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Journal articles on the topic 'Cluster algebra'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

Goodearl, K., and M. Yakimov. "Quantum cluster algebra structures on quantum nilpotent algebras." Memoirs of the American Mathematical Society 247, no. 1169 (2017): 0. http://dx.doi.org/10.1090/memo/1169.

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12

Assem, Ibrahim, Thomas Brüstle, and Ralf Schiffler. "Cluster-tilted algebras without clusters." Journal of Algebra 324, no. 9 (2010): 2475–502. http://dx.doi.org/10.1016/j.jalgebra.2010.07.035.

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13

Zuevsky, Alexander. "Cluster algebras based on vertex operator algebras." International Journal of Modern Physics B 30, no. 28n29 (2016): 1640030. http://dx.doi.org/10.1142/s0217979216400300.

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Starting from Zhu recursion formulas for correlation functions for vertex operator algebras with formal parameters associated to local coordinates around marked points on a Riemann surfaces, we introduce a cluster algebra structure over a noncommutative set of variables. Cluster elements and mutation rules are explicitly defined. In particular, we propose an elliptic version of vertex operator cluster algebras arising from correlation functions and Zhu reduction procedure for vertex operators on the torus.

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14

Liu, Siyang, and Fang Li. "Periodicities in Cluster Algebras and Cluster Automorphism Groups." Algebra Colloquium 28, no. 04 (2021): 601–24. http://dx.doi.org/10.1142/s100538672100047x.

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We study the relations between two groups related to cluster automorphism groups which are defined by Assem, Schiffler and Shamchenko. We establish the relationships among (strict) direct cluster automorphism groups and those groups consisting of periodicities of labeled seeds and exchange matrices, respectively, in the language of short exact sequences. As an application, we characterize automorphism-finite cluster algebras in the cases of bipartite seeds or finite mutation type. Finally, we study the relation between the group [Formula: see text] for a cluster algebra [Formula: see text] and

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15

Kashiwara, Masaki, Myungho Kim, Se-jin Oh, and Euiyong Park. "Laurent family of simple modules over quiver Hecke algebras." Compositio Mathematica 160, no. 8 (2024): 1916–40. http://dx.doi.org/10.1112/s0010437x24007310.

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We introduce the notions of quasi-Laurent and Laurent families of simple modules over quiver Hecke algebras of arbitrary symmetrizable types. We prove that such a family plays a similar role of a cluster in quantum cluster algebra theory and exhibits a quantum Laurent positivity phenomenon similar to the basis of the quantum unipotent coordinate ring $\mathcal {A}_q(\mathfrak {n}(w))$ , coming from the categorification. Then we show that the families of simple modules categorifying Geiß–Leclerc–Schröer (GLS) clusters are Laurent families by using the Poincaré–Birkhoff–Witt (PBW) decomposition

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16

Chen, Chin Chun, Yuan Horng Lin, Jeng Ming Yih, and Sue Fen Huang. "Construct Knowledge Structure of Linear Algebra." Advanced Materials Research 211-212 (February 2011): 793–97. http://dx.doi.org/10.4028/www.scientific.net/amr.211-212.793.

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Apply interpretive structural modeling to construct knowledge structure of linear algebra. New fuzzy clustering algorithms improved fuzzy c-means algorithm based on Mahalanobis distance has better performance than fuzzy c-means algorithm. Each cluster of data can easily describe features of knowledge structures individually. The results show that there are six clusters and each cluster has its own cognitive characteristics. The methodology can improve knowledge management in classroom more feasible.

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17

BARCELO, HÉLÈNE, CHRISTOPHER SEVERS, and JACOB A. WHITE. "THE DISCRETE FUNDAMENTAL GROUP OF THE ASSOCIAHEDRON, AND THE EXCHANGE MODULE." International Journal of Algebra and Computation 23, no. 04 (2013): 745–62. http://dx.doi.org/10.1142/s0218196713400079.

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The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory. We study the abelianization of the discrete fundamental group, and show that it is free abelian of rank [Formula: see text]. We also find a combinatorial description for a basis of this rank. We also introduce the exchange module of the type An cluster algebra, used to model the relat

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18

DUPONT, G. "POSITIVITY FOR REGULAR CLUSTER CHARACTERS IN ACYCLIC CLUSTER ALGEBRAS." Journal of Algebra and Its Applications 11, no. 04 (2012): 1250069. http://dx.doi.org/10.1142/s0219498812500697.

