scholarly journals Cluster algebras and cluster categories associated with triangulated surfaces: an introduction

2020 ◽  
Vol 5 ◽  
pp. 1-14
Author(s):  
Claire Amiot
Keyword(s):  
Cluster Algebras ◽  
Triangulated Surfaces ◽  
Cluster Categories

scholarly journals Density of g-Vector Cones From Triangulated Surfaces

2020 ◽  
Vol 2020 (21) ◽  
pp. 8081-8119
Author(s):  
Toshiya Yurikusa
Keyword(s):  
Asymptotic Behavior ◽  
Half Space ◽  
Closed Surface ◽  
Cluster Algebras ◽  
Connected Components ◽  
Cluster Category ◽  
Triangulated Surfaces ◽  
Cluster Tilting ◽  
Tilting Objects ◽  
Marked Surface

Abstract We study $g$-vector cones associated with clusters of cluster algebras defined from a marked surface $(S,M)$ of rank $n$. We determine the closure of the union of $g$-vector cones associated with all clusters. It is equal to $\mathbb{R}^n$ except for a closed surface with exactly one puncture, in which case it is equal to the half space of a certain explicit hyperplane in $\mathbb{R}^n$. Our main ingredients are laminations on $(S,M)$, their shear coordinates, and their asymptotic behavior under Dehn twists. As an application, if $(S,M)$ is not a closed surface with exactly one puncture, the exchange graph of cluster tilting objects in the corresponding cluster category is connected. If $(S,M)$ is a closed surface with exactly one puncture, it has precisely two connected components.

scholarly journals Cluster algebras via cluster categories with infinite-dimensional morphism spaces

2011 ◽  
Vol 147 (6) ◽  
pp. 1921-1954 ◽  
Author(s):  
Pierre-Guy Plamondon
Keyword(s):  
Cluster Algebras ◽  
Cluster Category ◽  
Infinite Cluster ◽  
Infinite Dimensional ◽  
Rigid Objects ◽  
Cluster Categories ◽  
Compositio Math

AbstractWe apply our previous work on cluster characters for Hom-infinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky’s Cluster algebras IV [Compositio Math. 143 (2007), 112–164] for skew-symmetric cluster algebras. We also construct an explicit bijection sending certain objects of the cluster category to the decorated representations of Derksen, Weyman and Zelevinsky, and show that it is compatible with mutations in both settings. Using this map, we give a categorical interpretation of the E-invariant and show that an arbitrary decorated representation with vanishing E-invariant is characterized by its g-vector. Finally, we obtain a substitution formula for cluster characters of not necessarily rigid objects.

scholarly journals Combinatorial Cluster Expansion Formulas from Triangulated Surfaces

10.37236/8351 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Toshiya Yurikusa
Keyword(s):  
Bipartite Graph ◽  
Cluster Expansion ◽  
Independent Sets ◽  
Perfect Matchings ◽  
Cluster Algebras ◽  
Expansion Formula ◽  
Triangulated Surfaces ◽  
Quiver With Potential ◽  
Maximal Independent Sets

We give a cluster expansion formula for cluster algebras with principal coefficients defined from triangulated surfaces in terms of maximal independent sets of angles. Our formula simplifies the cluster expansion formula given by Musiker, Schiffler and Williams in terms of perfect matchings of snake graphs. A key point of our proof is to give a bijection between maximal independent sets of angles in some triangulated polygon and perfect matchings of the corresponding snake graph. Moreover, they also correspond bijectively with perfect matchings of the corresponding bipartite graph and minimal cuts of the corresponding quiver with potential.

scholarly journals Cluster algebras and triangulated surfaces. Part I: Cluster complexes

2008 ◽  
Vol 201 (1) ◽  
pp. 83-146 ◽  
Author(s):  
Sergey Fomin ◽  
Michael Shapiro ◽  
Dylan Thurston
Keyword(s):  
Cluster Algebras ◽  
Cluster Complexes ◽  
Triangulated Surfaces

scholarly journals Group actions on cluster algebras and cluster categories

2019 ◽  
Vol 345 ◽  
pp. 161-221 ◽  
Author(s):  
Charles Paquette ◽  
Ralf Schiffler
Keyword(s):  
Group Actions ◽  
Cluster Algebras ◽  
Cluster Categories

scholarly journals CLUSTER CATEGORIES FROM GRASSMANNIANS AND ROOT COMBINATORICS

2019 ◽  
Vol 240 ◽  
pp. 322-354 ◽  
Author(s):  
KARIN BAUR ◽  
DUSKO BOGDANIC ◽  
ANA GARCIA ELSENER
Keyword(s):  
Cluster Algebra ◽  
Explicit Construction ◽  
Cluster Algebras ◽  
Algebra Structure ◽  
Coordinate Ring ◽  
Moody Algebra ◽  
Indecomposable Modules ◽  
Real Roots ◽  
Rank 2 ◽  
Cluster Categories

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 establish a correspondence between rigid indecomposable modules of rank 2 and real roots of degree 2 for the associated Kac–Moody algebra in the tame cases.

scholarly journals Tropical friezes and the index in higher homological algebra

2020 ◽  
pp. 1-27 ◽  
Author(s):  
PETER JØRGENSEN
Keyword(s):  
Homological Algebra ◽  
Cluster Algebras ◽  
Triangulated Categories ◽  
Two Dimensional ◽  
Cluster Category ◽  
Higher Dimensional ◽  
Cluster Categories ◽  
The Subject ◽  
Marked Points

Abstract Cluster categories and cluster algebras encode two dimensional structures. For instance, the Auslander–Reiten quiver of a cluster category can be drawn on a surface, and there is a class of cluster algebras determined by surfaces with marked points. Cluster characters are maps from cluster categories (and more general triangulated categories) to cluster algebras. They have a tropical shadow in the form of so-called tropical friezes, which are maps from cluster categories (and more general triangulated categories) to the integers. This paper will define higher dimensional tropical friezes. One of the motivations is the higher dimensional cluster categories of Oppermann and Thomas, which encode (d + 1)-dimensional structures for an integer d ⩾ 1. They are (d + 2)-angulated categories, which belong to the subject of higher homological algebra. We will define higher dimensional tropical friezes as maps from higher cluster categories (and more general (d + 2)-angulated categories) to the integers. Following Palu, we will define a notion of (d + 2)-angulated index, establish some of its properties, and use it to construct higher dimensional tropical friezes.

scholarly journals F-Matrices of Cluster Algebras from Triangulated Surfaces

2020 ◽  
Vol 24 (4) ◽  
pp. 649-695
Author(s):  
Yasuaki Gyoda ◽  
Toshiya Yurikusa
Keyword(s):  
Cluster Algebras ◽  
Triangulated Surfaces
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