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Cluster automorphisms and hyperbolic cluster algebrasSaleh, Ibrahim A. January 1900 (has links)
Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / Let A[subscript]n(S) be a coefficient free commutative cluster algebra over a field K. A cluster automorphism is an element of Aut.[subscript]KK(t[subscript]1,[dot, dot, dot],t[subscript]n) which leaves the set of all cluster variables, [chi][subscript]s invariant. In Chapter 2, the group of all such automorphisms is studied in terms of the orbits of the symmetric group action on the set of all seeds of the field K(t[subscript]1,[dot,dot, dot],t[subscript]n).
In Chapter 3, we set up for a new class of non-commutative algebras that carry a
non-commutative cluster structure. This structure is related naturally to some hyperbolic algebras such as, Weyl Algebras, classical and quantized
universal enveloping algebras of sl[subscript]2 and the quantum coordinate algebra of SL(2). The cluster structure gives rise to some combinatorial data, called cluster strings, which are used to introduce a class of representations of Weyl algebras. Irreducible and indecomposable
representations are also introduced from the same data.
The last section of Chapter 3 is devoted to introduce a class of categories that
carry a hyperbolic cluster structure. Examples of these categories are the categories of representations of certain algebras such as Weyl
algebras, the coordinate algebra of the Lie algebra sl[subscript]2, and the quantum coordinate algebra of SL(2).
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Geometrical and combinatorial generalizations of the associahedron / Généralisations géométriques et combinatoires de l'associaèdreManneville, Thibault 06 July 2017 (has links)
L'associaèdre se situe à l'interface de plusieurs domaines mathématiques. Combinatoirement, il s'agit du complexe simplicial des dissections d'un polygone convexe (ensembles de diagonales ne se croisant pas deux à deux). Géométriquement, il s'agit d'un polytope dont les sommets et les arêtes encodent le graphe dual du complexe des dissections. Enfin l'associaèdre décrit la structure combinatoire qui définit la présentation par générateurs et relations de certaines algèbres, dites << amassées >>. Du fait de son omniprésence, de nouvelles familles généralisant cet objet sont régulièrement découvertes. Cependant elles n'ont souvent que de faibles interactions. Leurs études respectives présentent de notre point de vue deux enjeux majeurs : chercher à les relier en se basant sur les propriétés connues de l'associaèdre ; et chercher pour chacune des cadres combinatoire, géométrique et algébrique dans le même esprit.Dans cette thèse, nous traitons le lien entre combinatoire et géométrie pour certaines de ces généralisations : les associaèdres de graphes, les complexes de sous-mots et les complexes d'accordéons. Nous suivons un fil rouge consistant à adapter, à ces trois familles, une méthode de construction des associaèdres comme éventails (ensembles de cônes polyédraux), dite méthode des d-vecteurs et issue de la théorie des algèbres amassées. De manière plus large, notre problématique principale consiste à réaliser, c'est-à-dire plonger géométriquement dans un espace vectoriel, des complexes abstraits. Nous obtenons trois familles de nouvelles réalisations, ainsi qu'une quatrième encore conjecturale dont les premières instances constituent déjà des avancées significatives.Enfin, en sus des résultats géométriques, nous démontrons des propriétés combinatoires spécifiques à chaque complexe simplicial abordé. / The associahedron is at the interface between several mathematical fields. Combinatorially, it is the simplicial complex of dissections of a convex polygon (sets of mutually noncrossing diagonals). Geometrically, it is a polytope whose vertices and edges encode the dual graph of the complex of dissections. Finally the associahedron describes the combinatorial structure defining a presentation by generators and relations of certain algebras, called ``cluster algebras''. Because of its ubiquity, we regularly come up with new families generalizing this object. However there often are only few interactions between them. From our perspective, there are two main issues when studying them: looking for relations on the basis of known properties of the associahedron; and, for each, looking for combinatorial, geometric and algebraic frameworks in the same spirit.In this thesis, we deal with the link between combinatorics and geometry for some of these generalizations: graph associahedra, subword complexes and accordion complexes. We follow a guidelight consisting in adapting, to these three families, a method for constructing associahedra as fans (sets of polyhedral cones), called the d-vector method and coming from cluster algebra theory. More generally, our main concern is to realize, that is geometrically embed in a vector space, abstract complexes. We obtain three new families of generalizations, and a fourth conjectural one whose first instances already constitute significant advances.Finally in addition to the geometric results, we prove combinatorial properties specific to each encountered simplicial complex.
