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Representations of álgebras of finite dimension

Grant number: 98/09757-5
Support type:Research Projects - Thematic Grants
Duration: December 01, 1998 - November 30, 2002
Field of knowledge:Physical Sciences and Mathematics - Mathematics
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: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 techniques like the Auslander-Reiten theory, and for other, through the application, in the context of noncommutative algebras, of techniques 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 TA of A. To each module in indA, it is assign a vertice of TA and its arrows represent the morphism in indA which are irreductible 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 radoo (modA) of modA, since any non-isomorphism in modA can be written as a sum of compositions of irreducible morphisms and a morphism in radoo (modA). Other aspects which has been much considered are the homological and cohomological. Many classes of algebras can be characterized 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 also 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 ideals, etc. (AU)