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Dissertations / Theses on the topic 'Cluster algebra'

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Published: 4 June 2021

Last updated: 25 July 2025

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1

Fomin, Sergey, Andrei Zelevinsky, and fomin@math lsa umich edu. "Cluster algebras I: Foundations." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1023.ps.

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2

Vaso, Laertis. "Cluster Tilting for Representation-Directed Algebras." Licentiate thesis, Uppsala universitet, Algebra och geometri, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-364224.

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3

Pressland, Matthew. "Frobenius categorification of cluster algebras." Thesis, University of Bath, 2015. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.678852.

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Cluster categories, introduced by Buan–Marsh–Reineke–Reiten–Todorov and later generalised by Amiot, are certain 2-Calabi–Yau triangulated categories that model the combinatorics of cluster algebras without frozen variables. When frozen variables do occur, it is natural to try to model the cluster combinatorics via a Frobenius category, with the indecomposable projective-injective objects corresponding to these special variables. Amiot–Iyama–Reiten show how Frobenius categories admitting (d-1)-cluster-tilting objects arise naturally from the data of a Noetherian bimodule d-Calabi–Yau algebra A

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4

Persson, Westin Elin. "Tilting modules in d-cluster tilting subcategories." Thesis, Uppsala universitet, Algebra och geometri, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-325539.

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5

Cerulli, Irelli Giovanni. "Structural theory of rank three cluster algebras of affine type." Doctoral thesis, Università degli studi di Padova, 2008. http://hdl.handle.net/11577/3425220.

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6

Lawson, John William. "On the combinatorics of quivers, mutations and cluster algebra exchange graphs." Thesis, Durham University, 2017. http://etheses.dur.ac.uk/12095/.

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Over the last 20 years, cluster algebras have been widely studied, with numerous links to different areas of mathematics and physics. These algebras have a cluster structure given by successively mutating seeds, which can be thought of as living on some graph or tree. In this way one can use various combinatorial tools to discover more about these cluster structures and the cluster algebras themselves. This thesis considers some of the combinatorics at play here. Mutation-finite quivers have been classified, with links to triangulations of surfaces and semi-simple Lie algebras, while comparati

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7

Schmidt, Karl. "Factorizable Module Algebras, Canonical Bases, and Clusters." Thesis, University of Oregon, 2018. http://hdl.handle.net/1794/23793.

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The present dissertation consists of four interconnected projects. In the first, we introduce and study what we call factorizable module algebras. These are $U_q(\mathfrak{g})$-module algebras $A$ which factor, potentially after localization, as the tensor product of the subalgebra $A^+$ of highest weight vectors of $A$ and a copy of the quantum coordinate algebra $\mathcal{A}_q[U]$, where $U$ is a maximal unipotent subgroup of $G$, a semisimple Lie group whose Lie algebra is $\mathfrak{g}$. The class of factorizable module algebras is surprisingly rich, in particular including the quantum

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8

Warkentin, Matthias. "Fadenmoduln über Ãn und Cluster-Kombinatorik." Master's thesis, Universitätsbibliothek Chemnitz, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-94793.

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Inspired by work of Hubery [Hub] and Fomin, Shapiro and Thurston [FST06] related to cluster algebras, we construct a bijection between certain curves on a cylinder and the string modules over a path algebra of type Ãn. We show that under this bijection irreducible maps and the Auslander-Reiten translation have a geometric interpretation. Furthermore we prove that the dimension of extension groups can be expressed in terms of intersection numbers. Finally we explain the connection to cluster algebras and apply our results to describe the exchange graph in type Ãn<br>Angeregt durch Arbeiten zu C

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9

Torrente, Maria Laura. "Applications of Algebra in the Oil Industry." Doctoral thesis, Scuola Normale Superiore, 2009. http://hdl.handle.net/11384/85681.

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10

Bittmann, Léa. "Quantum Grothendieck rings, cluster algebras and quantum affine category O." Thesis, Sorbonne Paris Cité, 2019. http://www.theses.fr/2019USPCC024.

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L'objectif de cette thèse est de construire et d'étudier une structure d'anneau de Grothendieck quantique pour une catégorie O de représentations de la sous-algèbre de Borel Uq(b) d'une algèbre affine quantique Uq(g). On s'intéresse dans un premier lieu à la construction de modules standards asymptotiques pour la catégorie O, qui sont des analogues des modules standards existant dans la catégorie des représentations de dimension finie de Uq(^g). Une construction complète de ces modules est proposée dans le cas où l'algèbre de Lie simple sous-jacente g est sl2. Ensuite, nous définissons un tore

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11

Warkentin, Matthias. "Exchange Graphs via Quiver Mutation." Doctoral thesis, Universitätsbibliothek Chemnitz, 2014. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-153172.