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Let Q be an acyclic quiver and let [Formula: see text] be the corresponding cluster algebra. Let H be the path algebra of Q over an algebraically closed field and let M be an indecomposable regular H-module. We prove the positivity of the cluster characters associated to M expressed in the initial seed of [Formula: see text] when either H is tame and M is any regular H-module, or H is wild and M is a regular Schur module which is not quasi-simple.

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19

Kashiwara, Masaki, Myungho Kim, Se‐jin Oh, and Euiyong Park. "Cluster algebra structures on module categories over quantum affine algebras." Proceedings of the London Mathematical Society 124, no. 3 (2022): 301–72. http://dx.doi.org/10.1112/plms.12428.

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20

Adachi, Takahide, Osamu Iyama, and Idun Reiten. "-tilting theory." Compositio Mathematica 150, no. 3 (2013): 415–52. http://dx.doi.org/10.1112/s0010437x13007422.

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AbstractThe aim of this paper is to introduce $\tau $-tilting theory, which ‘completes’ (classical) tilting theory from the viewpoint of mutation. It is well known in tilting theory that an almost complete tilting module for any finite-dimensional algebra over a field $k$ is a direct summand of exactly one or two tilting modules. An important property in cluster-tilting theory is that an almost complete cluster-tilting object in a 2-CY triangulated category is a direct summand of exactly two cluster-tilting objects. Reformulated for path algebras $kQ$, this says that an almost complete support

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21

Cao, Peigen, and Fang Li. "Unistructurality of cluster algebras." Compositio Mathematica 156, no. 5 (2020): 946–58. http://dx.doi.org/10.1112/s0010437x20007113.

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We prove that any skew-symmetrizable cluster algebra is unistructural, which is a conjecture by Assem, Schiffler and Shramchenko. As a corollary, we obtain that a cluster automorphism of a cluster algebra ${\mathcal{A}}({\mathcal{S}})$ is just an automorphism of the ambient field ${\mathcal{F}}$ which restricts to a permutation of the cluster variables of ${\mathcal{A}}({\mathcal{S}})$.

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22

Gubitosi, Viviana. "Open Frobenius Cluster-Tilted Algebras." Algebra Colloquium 29, no. 01 (2022): 1–22. http://dx.doi.org/10.1142/s1005386722000025.

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In this paper, we compute the Frobenius dimension of any cluster-tilted algebra of finite type. Moreover, we give conditions on the bound quiver of a cluster-tilted algebra [Formula: see text] such that [Formula: see text] has non-trivial open Frobenius structures.

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23

Keller, Bernhard, and Idun Reiten. "Acyclic Calabi–Yau categories." Compositio Mathematica 144, no. 5 (2008): 1332–48. http://dx.doi.org/10.1112/s0010437x08003540.

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AbstractWe prove a structure theorem for triangulated Calabi–Yau categories: an algebraic 2-Calabi–Yau triangulated category over an algebraically closed field is a cluster category if and only if it contains a cluster-tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As an application to commutative algebra, we show that the stable category of maximal Cohen–Macaulay modules over a certain isolated singularity of dimension 3 is a cluster category. This implies the classification of the rigid Cohen–Macaulay modules first

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24

Pérez, Claudia, and Daniel Rivera. "Polynomial-time Classification of Skew-symmetrizable Matrices with a Positive Definite Quasi-Cartan Companion." Fundamenta Informaticae 181, no. 4 (2021): 313–37. http://dx.doi.org/10.3233/fi-2021-2061.

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Skew-symmetrizable matrices play an essential role in the classification of cluster algebras. We prove that the problem of assigning a positive definite quasi-Cartan companion to a skew-symmetrizable matrix is in polynomial class P. We also present an algorithm to determine the finite type Δ ∈ {𝔸n; 𝔻n; 𝔹n; ℂn; 𝔼6; 𝔼7; 𝔼8; 𝔽4; 𝔾2} of a cluster algebra associated to the mutation-equivalence class of a connected skew-symmetrizable matrix B, if it has one.

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25

Hikami, Kazuhiro, and Rei Inoue. "Braiding operator via quantum cluster algebra." Journal of Physics A: Mathematical and Theoretical 47, no. 47 (2014): 474006. http://dx.doi.org/10.1088/1751-8113/47/47/474006.

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26

Kashiwara, Masaki, Myungho Kim, Se-jin Oh, and Euiyong Park. "Monoidal categorification and quantum affine algebras." Compositio Mathematica 156, no. 5 (2020): 1039–77. http://dx.doi.org/10.1112/s0010437x20007137.

