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Relevant bibliographies by topics / Cluster algebra

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Academic literature on the topic 'Cluster algebra'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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Journal articles on the topic "Cluster algebra"

1

ASSEM, IBRAHIM, VASILISA SHRAMCHENKO, and RALF SCHIFFLER. "CLUSTER AUTOMORPHISMS AND COMPATIBILITY OF CLUSTER VARIABLES." Glasgow Mathematical Journal 56, no. 3 (2014): 705–20. http://dx.doi.org/10.1017/s0017089514000214.

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AbstractIn this paper, we introduce a notion of unistructural cluster algebras, for which the set of cluster variables uniquely determines the clusters, as well as the notion of weak unistructural cluster algebras, for which the set of cluster variables determines the clusters, provided that the type of the cluster algebra is fixed. We prove that, for cluster algebras of the Dynkin type, the two notions of unistructural and weakly unistructural coincide, and that cluster algebras of rank 2 are always unistructural. We then prove that a cluster algebra $\mathcal A$ is weakly unistructural if an

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Bai, Liqian, Xueqing Chen, Ming Ding, and Fan Xu. "A generalized quantum cluster algebra of Kronecker type." Electronic Research Archive 32, no. 1 (2024): 670–85. http://dx.doi.org/10.3934/era.2024032.

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<abstract><p>The notion of generalized quantum cluster algebras was introduced as a natural generalization of Berenstein and Zelevinsky's quantum cluster algebras as well as Chekhov and Shapiro's generalized cluster algebras. In this paper, we focus on a generalized quantum cluster algebra of Kronecker type which possesses infinitely many cluster variables. We obtain the cluster multiplication formulas for this algebra. As an application of these formulas, a positive bar-invariant basis is explicitly constructed. Both results generalize those known for the Kronecker cluster algebra

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Amiot, Claire, and Steffen Oppermann. "Algebras of acyclic cluster type: Tree type and type Ã." Nagoya Mathematical Journal 211 (September 2013): 1–50. http://dx.doi.org/10.1017/s0027763000010771.

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AbstractIn this paper, we study algebras of global dimension at most 2 whose generalized cluster category is equivalent to the cluster category of an acyclic quiver which is either a tree or of typeÃ.We are particularly interested in their derived equivalence classification. We prove that each algebra which is cluster equivalent to a tree quiver is derived equivalent to the path algebra of this tree. Then we describe explicitly the algebras of cluster typeÃnfor each possible orientation ofÃn.We give an explicit way to read off the derived equivalence class in which such an algebra lies, and we

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Amiot, Claire, and Steffen Oppermann. "Algebras of acyclic cluster type: Tree type and type Ã." Nagoya Mathematical Journal 211 (September 2013): 1–50. http://dx.doi.org/10.1215/00277630-2083124.

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AbstractIn this paper, we study algebras of global dimension at most 2 whose generalized cluster category is equivalent to the cluster category of an acyclic quiver which is either a tree or of type Ã. We are particularly interested in their derived equivalence classification. We prove that each algebra which is cluster equivalent to a tree quiver is derived equivalent to the path algebra of this tree. Then we describe explicitly the algebras of cluster type Ãn for each possible orientation of Ãn. We give an explicit way to read off the derived equivalence class in which such an algebra lies,

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Cerulli Irelli, Giovanni, Bernhard Keller, Daniel Labardini-Fragoso, and Pierre-Guy Plamondon. "Linear independence of cluster monomials for skew-symmetric cluster algebras." Compositio Mathematica 149, no. 10 (2013): 1753–64. http://dx.doi.org/10.1112/s0010437x1300732x.

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AbstractFomin–Zelevinsky conjectured that in any cluster algebra, the cluster monomials are linearly independent and that the exchange graph and cluster complex are independent of the choice of coefficients. We confirm these conjectures for all skew-symmetric cluster algebras.

