalizes all the previous classes.

We define an Artin algebra A to be a laura algebra if all but at most finitely many non-isomorphic indecomposable A-modules are such that all their predecessors have projective dimension at most one, or all their successors have injective dimension at most one. We start by giving various examples and characterizations of laura algebras. We then study the representation theory of laura algebras, and our main theorem gives a full description of the Auslander-Reiten quiver of a laura algebra. The class of laura algebras is then characterized in the spirit of [A-C1] as a double gluing of tilted algebras.

Bibliography

[A-C1] I. Assem, F. U. Coelho, Glueings of tilted algebras, J. Pure Appl. Algebra 96(3) (1994), 225-243.

[A-C2] I. Assem, F. U. Coelho, Two-sided gluing of tilted algebras.

[C-L1] F. U. Coelho, M. Lanzilotta, Algebras with small homological dimensions, Manuscripta Mathematica 100 (1999) 1-11.

[C-L2] F. U. Coelho, M. Lanzilotta, On semiregular components with paths from injective to projective modules, Comm. Algebra 30, 10 (2002).

[C-L3] F. U. Coelho, M. Lanzilotta, Weakly shod algebras, J. Algebra, to appear.

[H-R-S] D. Happel, I. Reiten, S. Smal\o, Tilting in abelian categories and quasitilted algebras, Mem. Am. Math. Soc. 120 (1996), No. 575.

[H-R] D. Happel, C. Ringel, Tilted algebras, Trans. AMS 274 (1982), 399-443.

[R] C. Ringel, Tame algebras and integral quadratic forms, Springer Lect. Notes Math. 1099 (1984).

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Juan José Guccione

Teoría de productos cruzados trenzados

Esta es una comunicación acerca de dos trabajos. Uno en colaboración con Jorge Guccione y otro con Jorge Guccione y Constanza Di Luigi. Nosotros definimos un tipo de producto cruzado sobre álgebras de Hopf trenzadas, que generaliza los ahora clásicos introducidos independientemente por Blattner, Cohen y Montgomery y por Doi and Takeuchi, y estudiamos algunas de sus propiedades. Por ejemplo, probamos un Teorema de Maschke para estos nuevos productos cruzados y bajo hipótesis adecuadas construímos un contexto Morita que extiende el encontrado por Cohen, Fischman y Montgomery en el caso clásico. Por último caracterizamos las álgebras de Hopf cuyo doble de Drinfel'd es un producto cruzado en nuestro sentido, obteniendo una versión para este nuevo contexto de un conocido resultado de Majid.

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Matías Graña

Quandles y álgebras de Nichols.

Un quandle es un conjunto con una operación binaria, generalización de un grupo con la conjugación. Los quandles fueron utilizados sobre todo en topología para dar invariantes de nudos. En esta charla contaré cómo se puede usar un quandle en álgebra, junto a un 2-cociclo, para dar un módulo de Yetter-Drinfeld sobre un grupo. Así se producen de manera sistemática espacios trenzados, sobre los cuales se pueden (se intentan) calcular las álgebras de Nichols. La charla estará sazonada con definiciones y ejemplos.

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Alejandra Patricia Jancsa (FaMAF, Univeridad Nacional de Córdoba)

Biálgebras de Lie reales simples

Describiré las estructuras de biálgebras de Lie casi factorizables en álgebras de Lie reales g₀ absolutamente simples, es decir, aquellas cuya complexificación g=g₀\otimes _RC es una biálgebra de Lie simple factorizable. Una de las herramientas esenciales es el teorema de Belavin - Drinfel'd, que da una descripción exhaustiva de las estructuras de biálgebras de Lie factorizables en álgebras de Lie complejas simples. La determinación de las clases de isomorfismo requiere además de la clasificación de las álgebras de Lie reales simples, obtenida a partir de diagramas de Vogan e involuciones antilineales.

Por otra parte, describiré los dobles de Drinfel'd y los triples de Manin asociados a las biálgebras de Lie reales absolutamente simples.

Este trabajo está contenido en mi tesis doctoral y ha sido aceptado para su publicación en Int. Math. Research Notices, en coautoria con mi director N. Andruskiewitsch.

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Marcelo Lanzilotta

Álgebras Toupie

Trabajo en conjunto con: Diane Castonguay; Francois Huard.

Con el objetivo final de clasificar los carcajes de las álgebras Shod, Weakly-Shod, y Laura, definimos en el trabajo una vasta familia que llamamos: Carcajes Toupie. Obtenemos la clasificación de los carcajes de esta familia (con cualquier ideal admisible) según representen un álgebra Shod, Weakly Shod o Laura.

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Alejandro Lopez (Universidad Nacional Andres Bello)

Local Cohomology for Graded - commutative Rings

A graded ring satisfying the identity rs=(-1)^deg(r)deg(s) for all homogeneous elements r and s is called a graded-commutative ring. Such rings appear as cohomology rings of groups and in a number of other contexts. Their homological properties seem to be similar to the well-known properties of commutative rings. Although one may treat these rings as modules over the even part of the ring, it is interesting to study them as modules over themselves and to compare the structures obtained.

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Eduardo do Nascimento Marcos

Estructura estandar estratificada en Álgebras

Esta plática en Portunhol, sera sobre el trabajo conjunto que estamos desenvolviendo Eduardo N. Marcos y Edith Corina Saenz Valadez.

Consideraremos un conjunto de R-modulos \theta={\theta(1),... ,\theta(n)}. Mas aun sea \Y={Y(1),... , Y(n)} un conjunto de R-modulos inescindibles. Decimos que (\theta, \Y) es un sistema estratificante si las siguientes 3 condiciones se satisfacen.