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Inspired by Happel's question, whether the exchange graph and the simplicial complex of tilting modules over a quiver algebra are independent from the multiplicities of multiple arrows in the quiver, we study quantitative aspects of Fomin and Zelevinsky's quiver mutation rule. Our results turn out to be very useful in the mutation-infinite case for understanding combinatorial structures as the cluster exchange graph or the simplicial complex of tilting modules, which are governed by quiver mutation. Using a class of quivers we call forks we can show that any such quiver yields a tree in the ex

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12

Saleh, Ibrahim A. "Cluster automorphisms and hyperbolic cluster algebras." Diss., Kansas State University, 2012. http://hdl.handle.net/2097/14195.

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Doctor of Philosophy<br>Department of Mathematics<br>Zongzhu Lin<br>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 algeb

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13

Fish, C. D. "Connected quantized Weyl algebras and quantum cluster algebras." Thesis, University of Sheffield, 2016. http://etheses.whiterose.ac.uk/16636/.

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We investigate a class of noncommutative algebras, which we call connected quantized Weyl algebras, with a simple description in terms of generators and relations. We already knew of two families, both of which arise from cluster mutation in mutationperiodic quivers, and we show that for generic values of a scalar parameter q these are the only examples. We then investigate the ring-theoretic properties of these two families, determining their prime spectra, automorphism groups and some results on their Krull and global dimensions. The theory of ambiskew polynomial rings and generalised Weyl a

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14

Nakanishi, Tomoki, and 知樹中西. "DILOGARITHM IDENTITIES AND CLUSTER ALGEBRAS." 日本数学会代数学分科会, 2010. http://hdl.handle.net/2237/16356.

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15

Murphy, Graham James. "Cluster combinatorics and derived equivalences for m-cluster tilted algebras." Thesis, University of Leeds, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.490965.

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Using the polygonal models for the m-cluster complexes developed in [25] we classify maximal m-orthogonal subsets, defined in [38], of the translation quiver Z~ where t1 is a Dynkin diagram of type Am En/en or Dn by describing explicitly a bijective correspondence between the maximal facets of the m-cluster complexes and the maximal m-orthogonal subsets of Z~. This generalizes results appearing in [38]. We then apply our result to classify the blocks of group algebras whose defect group is not quatemion and for which a maximal m-orthogonal module exists. We also describe the m-cluster tilted a

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16

Casbi, Elie. "Categorifications of cluster algebras and representations of quiver Hecke algebras." Thesis, Université de Paris (2019-....), 2020. http://www.theses.fr/2020UNIP7032.

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Cette thèse porte sur l’étude de diverses conséquences des résultats de catégorifications monoïdales d'algèbres amassées par les algèbres de Hecke carquois, établis dans les travaux de Kang-Kashiwara-Kim-Oh [69]. Nous nous intéresserons en particulier à trois aspects de cette théorie: en premier lieu celui de la combinatoire, puis de la géométrie polytopale, et enfin celui de la théorie des représentations géométrique. Nous étudierons tout d'abord certaines relations combinatoires entre objets de nature a priori différentes: d'une part, les g-vecteurs au sens de Fomin-Zelevinsky, et d'autre pa

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17

Fisher, Thomas Andrew. "On the homological algebra of clusters, quivers, and triangulations." Thesis, University of Newcastle upon Tyne, 2017. http://hdl.handle.net/10443/3634.

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This thesis is comprised of three parts. Chapter one contains background material detailing some important aspects of category theory and homological algebra. Beginning with abelian categories, we introduce triangulated categories, the homotopy category and give the construction of the derived category. Differential graded algebras, basic Auslander-Reiten theory, and the Cluster Category of Dynkin Type An are also introduced, which all play a major role in chapters two and three. In [21], the cluster category D of type A1, with Auslander-Reiten quiver ZA1, is introduced. Slices in the Auslande

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18

Lampe, Philipp [Verfasser]. "Quantum cluster algebras and the dual canonical basis / Philipp Lampe." Bonn : Universitäts- und Landesbibliothek Bonn, 2011. http://d-nb.info/1016181450/34.

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19

Parsons, Mark James. "On indecomposable modules over cluster-tilted algebras of type A." Thesis, University of Leicester, 2007. http://hdl.handle.net/2381/30538.

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Gabriel's Theorem describes the dimension vectors of the finitely generated indecomposable modules over the path algebra of a simply-laced Dynkin quiver. It shows that they can be obtained from the expressions for the positive roots of the corresponding root system in terms of the simple roots. Here, we present a method for finding the dimension vectors of the finitely generated indecomposable modules over a cluster-tilted algebra of Dynkin type A.;It is known that the quiver of a cluster-tilted algebra of Dynkin type A is given by an exchange matrix of the corresponding cluster algebra. We de

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20

Vaso, Laertis. "d-cluster tilting modules over quotients of path algebras of type An." Thesis, Uppsala universitet, Algebra och geometri, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-254673.