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BOBIŃSKI, GRZEGORZ, and ASLAK BAKKE BUAN. "THE ALGEBRAS DERIVED EQUIVALENT TO GENTLE CLUSTER TILTED ALGEBRAS." Journal of Algebra and Its Applications 11, no. 01 (2012): 1250012. http://dx.doi.org/10.1142/s021949881100535x.

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A cluster tilted algebra is known to be gentle if and only if it is cluster tilted of Dynkin type 𝔸 or Euclidean type [Formula: see text]. We classify all finite-dimensional algebras which are derived equivalent to gentle cluster tilted algebras.

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Çanakçı, İlke, and Ralf Schiffler. "Cluster algebras and continued fractions." Compositio Mathematica 154, no. 3 (2017): 565–93. http://dx.doi.org/10.1112/s0010437x17007631.

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We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To a continued fraction $[a_{1},a_{2},\ldots ,a_{n}]$ we associate a snake graph ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ such that the continued fraction is the quotient of the number of perfect matchings of ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ and ${\mathcal{G}}[a_{2},\ldots ,a_{n}]$. We also show that snake graphs are in bijection with continued fra

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Gubitosi, Viviana. "Derived class of m-cluster tilted algebras of type 𝔸̃". Journal of Algebra and Its Applications 17, № 11 (2018): 1850216. http://dx.doi.org/10.1142/s021949881850216x.

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In this paper, we characterize all the finite-dimensional algebras that are derived equivalent to an [Formula: see text]-cluster tilted algebras of type [Formula: see text]. This generalizes a result of Bobiński and Buan [The algebras derived equivalent to gentle cluster tilted algebras, J. Algebra Appl. 11(1) (2012), Article ID:1250012, 26 pp.].

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BAUR, KARIN, DUSKO BOGDANIC, and ANA GARCIA ELSENER. "CLUSTER CATEGORIES FROM GRASSMANNIANS AND ROOT COMBINATORICS." Nagoya Mathematical Journal 240 (June 3, 2019): 322–54. http://dx.doi.org/10.1017/nmj.2019.14.

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The category of Cohen–Macaulay modules of an algebra $B_{k,n}$ is used in Jensen et al. (A categorification of Grassmannian cluster algebras, Proc. Lond. Math. Soc. (3) 113(2) (2016), 185–212) to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$-planes in $n$-space. In this paper, we find canonical Auslander–Reiten sequences and study the Auslander–Reiten translation periodicity for this category. Furthermore, we give an explicit construction of Cohen–Macaulay modules of arbitrary rank. We then use our results to es

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Basharov, A. M. "Atom-photon cluster in nonlinear and quantum optics." Izvestiâ Akademii nauk SSSR. Seriâ fizičeskaâ 88, no. 6 (2024): 876–83. https://doi.org/10.31857/s0367676524060051.

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We described the constructed models of radiative quantum systems analogous of multiphoton and Raman resonances of classical field quanta on an atomic system with the participation of resonator mode quanta. A distinctive feature of the models is the possibility to describe an atomic ensemble and quanta using either the generators of polynomial algebra or the two-mode Jordan-Schwinger representation of the su(2) algebra, that could point to atomic-photon and/or photon clusters. The algebras are arising that are mathematically insoluble, in contrast to the Heisenberg — Weyl algebra, which makes i

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Dissertations / Theses on the topic "Cluster algebra"

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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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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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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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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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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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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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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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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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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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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Books on the topic "Cluster algebra"

1

Society, European Mathematical, ed. Lecture notes on cluster algebras. European Mathematical Society, 2013.

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2

1963-, Shapiro Michael, and Vainshtein Alek 1958-, eds. Cluster algebra and Poisson geometry. American Mathematical Society, 2010.

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3

Greenstein, Jacob, David Hernandez, Kailash C. Misra, and Prasad Senesi, eds. Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-63849-8.

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Assem, Ibrahim, and Sonia Trepode, eds. Homological Methods, Representation Theory, and Cluster Algebras. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-74585-5.