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We introduce and investigate new invariants of pairs of modules $M$ and $N$ over quantum affine algebras $U_{q}^{\prime }(\mathfrak{g})$ by analyzing their associated $R$-matrices. Using these new invariants, we provide a criterion for a monoidal category of finite-dimensional integrable $U_{q}^{\prime }(\mathfrak{g})$-modules to become a monoidal categorification of a cluster algebra.

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27

CAO, PEIGEN, MIN HUANG, and FANG LI. "A CONJECTURE ON -MATRICES OF CLUSTER ALGEBRAS." Nagoya Mathematical Journal 238 (June 19, 2018): 37–46. http://dx.doi.org/10.1017/nmj.2018.18.

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For a skew-symmetrizable cluster algebra ${\mathcal{A}}_{t_{0}}$ with principal coefficients at $t_{0}$, we prove that each seed $\unicode[STIX]{x1D6F4}_{t}$ of ${\mathcal{A}}_{t_{0}}$ is uniquely determined by its $C$-matrix, which was proposed by Fomin and Zelevinsky (Compos. Math. 143 (2007), 112–164) as a conjecture. Our proof is based on the fact that the positivity of cluster variables and sign coherence of $c$-vectors hold for ${\mathcal{A}}_{t_{0}}$, which was actually verified in Gross et al. (Canonical bases for cluster algebras, J. Amer. Math. Soc. 31(2) (2018), 497–608). Further di

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28

BARNES, DONALD W. "CHARACTER CLUSTERS FOR LIE ALGEBRA MODULES OVER A FIELD OF NONZERO CHARACTERISTIC." Bulletin of the Australian Mathematical Society 89, no. 2 (2013): 234–42. http://dx.doi.org/10.1017/s0004972713000312.

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AbstractFor a Lie algebra $L$ over an algebraically closed field $F$ of nonzero characteristic, every finite dimensional $L$-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Also, it is shown that if the soluble Lie algebra $L$ is in the saturated formation $\mathfrak{F}$ and if $V, W$ are irreducible $L$-modules with the same cluster and the $p$-operation vanishes on the centre of the $p$-envelope use

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29

Zhu, Bin. "Preprojective Cluster Variables of Acyclic Cluster Algebras." Communications in Algebra 35, no. 9 (2007): 2857–71. http://dx.doi.org/10.1080/00927870701451334.

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30

Liu, Pin. "Lifting to Cluster-tilting Objects in Higher Cluster Categories." Algebra Colloquium 19, no. 04 (2012): 707–12. http://dx.doi.org/10.1142/s1005386712000582.

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Let d > 1 be a positive integer. In this note, we consider the d-cluster-tilted algebras, i.e., algebras which appear as endomorphism rings of d-cluster-tilting objects in higher cluster categories (d-cluster categories). We show that tilting modules over such algebras lift to d-cluster-tilting objects in the corresponding higher cluster category.

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31

Zou, Teng, and Bin Zhu. "Finite Repetitive Generalized Cluster Complexes and d-Cluster Categories." Algebra Colloquium 20, no. 01 (2013): 123–40. http://dx.doi.org/10.1142/s1005386713000114.

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For any positive integer n, we construct an n-repetitive generalized cluster complex (a simplicial complex) associated with a given finite root system by defining a compatibility degree on the n-repetitive set of the colored root system. This simplicial complex includes Fomin-Reading's generalized cluster complex as a special case when n=1. We also introduce the intermediate coverings (called generalized d-cluster categories) of d-cluster categories of hereditary algebras, and study the d-cluster tilting objects and their endomorphism algebras in those categories. In particular, we show that t

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32

Chen, Chin Chun, Yuan Horng Lin, Jeng Ming Yih, and Shu Yi Juan. "Construct Concept Structure for Linear Algebra Based on Cognition Diagnosis and Clustering with Mahalanobis Distances." Advanced Materials Research 211-212 (February 2011): 756–60. http://dx.doi.org/10.4028/www.scientific.net/amr.211-212.756.

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Euclidean distance function based fuzzy clustering algorithms can only be used to detect spherical structural clusters. The purpose of this study is improved Fuzzy C-Means algorithm based on Mahalanobis distance to identify concept structure for Linear Algebra. In addition, Concept structure analysis (CSA) could provide individualized knowledge structure. CSA algorithm is the major methodology and it is based on fuzzy logic model of perception (FLMP) and interpretive structural modeling (ISM). CSA could display individualized knowledge structure and clearly represent hierarchies and linkage am

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33

Mandel, Travis. "Theta bases are atomic." Compositio Mathematica 153, no. 6 (2017): 1217–19. http://dx.doi.org/10.1112/s0010437x17007060.

