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Cours basiques et avancés de l’école
Introducción a la teorı́a de representaciones de álgebras.
Ibrahim Assem
Université de Sherbrooke, Québec
El objetivo del curso es estudiar los fundamentos de la teorı́a de representaciones de álgebras de dimensión finita sobre un cuerpo. El curso consiste
en los siguientes capı́tulos: 1. Carcajes y álgebras: álgebras de caminos, ideales admisibles y cocientes de álgebras de caminos, el carcaj de un álgebra
de dimensión finita; 2. Representaciones y módulos: representaciones de carcajes con relaciones, los módulos simples, proyectivos e inyectivos; 3. Teorı́a
de Auslander-Reiten: morfismos irreducibles y sucesiones casi escindidas, las
traslaciones de Auslander-Reiten, existencia de sucesiones casi escindidas; 4.
El carcaj de Auslander-Reiten: el carcaj de Auslander-Reiten, la primera
conjetura de Brauer-Thrall y el teorema de Auslander, un ejemplo: álgebras
de Nakayama; 5. Teorı́a de inclinación: módulos inclinantes, el teorema
de Brenner-Butler y sus consecuencias; 6. Algebras hereditarias y álgebras
inclinadas: álgebras hereditarias, grafos de Dynkin y grafos euclidianos, el
teorema de Gabriel, álgebras inclinadas y el criterio de Liu-Skowronski.
===========================================
Polynomial invariants, an introduction to some classical
commutative topics and an overview of some current
noncommutative developments.
François Dumas
Université de Clermont-Ferrand, France
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The course will be organized in the following way: 1. Linear invariants:
actions of subgroups of GL(n, C) on polynomial algebras; some classical results in commutative invariant theory; analogues for noncommutative polynomials and Weyl algebras; applications in Lie theory and differential operators algebras. 2. Multiplicative invariants: actions of subgroups of GL(n, Z)
on Laurent polynomial algebras; analogues for quantum tori and Hopf algebras; applications in quantum groups theory. 3. Rational invariants: classical
Noether’s problem and related results; analogues for noncommutative rational functions; applications to the Weyl skewfields and quantum Weyl skewfields; links with the Gelfand-Kirillov problem. 4. Completion and invariants:
automorphisms actions on commutative and non-commutative power series;
applications to pseudodifferential operators; links with the modular forms
theory.
===========================================
Generic modules and complexes.
Raymundo Bautista
Universidad Nacional Autónoma de México, México
The main aim of the course is to present the research made by many
authors about properties of modules of finite endolength, which means that
the module has finite length when considered as module over its endomorphism ring. He will present properties of these modules, and their connection
with the pure-injective modules. The indecomposable not finitely generated
finite endolength modules are the so called generic modules. When the field
is algebraically closed and the algebra is of infinite representation type, he
will show that there exists a bijection between isomorphism classes of generic
modules and families of Auslander-Reiten components, parametrized by one
parameter. Finally, he will present the generic complexes in the derived category of the algebra.
===========================================
La structure de Poisson de certaines varietés quotient.
Jacques Alev
Université de Reims, France
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Soit V un espace symplectique et G un sous-groupe fini de Sp(V ). La
variété quotient V /G porte une riche structure de Poisson. Dans ce cours,
nous étudierons cette structure et rappellerons les résultats fondamentaux
sur les variétés de Poisson. Puis, nous étudierons les déformations non commutatives en vue d’une quantification et de la comparaison des homologies
de Poisson de V /G et de l’homologie de Hochschild de sa quantification.
===========================================
Koszul algebras and applications.
Roberto Martı́nez-Villa
Universidad Nacional Autónoma de México, México
We will present first some basic facts about Koszul algebras and their
representations. We will then proceed and talk about the Koszul duality
and the Bernstein-Gelfand-Gelfand theorem and some of its uses. One of
the main aims of this minicourse is to present examples of situations where
Koszul algebras appear or can be used. Among the applications that will be
discussed will be the preprojective algebra, selfinjective Koszul algebras and
applications to the study of vector bundles over the projective n-space.
===========================================
Group actions on finite dimensional algebras and their categories
of modules.
José Antonio de la Peña
Universidad Nacional Autónoma de México, México
We will consider the automorphism groups of algebras and the associated
locally finite categories, and different applications in the representation theory. We will present the following subjects: 1. Galois coverings: modules
of first and second type; invariance of the representation type; 2. Induced
actions on the module categories; 3. Symmetries of algebras and their consequences in the theory of associated invariants (quadratic forms, Coxeter
polynomial); 4. Computation of the simplicial cohomology and Hochschild
cohomology using Galois coverings.
===========================================
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Derived categories and their applications.
Marı́a Julia Redondo and Andrea Solotar
Universidad Nacional del Sur, Argentina
Universidad de Buenos Aires, Argentina
We will start with the basic definitions of derived categories, derived functors, tilting complexes and stable equivalences of Morita type. With several
examples, we will show that this is the best framework to do homological
algebra and will exhibit their usefulness for getting new proofs of well known
results. We will consider the invariants of a ring under derived equivalences:
Grothendieck group, Hochschild cohomology.
===========================================
Homological conjectures and degenerations of modules.
Sverre Smalø
University of Trondheim, Norway
The first two hours will be dedicated to some homological conjectures
and an example giving a negative answer to a question of Maurice Auslander. In addition, Prof. Smalø will give a series of lectures on degenerations
of modules over the same algebra.
===========================================
Cluster categories and their relation to Cluster algebras,
Semi-invariants and Homology of torsion free nilpotent groups.
Gordana Todorov
Northeastern University, Boston, USA
1. Cluster categories will be defined and their basic properties stated as
done by Buan, Marsh, Reineke, Reiten, T.
2. Cluster algebras were introduced by Fomin and Zelevinsky. Known
relations between cluster categories and combinatorics of cluster algebras will
be stated, as well as some of the open questions.
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3. Semi-invariants for quivers were studied by Schofield, Derksen and
Weyman. Generalized semi-invariants will be defined and the theorems relating domains of such semi-invariants and the simplicial complexes associated
to cluster categories will be given.
4. The same simplicial complexes associated to cluster categories are related to the Igusa-Orr pictures in the homology of nilpotent groups. Results,
and many open questions in this direction will be stated.
===========================================
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