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Bárcenas, Noé, and Monica Moreno Rocha. Mexican mathematicians abroad: Recent contributions : first workshop, Matematicos Mexicanos Jovenes en el Mundo, August 22-24, 2012, Centro de Investigacion en Matematicas, A.C., Guanajuato, Mexico. Edited by Galaz-García Fernando editor. American Mathematical Society, 2016.

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Arkady, Berenstein, and Retakh Vladimir, eds. Noncommutative birational geometry, representations and combinatorics: AMS Special Session on Noncommutative Birational Geometry, Representations and Cluster Algebras, January 6-7, 2012, Boston, MA. American Mathematical Society, 2013.

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7

Cluster Algebra Structures on Poisson Nilpotent Algebras. American Mathematical Society, 2024.

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8

Gekhtman, M., A. Vainshtein, and M. Shapiro. Exotic Cluster Structures on $SL_n$: The Cremmer-Gervais Case. American Mathematical Society, 2017.

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Homological Methods, Representation Theory, and Cluster Algebras. Springer, 2018.

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10

Fomin, Sergey, and Dylan Thurston. Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths. American Mathematical Society, 2018.

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Book chapters on the topic "Cluster algebra"

1

Vinitsky, Sergue, Alexander Gusev, Ochbadrakh Chuluunbaatar, et al. "Symbolic-Numerical Algorithm for Generating Cluster Eigenfunctions: Tunneling of Clusters through Repulsive Barriers." In Computer Algebra in Scientific Computing. Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-02297-0_35.

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Fock, V. V., and A. B. Goncharov. "Cluster Ensembles, Quantization and the Dilogarithm II: The Intertwiner." In Algebra, Arithmetic, and Geometry. Birkhäuser Boston, 2009. http://dx.doi.org/10.1007/978-0-8176-4745-2_15.

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Gekhtman, Michael, Michael Shapiro, and Alek Vainshtein. "Poisson structures compatible with the cluster algebra structure." In Mathematical Surveys and Monographs. American Mathematical Society, 2010. http://dx.doi.org/10.1090/surv/167/04.

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Gekhtman, Michael, Michael Shapiro, and Alek Vainshtein. "Pre-symplectic structures compatible with the cluster algebra structure." In Mathematical Surveys and Monographs. American Mathematical Society, 2010. http://dx.doi.org/10.1090/surv/167/06.

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Gusev, Alexander, Sergue Vinitsky, Ochbadrakh Chuluunbaatar, et al. "Symbolic-Numerical Algorithm for Generating Cluster Eigenfunctions: Identical Particles with Pair Oscillator Interactions." In Computer Algebra in Scientific Computing. Springer International Publishing, 2013. http://dx.doi.org/10.1007/978-3-319-02297-0_14.

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Gekhtman, Michael, Michael Shapiro, and Alek Vainshtein. "Generalized Bäcklund-Darboux transforms for Coxeter-Toda flows from a cluster algebra perspective." In Mathematical Surveys and Monographs. American Mathematical Society, 2010. http://dx.doi.org/10.1090/surv/167/10.

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You, Xin, Hailong Yang, Zhongzhi Luan, Yi Liu, and Depei Qian. "Performance Evaluation and Analysis of Linear Algebra Kernels in the Prototype Tianhe-3 Cluster." In Supercomputing Frontiers. Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-18645-6_6.

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Gekhtman, Michael, Michael Shapiro, and Alek Vainshtein. "Cluster algebras." In Mathematical Surveys and Monographs. American Mathematical Society, 2010. http://dx.doi.org/10.1090/surv/167/03.

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Hernandez, David, and Bernard Leclerc. "Quantum Affine Algebras and Cluster Algebras." In Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification. Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-63849-8_2.

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Glick, Max, and Dylan Rupel. "Introduction to Cluster Algebras." In Symmetries and Integrability of Difference Equations. Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-56666-5_7.

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Conference papers on the topic "Cluster algebra"

1

Shen, Yadong, Jami J. Shah, and Joseph K. Davidson. "Feature Cluster Algebra for Geometric Tolerancing." In ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2011. http://dx.doi.org/10.1115/detc2011-47937.