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Fock and Goncharov conjectured that the indecomposable universally positive (i.e. atomic) elements of a cluster algebra should form a basis for the algebra. This was shown to be false by Lee, Li and Zelevinsky. However, we find that the theta bases of Gross, Hacking, Keel and Kontsevich do satisfy this conjecture for a slightly modified definition of universal positivity in which one replaces the positive atlas consisting of the clusters by an enlargement we call the scattering atlas. In particular, this uniquely characterizes the theta functions.

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34

SCOTT, JOSHUA S. "GRASSMANNIANS AND CLUSTER ALGEBRAS." Proceedings of the London Mathematical Society 92, no. 2 (2006): 345–80. http://dx.doi.org/10.1112/s0024611505015571.

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This paper follows the program of study initiated by S. Fomin and A. Zelevinsky, and demonstrates that the homogeneous coordinate ring of the Grassmannian $\mathbb{G}(k, n)$ is a {\it cluster algebra of geometric type}. Those Grassmannians that are of {\it finite cluster type} are identified and their cluster variables are interpreted geometrically in terms of configurations of points in $\mathbb{C}\mathbb{P}^2$.

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35

Casbi, Elie. "Equivariant multiplicities of simply-laced type flag minors." Representation Theory of the American Mathematical Society 25, no. 37 (2021): 1049–92. http://dx.doi.org/10.1090/ert/589.

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Let g \mathfrak {g} be a finite simply-laced type simple Lie algebra. Baumann-Kamnitzer-Knutson recently defined an algebra morphism D ¯ \overline {D} on the coordinate ring C [ N ] \mathbb {C}[N] related to Brion’s equivariant multiplicities via the geometric Satake correspondence. This map is known to take distinguished values on the elements of the MV basis corresponding to smooth MV cycles, as well as on the elements of the dual canonical basis corresponding to Kleshchev-Ram’s strongly homogeneous modules over quiver Hecke algebras. In this paper we show that when g \mathfrak {g} is of typ

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36

Goodearl, Kenneth, and M. T. Yakimov. "The Berenstein–Zelevinsky quantum cluster algebra conjecture." Journal of the European Mathematical Society 22, no. 8 (2020): 2453–509. http://dx.doi.org/10.4171/jems/969.

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37

Barot, M., D. Kussin, and H. Lenzing. "The cluster category of a canonical algebra." Transactions of the American Mathematical Society 362, no. 08 (2010): 4313–30. http://dx.doi.org/10.1090/s0002-9947-10-04998-6.

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38

Mather, Laura A. "A linear algebra measure of cluster quality." Journal of the American Society for Information Science 51, no. 7 (2000): 602–13. http://dx.doi.org/10.1002/(sici)1097-4571(2000)51:7<602::aid-asi3>3.0.co;2-1.

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39

Zito, Stephen. "τ-Tilting finite cluster-tilted algebras". Proceedings of the Edinburgh Mathematical Society 63, № 4 (2020): 950–55. http://dx.doi.org/10.1017/s0013091520000255.

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40

Saleh, Ibrahim. "Clustered Hyperbolic Categories." Algebra Colloquium 25, no. 01 (2018): 81–106. http://dx.doi.org/10.1142/s1005386718000068.

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We introduce a class of categories, called clustered hyperbolic categories, which are constructed from a categorized version of preseeds called categorical preseeds using categorical mutations that are a “functorial” edition of preseed mutations. Every Weyl preseed p gives rise to a categorical preseed [Formula: see text] which generates a clustered hyperbolic category; this is formed by copies of categories each one of which is equivalent to the category of representations of the Weyl cluster algebra [Formula: see text]. A “categorical realization” of Weyl cluster algebra is provided in the s

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41

Grabowski, Jan E. "Graded cluster algebras." Journal of Algebraic Combinatorics 42, no. 4 (2015): 1111–34. http://dx.doi.org/10.1007/s10801-015-0619-9.

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42

Kimura, Yuta. "Tilting and Cluster Tilting for Preprojective Algebras and Coxeter Groups." International Mathematics Research Notices 2019, no. 18 (2017): 5597–634. http://dx.doi.org/10.1093/imrn/rnx265.

