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Right ada algebra

Grant number: 13/17903-3
Support type:Scholarships abroad - Research
Effective date (Start): October 30, 2013
Effective date (End): November 30, 2013
Field of knowledge:Physical Sciences and Mathematics - Mathematics
Principal Investigator:Rosana R. Signorelli Vargas
Grantee:Rosana R. Signorelli Vargas
Host: Ibrahim Assem
Home Institution: Escola de Artes, Ciências e Humanidades (EACH). Universidade de São Paulo (USP). São Paulo , SP, Brazil
Local de pesquisa : Université de Sherbrooke (UdeS), Canada  

Abstract

Theory of Algebras, we are interested in studying the category modA of finitely generated right A-modules where A is an artin algebra. For this we consider a full subcategory of modA as indA consisting of one representative from each isomorphism class of indecomposable modules. We are studying techniques related to classify algebras characterized by the subcategories LA and RA in indA. When Happel, Reiten and Smalo defined the class of quasi-tilted algebras, they introduced the notion of left part of A, LA as the full subcategory of objects being in indA with modules whose predecessors have projective dimension at most 1. The right part is defined dually and denoted by RA. Since then, these subcategories have been quite studied and applied. An artin algebra is quasi-tilted if and only if every indecomposable projective module lies in the left part of module category. Other algebras have also been introduced by these subcategories such as shod algebras (indA = LA U RA) and laura algebras (LA U RA is cofinita in indA). Through these motivations we introduce and study a new class of algebras that we call ada algebras in which any indecomposable projective module and any indecomposable injective module lies in the union LA U RA. This generalizes quasi-tilted algebras. We describe the components of the Auslander-Reiten quiver of an ada algebra. We show that their representation theory is entirely contained in the left and right supports which in this case are tilted algebras. We observe that the left support of an artin algebra is the endomorphism ring of the direct sum of all indecomposable projective modules lying in the left part of modA, and right support is defined dually. This project aims to study the class of algebras with the property that all indecomposable projective lies in LA U RA. We call these algebras as right ada algebras. We have obtained that right ada algebras have global dimension less than or equal to 4 and are triangular algebras among other interesting properties. We know that some properties of ada algebras are kept in right ada algebras. But in many aspects the right ada algebra has other distinct descriptions from the ada algebras such as their Auslander-Reiten components. One goal of this project is to obtain a complete description of the structure of the category of modules of right ada algebras. When we consider a strict right ada algebra A, we can define as in the case of ada algebras the connected component of Auslader-Reiten quiver of A containing the Ext-projective in addRa of which are in the same connected component. In ada algebras this components are directed and generalized standard. However, in the case of a right ada algebra this may not happen, causing some problems that we have to deal. We also interested in study the right ada algebras which are simply connected, that is every presentation A = kQ/I as a bound quiver algebra, the fundamental group (Q,I) is trivial. One of the classical problems in the representation theory of algebras is to discover the representation type of the algebra. This problem consists in know if the isomorphism classes number of indecomposable modules over the algebra is finite or infinite. We are interested in describing how are ada algebras and right ada algebras of finite representation type. Through this description we are taking another step in understanding the behavior of these classes of algebras using another approach and enabling the discovery of other related properties in theory. (AU)

Scientific publications
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
ALVARES, EDSON RIBEIRO; ASSEM, IBRAHIM; CASTONGUAY, DIANE; VARGAS, ROSANA R. S. Right ADA algebras. JOURNAL OF ALGEBRA AND ITS APPLICATIONS, v. 16, n. 11 NOV 2017. Web of Science Citations: 0.

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