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Doctoral Thesis
DOI
Document
Author
Full name
Fernando Araujo Borges
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2015
Supervisor
Committee
Marcos, Eduardo do Nascimento (President)
Alvares, Edson Ribeiro
Iusenko, Kostiantyn
Salazar, Hernan Alonso Giraldo
Trepode, Sonia Elizabeth
Title in Portuguese
Álgebra c-conglomerada e c-frisos
Keywords in Portuguese
Álgebra de conglomerado
Aplicação de Caldero-Chapoton
c-friso
Abstract in Portuguese
Neste trabalho introduzimos uma nova classe de álgebra de conglomerado com coeficientes do tipo Dynkin A_n, a qual denominaremos álgebra c-conglomerada. Desenvolvemos a teoria dos c-frisos, a qual foi introduzida por Matte, Desloges e Sanchez, para o estudo das propriedades combinatórias da álgebra c-conglomerada. Usando c-frisos, obtemos uma fórmula explícita para as variáveis de conglomerado de uma álgebra c-conglomerada que explica simultaneamente o fenômeno de Laurent e a positividade. Interpretamos geometricamente a álgebra c-conglomerada em termos de triangulações de polígonos, em que triangulações correspondem aos conglomerados e diagonais correspondem às variáveis de conglomerado de uma álgebra c-conglomerada. Além disso, generalizamos a aplicação de Caldero-Chapoton e utilizamos esta versão mais geral para obter as variáveis de conglomerado de uma álgebra c-conglomerada em função dos objetos indecomponíveis da categoria de conglomerado do tipo A_n.
Title in English
c-Cluster algebra and c-friezes
Keywords in English
c-frieze
Caldero-Chapoton map
Cluster algebra
Abstract in English
In this work we introduce a new class of cluster algebra with coefficients of Dynkin type A_n, which we call c-cluster algebra. In order to study the combinatorics of the c-cluster algebra, we develop the theory of c-friezes introduced by Matte, Desloges and Sanchez. Using c-friezes, we give an explicit formula for all cluster variables of a c-cluster algebra, which explains simultaneously the Laurent phenomenon and the positivity. A c-cluster algebra also has a geometric interpretation in terms of triangulations of a polygon, where clusters are in one-to-one correspondence with triangulations and the cluster variables are in one-to-one correspondence with diagonals. Finally, we give a generalization of the Caldero-Chapoton map which we use to obtain the cluster variables of a c-cluster algebra in terms of the indecomposable objects of the cluster category of type A_n.
 
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frisos_tese.pdf (946.64 Kbytes)
Publishing Date
2019-08-21
 
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