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Cluster-tilting theory
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The notion of Cluster Algebra first appeared in 2001, in a paper by S. Fomin and A. Zelevinsky, studying the dual canonical basis of the quantum group of a finite dimensional simple Lie algebra over the complex numbers, and also total positivity for algebraic groups. Cluster categories, introduced by A. Buan, R. Marsh, M. Reineke, I. Reiten and G.Todorov [11] in 2004, are certain quotients of the derived category of the module category of a finite dimensional path algebra, and the cluster-tilted algebras, defined by A. Buan, R. Marsh and I. Reiten [8], are the endomorphism algebras of certain objects in a cluster category. Our aim is to study the cluster-tilted algebras associated to a Dynkin quiver of type A, using the graphical definition, involving diagonals of polygons, of the cluster category and its associated cluster-tilted algebras, introduced by P. Caldero, F. Chapoton and R. Schiffler, in 2006. In order to understand the cluster-tilted algebras, we need some background on the theory of representations of algebras (with emphasis on representations of quivers and the Auslander-Reiten theory) and on the theory of cluster algebras (with emphasis on the description of cluster algebras of type A in terms of diagonals and triangulations of a regular polygon and the set of almost positive roots in the root system of type A).
The notion of Cluster Algebra first appeared in 2001, in a paper by S. Fomin and A. Zelevinsky, studying the dual canonical basis of the quantum group of a finite dimensional simple Lie algebra over the complex numbers, and also total positivity for algebraic groups. Cluster categories, introduced by A. Buan, R. Marsh, M. Reineke, I. Reiten and G.Todorov [11] in 2004, are certain quotients of the derived category of the module category of a finite dimensional path algebra, and the cluster-tilted algebras, defined by A. Buan, R. Marsh and I. Reiten [8], are the endomorphism algebras of certain objects in a cluster category. Our aim is to study the cluster-tilted algebras associated to a Dynkin quiver of type A, using the graphical definition, involving diagonals of polygons, of the cluster category and its associated cluster-tilted algebras, introduced by P. Caldero, F. Chapoton and R. Schiffler, in 2006. In order to understand the cluster-tilted algebras, we need some background on the theory of representations of algebras (with emphasis on representations of quivers and the Auslander-Reiten theory) and on the theory of cluster algebras (with emphasis on the description of cluster algebras of type A in terms of diagonals and triangulations of a regular polygon and the set of almost positive roots in the root system of type A).
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Tese de mestado em Matemática (Álgebra, Lógica e Fundamentos) apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2008
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Matemática Teses de mestrado