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Cluster structures for 2-Calabi-Yau categories and unipotent groupsScott, J, Reiten, I, Iyama, O, Buan, A.B. 12 1900 (has links)
No description available.
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Reconhecimento polinomial de álgebras cluster de tipo finito / Polynomial recognition of cluster algebras of finite typeDias, Elisângela SIlva 09 September 2015 (has links)
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Previous issue date: 2015-09-09 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / Cluster algebras form a class of commutative algebra, introduced at the beginning of the
millennium by Fomin and Zelevinsky. They are defined constructively from a set of generating
variables (cluster variables) grouped into overlapping subsets (clusters) of fixed
cardinality. Since its inception, the theory of cluster algebras found applications in many
areas of science, specially in mathematics. In this thesis, we study, with computational focus,
the recognition of cluster algebras of finite type. In 2006, Barot, Geiss and Zelevinsky
showed that a cluster algebra is of finite type whether the associated graph is cyclically
oriented, i.e., all chordless cycles of the graph are cyclically oriented, and whether the
skew-symmetrizable matrix associated has a positive quasi-Cartan companion. At first,
we studied the two topics independently. Related to the first part of the criteria, we developed
an algorithm that lists all chordless cycles (polynomial on the length of those
cycles) and another that checks whether a graph is cyclically oriented and, if so, list all
their chordless cycles (polynomial on the number of vertices). Related to the second part
of the criteria, we developed some theoretical results and we also developed a polynomial
algorithm that checks whether a quasi-Cartan companion matrix is positive. The latter
algorithm is used to prove that the problem of deciding whether a skew-symmetrizable
matrix has a positive quasi-Cartan companion for general graphs is in NP class. We conjecture
that this problem is in NP-complete class.We show that the same problem belongs
to the class of polynomial problems for cyclically oriented graphs and, finally, we show
that deciding whether a cluster algebra is of finite type also belongs to this class. / As álgebras cluster formam uma classe de álgebras comutativas introduzida no início
do milênio por Fomin e Zelevinsky. Elas são definidas de forma construtiva a partir de
um conjunto de variáveis geradoras (variáveis cluster) agrupadas em subconjuntos sobrepostos
(clusters) de cardinalidade fixa. Desde a sua criação, a teoria das álgebras cluster
encontrou aplicações em diversas áreas da matemática e afins. Nesta tese, estudamos,
com foco computacional, o reconhecimento das álgebras cluster de tipo finito. Em 2006,
Barot, Geiss e Zelevinsky mostraram que uma álgebra cluster é de tipo finito se o grafo
associado é ciclicamente orientado, isto é, todos os ciclos sem corda do grafo são ciclicamente
orientados, e se a matriz antissimetrizável associada possui uma companheira
quase-Cartan positiva. Em um primeiro momento, estudamos os dois tópicos de forma
independente. Em relação à primeira parte do critério, elaboramos um algoritmo que lista
todos os ciclos sem corda (polinomial no tamanho destes ciclos) e outro que verifica se
um grafo é ciclicamente orientado e, em caso positivo, lista todos os seus ciclos sem corda
(polinomial na quantidade de vértices). Relacionado à segunda parte do critério, desenvolvemos
alguns resultados teóricos e elaboramos um algoritmo polinomial que verifica
se uma matriz companheira quase-Cartan é positiva. Este último algoritmo é utilizado
para provar que o problema de decidir se uma matriz antissimetrizável tem uma companheira
quase-Cartan positiva para grafos gerais está na classe NP. Conjecturamos que
este problema pertence à classe NP-completa. Mostramos que o mesmo pertence à classe
de problemas polinomiais para grafos ciclicamente orientados e, por fim, mostramos que
decidir se uma álgebra cluster é de tipo finito também pertence a esta classe.
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