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The ASME Y14.5.1 companion to the Geometric Dimensioning and Tolerancing (GD&T) standard gives an exhaustive tabulation of active and invariant degrees of freedom (DOF) for Datum Reference Frames (DRF). These DRFs are entity clusters of points, lines, and planes with different geometric relations to each other (coincident, parallel, perpendicular etc.). This paper investigates the systematic derivation of DRF & target clusters’ DOFs and associates them with tolerance classes. A vector representation of geometric entities and Boolean operations are proposed. This algebra was validated b

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van de Geijn, Robert. "The science of programming dense linear algebra libraries." In 2007 IEEE International Conference on Cluster Computing (CLUSTER). IEEE, 2007. http://dx.doi.org/10.1109/clustr.2007.4629198.

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Azad, Ariful, and Aydin Buluc. "Distributed-Memory Algorithms for Maximal Cardinality Matching Using Matrix Algebra." In 2015 IEEE International Conference on Cluster Computing (CLUSTER). IEEE, 2015. http://dx.doi.org/10.1109/cluster.2015.62.

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Zhang, Jianbo, Jiping Liu, and Bei Wang. "Spatial cluster based on map algebra." In 2010 International Conference on E-Business and E-Government (ICEE). IEEE, 2010. http://dx.doi.org/10.1109/icee.2010.590.

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Heuveline, Vince, Chandramowli Subramanian, Dimitar Lukarski, and Jan-Philipp Weiss. "A multi-platform linear algebra toolbox for finite element solvers on heterogeneous clusters." In 2010 IEEE International Conference On Cluster Computing Workshops and Posters (CLUSTER WORKSHOPS). IEEE, 2010. http://dx.doi.org/10.1109/clusterwksp.2010.5613084.

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Cuenca, Javier, Luis Pedro Garcia, Domingo Gimenez, and Jack Dongarra. "Processes Distribution of Homogeneous Parallel Linear Algebra Routines on Heterogeneous Clusters." In 2005 IEEE International Conference on Cluster Computing. IEEE, 2005. http://dx.doi.org/10.1109/clustr.2005.347021.

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Choy, Laurent, and Serge G. Petiton. "Toward global and grid computing for large scale linear algebra problems." In 2005 IEEE International Conference on Cluster Computing. IEEE, 2005. http://dx.doi.org/10.1109/clustr.2005.347026.

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Aouad, Lamine M., and Serge G. Petiton. "Parallel Basic Matrix Algebra on the Grid'5000 Large Scale Distributed Platform." In 2006 IEEE International Conference on Cluster Computing. IEEE, 2006. http://dx.doi.org/10.1109/clustr.2006.311894.

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Pinto, Vinicius Garcia, João V. F. Lima, Vanderlei Munhoz, Daniel Cordeiro, Emilio Francesquini, and Márcio Castro. "Performance Evaluation of Dense Linear Algebra Kernels using Chameleon and StarPU on AWS." In Simpósio em Sistemas Computacionais de Alto Desempenho. Sociedade Brasileira de Computação, 2024. http://dx.doi.org/10.5753/sscad.2024.244405.

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Due to recent advances and investments in cloud computing, public cloud providers now offer GPU-accelerated and compute-optimized Virtual Machine (VM) instances, allowing researchers to execute parallel workloads in virtual heterogeneous clusters in the cloud. This paper evaluates the performance and monetary costs of running dense linear algebra algorithms extracted from the Chameleon package implemented using StarPU on Amazon Elastic Compute Cloud (EC2) instances. We evaluated these metrics with a single powerful/costly instance with four NVIDIA GPUs (fat node) and with a cluster of five les

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Gottschling, Peter, and Torsten Hoefler. "Productive Parallel Linear Algebra Programming with Unstructured Topology Adaption." In 2012 12th IEEE/ACM International Symposium on Cluster, Cloud and Grid Computing (CCGrid). IEEE, 2012. http://dx.doi.org/10.1109/ccgrid.2012.51.

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