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21

Gratz, Sira Helena [Verfasser]. "From finite to infinite : cluster algebras as colimits, and mutating torsion pairs in discrete cluster categrories / Sira Helena Gratz." Hannover : Technische Informationsbibliothek und Universitätsbibliothek Hannover (TIB), 2015. http://d-nb.info/1075261287/34.

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22

Bastian, Janine [Verfasser]. "Derived equivalences for cluster-tilted algebras of types Ãn and Dn / Janine Bastian." Hannover : Technische Informationsbibliothek und Universitätsbibliothek Hannover (TIB), 2012. http://d-nb.info/1021441562/34.

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23

Gellert, Florian Verfasser], and Henning [Akademischer Betreuer] [Krause. "Sequential structures in cluster algebras and representation theory / Florian Gellert ; Betreuer: Henning Krause." Bielefeld : Universitätsbibliothek Bielefeld, 2017. http://d-nb.info/1135724628/34.

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24

Scott, J., I. Reiten, O. Iyama, and A. B. Buan. "Cluster structures for 2-Calabi-Yau categories and unipotent groups." Cambridge University Press, 2009. http://hdl.handle.net/2237/14400.

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25

Brito, Matheus Batagini 1985. "Álgebras de cluster e teoria de representações." [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306965.

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Orientador: Adriano Adrega de Moura<br>Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Cientifica<br>Made available in DSpace on 2018-08-18T00:34:15Z (GMT). No. of bitstreams: 1 Brito_MatheusBatagini_M.pdf: 1577922 bytes, checksum: 6e151b2dd08dad43a9a5c9840dd36514 (MD5) Previous issue date: 2011<br>Resumo: Na presente dissertação estudamos dois exemplos de relacionamento da teoria de álgebras de cluster com teoria de representações. A saber, estudamos os principais resultados dos artigos [5, 26]. O primeiro é uma relação entre álg

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26

Brodsky, Sarah [Verfasser], Olga [Akademischer Betreuer] Holtz, Olga [Gutachter] Holtz, Michael [Gutachter] Joswig, and Felipe [Gutachter] Rincón. "On relating aspects of tropical geometry, cluster algebras, and zonotopal algebras / Sarah Brodsky ; Gutachter: Olga Holtz, Michael Joswig, Felipe Rincón ; Betreuer: Olga Holtz." Berlin : Technische Universität Berlin, 2016. http://d-nb.info/1156011930/34.

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27

Bossinger, Lara [Verfasser], and Peter [Gutachter] Littelmann. "Toric degenerations: a bridge between representation theory, tropical geometry and cluster algebras / Lara Bossinger ; Gutachter: Peter Littelmann." Köln : Universitäts- und Stadtbibliothek Köln, 2018. http://d-nb.info/1163728381/34.

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28

Pape, Daniel [Verfasser]. "An Alternative Algebraic Framework for the Simplification of Coupled Cluster Type Expectation Values / Daniel Pape." München : Verlag Dr. Hut, 2013. http://d-nb.info/1045125881/34.

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29

Sowdi, Ravindra Bose Karthik. "A study of the properties of mesoscopic consensus clusters that arise due to Ising dynamics on graphs." Thesis, University of Nottingham, 2016. http://eprints.nottingham.ac.uk/32598/.

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We studied the impact of graph structure and temperature has on the sizes and lifetimes of crowds (mesoscopic clusters of consensus) that form when due to Ising Spin dynamics. Spin models have proven successful in modelling the formation of global consensus in opinion for large groups of people. We focus on the case where lots of small groups form consensuses but no global consensus forms. Our results can be interpreted in the light of consumer opinion and give an explanation for fads and brand reputation. Data was obtained by performing Monte Carlo simulations of the Ising Model on various gr

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30

Dias, Elisângela SIlva. "Reconhecimento polinomial de álgebras cluster de tipo finito." Universidade Federal de Goiás, 2015. http://repositorio.bc.ufg.br/tede/handle/tede/4848.

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Submitted by Cláudia Bueno (claudiamoura18@gmail.com) on 2015-10-29T19:17:43Z No. of bitstreams: 2 Tese - Elisângela Silva Dias - 2015.pdf: 1107380 bytes, checksum: e288bc934158fa879639c403bb15ba54 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5)<br>Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-11-03T14:30:02Z (GMT) No. of bitstreams: 2 Tese - Elisângela Silva Dias - 2015.pdf: 1107380 bytes, checksum: e288bc934158fa879639c403bb15ba54 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5)<br>Made availa

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31

Haga, Tim [Verfasser], Eva-Maria [Akademischer Betreuer] [Gutachter] Feichtner, and Josephine [Gutachter] Yu. "On Polytopes Arising in Cluster Algebras & Finite Frames / Tim Haga. Betreuer: Eva-Maria Feichtner. Gutachter: Eva-Maria Feichtner ; Josephine Yu." Bremen : Staats- und Universitätsbibliothek Bremen, 2016. http://d-nb.info/1106374606/34.

