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Representations of Artin Algebras

Grant number: 03/06001-7
Support type:Research Projects - Thematic Grants
Duration: February 01, 2004 - January 31, 2008
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Algebra
Principal Investigator:Flavio Ulhoa Coelho
Grantee:Flavio Ulhoa Coelho
Home Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
Co-Principal Investigators:Eduardo Do Nascimento Marcos ; Hector Alfredo Merklen Goldschmidt

Abstract

Let A be an Artin algebra, for instance a finite dimensional algebra over a field. The main aim in representation theory of algebras is to describe the category modA consisting of the finitely generated A-modules. We shall denote by indA the subcategory of modA consisting of the nonisomorphic indecomposable modules. In such study, the theory had developed in several direction, for one hand through the introduction of new tecniques like the Auslander-Reiten theory, and for other, through the application, in the context of noncommutative algebras, of tecniques which were very useful in other research lines. Auslander-Reiten theory allows us the organization, through the so-called almost split sequences, of the category modA with the aim of a better understanding of its morphisms. One of the most useful way of visualizing such organization is through the so-called Auslander-Reiten quiver ?A of A. To each module in indA, it is assign a vertice of ?A and its arrows represent the morphism in indA which are irredutcible in the sense that they do not split neither factor non-trivially through any other module. In addition to this study, it is essential to consider the morphisms in the ideal rad8 (modA) of modA, since any non-isomorphism in modA can be written as a sum of compositions of irreducible morphisms and a morphism in rad8 (modA).Other aspects which has been much considered are the homological and cohomological. Many classes of algebras can be caracterized through certain homological properties as, for instance, projective and injective dimensions. We mention, in particular, the classes of tilted, quasitilted, shod as well as the Auslander algebras. The cohomology, mainly the Hochschild cohomology, has algo been shown very useful for the area. Most of the work done by the group of representation theory in São Paulo has been in the above mentioned directions. Some other close questions had also been investigated, for instance, the postprojective and preinjective partitions, Hopf algebras, the study of algebras through the structure of its idempotents ideaIs, etc. (AU)