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AbstractWe study the stable category of the graded Cohen–Macaulay modules of the factor algebra of the preprojective algebra associated with an element $w$ of the Coxeter group of a quiver. We show that there exists a silting object $M(\boldsymbol{w})$ of this category associated with each reduced expression $\boldsymbol{w}$ of $w$ and give a sufficient condition on $\boldsymbol{w}$ such that $M(\boldsymbol{w})$ is a tilting object. In particular, the stable category is triangle equivalent to the derived category of the endomorphism algebra of $M(\boldsymbol{w})$. Moreover, we compare it with

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43

Musiker, Gregg, Ralf Schiffler, and Lauren Williams. "Bases for cluster algebras from surfaces." Compositio Mathematica 149, no. 2 (2012): 217–63. http://dx.doi.org/10.1112/s0010437x12000450.

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AbstractWe construct two bases for each cluster algebra coming from a triangulated surface without punctures. We work in the context of a coefficient system coming from a full-rank exchange matrix, such asprincipal coefficients.

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44

ASSEM, IBRAHIM, GRÉGOIRE DUPONT, RALF SCHIFFLER, and DAVID SMITH. "FRIEZES, STRINGS AND CLUSTER VARIABLES." Glasgow Mathematical Journal 54, no. 1 (2011): 27–60. http://dx.doi.org/10.1017/s0017089511000322.

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AbstractTo any walk in a quiver, we associate a Laurent polynomial. When the walk is the string of a string module over a 2-Calabi–Yau tilted algebra, we prove that this Laurent polynomial coincides with the corresponding cluster character of the string module up to an explicit normalising monomial factor.

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Bertani-Økland, M. A., S. Oppermann, and A. Wrålsen. "Finding a cluster-tilting object for a representation finite cluster-tilted algebra." Colloquium Mathematicum 121, no. 2 (2010): 249–63. http://dx.doi.org/10.4064/cm121-2-7.

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Chang, Wen, and Bin Zhu. "Cluster automorphism groups of cluster algebras of finite type." Journal of Algebra 447 (February 2016): 490–515. http://dx.doi.org/10.1016/j.jalgebra.2015.09.045.

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Zhou, Yu, and Bin Zhu. "Cluster combinatorics of d-cluster categories." Journal of Algebra 321, no. 10 (2009): 2898–915. http://dx.doi.org/10.1016/j.jalgebra.2009.01.032.

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Cañadas, Agustín Moreno, Robinson-Julian Serna, and Isaías David Marín Gaviria. "Zavadskij modules over cluster-tilted algebras of type $ \mathbb{A} $." Electronic Research Archive 30, no. 9 (2022): 3435–51. http://dx.doi.org/10.3934/era.2022175.

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<abstract><p>Zavadskij modules are uniserial tame modules. They arose from interactions between the poset representation theory and the classification of general orders. The main problem is to characterize Zavadskij modules over a finite-dimensional algebra $ A $. In this setting, we prove that the indecomposable uniserial $ A $-modules with a mast of multiplicity one in each vertex are Zavadskij modules. Since the Zavadskij property carries over to direct summands, but it is not invariant under the formation of direct sums, we give a criterion to determine when the direct sum of i

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Cerulli Irelli, Giovanni, and Daniel Labardini-Fragoso. "Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials." Compositio Mathematica 148, no. 6 (2012): 1833–66. http://dx.doi.org/10.1112/s0010437x12000528.

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AbstractTo each tagged triangulation of a surface with marked points and non-empty boundary we associate a quiver with potential in such a way that whenever we apply a flip to a tagged triangulation the Jacobian algebra of the quiver with potential (QP) associated to the resulting tagged triangulation is isomorphic to the Jacobian algebra of the QP obtained by mutating the QP of the original one. Furthermore, we show that any two tagged triangulations are related by a sequence of flips compatible with QP-mutation. We also prove that, for each of the QPs constructed, the ideal of the non-comple

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Du, Qiuning, and Fang Li. "Some elementary properties of Laurent phenomenon algebras." Electronic Research Archive 30, no. 8 (2022): 3019–41. http://dx.doi.org/10.3934/era.2022153.

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<abstract><p>Let $ \Sigma $ be a Laurent phenomenon (LP) seed of rank $ n $, $ \mathcal{A}(\Sigma) $, $ \mathcal{U}(\Sigma) $, and $ \mathcal{L}(\Sigma) $ be its corresponding Laurent phenomenon algebra, upper bound and lower bound respectively. We prove that each seed of $ \mathcal{A}(\Sigma) $ is uniquely defined by its cluster and any two seeds of $ \mathcal{A}(\Sigma) $ with $ n-1 $ common cluster variables are connected with each other by one step of mutation. The method in this paper also works for (totally sign-skew-symmetric) cluster algebras. Moreover, we show that $ \math

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