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32

Del, Zotto Michele. "Four-dimensional N=2 superconformal quantum field theories and BPS-quivers." Doctoral thesis, SISSA, 2013. http://hdl.handle.net/20.500.11767/4813.

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33

Pape, Daniel [Verfasser], Michael [Akademischer Betreuer] Dolg, and Michael [Akademischer Betreuer] Hanrath. "An Alternative Algebraic Framework for the Simplification of Coupled Cluster Type Expectation Values / Daniel Pape. Gutachter: Michael Dolg ; Michael Hanrath." Köln : Universitäts- und Stadtbibliothek Köln, 2013. http://d-nb.info/1045023361/34.

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34

Manneville, Thibault. "Geometrical and combinatorial generalizations of the associahedron." Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLX019/document.

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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 régulièr

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35

Menard, Etienne. "Algèbres amassées associées aux variétés de Richardson ouvertes : un algorithme de calcul de graines initiales." Thesis, Normandie, 2021. http://www.theses.fr/2021NORMC211.

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Les algèbres amassées sont des anneaux commutatifs intègres avec une structure combinatoire particulière.Cette structure consiste en la donnée d’une famille de graines, liées entre elles par une opération appelée mutation.Chaque graine est composée de deux parties : un amas et un carquois.Les variétés de Richardson ouvertes sont des strates de la variété de drapeaux associée à un groupe linéairealgébrique de type simplement lacé. Elles sont l’intersection de cellules de Schubert respectivement à deux sous-groupes de Borel opposés. Dans [Lec16], une sous-algèbre amassée de rang maximal sur l’an

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36

"Feature Cluster Algebra and Its Application for Geometric Tolerancing." Master's thesis, 2013. http://hdl.handle.net/2286/R.I.18672.

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abstract: The goal of this research project is to develop a DOF (degree of freedom) algebra for entity clusters to support tolerance specification, validation, and tolerance automation. This representation is required to capture the relation between geometric entities, metric constraints and tolerance specification. This research project is a part of an on-going project on creating a bi-level model of GD&T; (Geometric Dimensioning and Tolerancing). This thesis presents the systematic derivation of degree of freedoms of entity clusters corresponding to tolerance classes. The clusters can be dat

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37

Warkentin, Matthias. "Fadenmoduln über Ãn und Cluster-Kombinatorik." Master's thesis, 2008. https://monarch.qucosa.de/id/qucosa%3A19762.

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Inspired by work of Hubery [Hub] and Fomin, Shapiro and Thurston [FST06] related to cluster algebras, we construct a bijection between certain curves on a cylinder and the string modules over a path algebra of type Ãn. We show that under this bijection irreducible maps and the Auslander-Reiten translation have a geometric interpretation. Furthermore we prove that the dimension of extension groups can be expressed in terms of intersection numbers. Finally we explain the connection to cluster algebras and apply our results to describe the exchange graph in type Ãn.<br>Angeregt durch Arbeiten zu

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38

Wang, Victor Zhenyi. "Geometry & Cluster Algebras: Finite Type Classification & Double Bruhat Cells." Thesis, 2016. http://hdl.handle.net/1885/173669.

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In 2000, Fomin and Zelevinsky introduced a new language called cluster algebras for describing rings with certain combinatorial structures. Cluster algebras enjoy a variety of nice properties such as well-established collection of classification results and interesting geometric properties with the upper cluster algebra. We approach cluster algebras rst from the perspective of operations on quivers, then reacquaint ourselves with a more general definition. We then present the classification of cluster algebras of nite type and explore cluster algebra structures on the ring of regular functio

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39

Denham, Mónica Malén. "Paralelización de la factorización LU de matrices para clusters heterogéneos." Tesis, 2005. http://hdl.handle.net/10915/3926.

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40

Rykhlinskaya, Ekaterina [Verfasser]. "A computer-algebraic approach to the study of the symmetry properties of molecules and clusters / von Ekaterina Rykhlinskaya." 2006. http://d-nb.info/980981646/34.

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41

Nagel, Dominik. "The condition number of Vandermonde matrices and its application to the stability analysis of a subspace method." Doctoral thesis, 2021. https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-202103194121.

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This thesis consists of two main parts. First of all, the condition number of rectangular Vandermonde matrices with nodes on the complex unit circle is studied. The first time quantitative bounds for the extreme singular values are proven in the multivariate setting and when nodes of the Vandermonde matrix form clusters. In the second part, an optimized presentation of the deterministic stability analysis of the subspace method ESPRIT is given and results from the first part are applied.

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