scholarly journals The cluster category of a canonical algebra

2010 ◽  
Vol 362 (08) ◽  
pp. 4313-4330 ◽  
Author(s):  
M. Barot ◽  
D. Kussin ◽  
H. Lenzing
Keyword(s):  
Cluster Category ◽  
Canonical Algebra

scholarly journals The cluster category of a surface with punctures via group actions

2021 ◽  
Vol 389 ◽  
pp. 107884
Author(s):  
Claire Amiot ◽  
Pierre-Guy Plamondon
Keyword(s):  
Group Actions ◽  
Cluster Category

scholarly journals Kac Polynomials for Canonical Algebras

2017 ◽  
Vol 2019 (13) ◽  
pp. 3981-4003
Author(s):  
Pierre-Guy Plamondon ◽  
Olivier Schiffmann
Keyword(s):  
Finite Field ◽  
Dimension Vector ◽  
Indecomposable Representations ◽  
Moduli Stacks ◽  
Fixed Dimension ◽  
Canonical Algebra

Abstract We prove that the number of geometrically indecomposable representations of fixed dimension vector $\mathbf{d}$ of a canonical algebra $C$ defined over a finite field $\mathbb{F}_q$ is given by a polynomial in $q$ (depending on $C$ and $\mathbf{d}$). We prove a similar result for squid algebras. Finally, we express the volume of the moduli stacks of representations of these algebras of a fixed dimension vector in terms of the corresponding Kac polynomials.

scholarly journals Lifting to Cluster-tilting Objects in Higher Cluster Categories

2012 ◽  
Vol 19 (04) ◽  
pp. 707-712
Author(s):  
Pin Liu
Keyword(s):  
Positive Integer ◽  
Cluster Category ◽  
Tilting Modules ◽  
Endomorphism Rings ◽  
Tilted Algebras ◽  
Cluster Categories ◽  
Cluster Tilting ◽  
Tilting Objects

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.

scholarly journals Cotorsion pairs in the cluster category of a marked surface

2013 ◽  
Vol 391 ◽  
pp. 209-226 ◽  
Author(s):  
Jie Zhang ◽  
Yu Zhou ◽  
Bin Zhu
Keyword(s):  
Cluster Category ◽  
Cotorsion Pairs ◽  
Marked Surface

INFORMATION GEOMETRY FOR SOME LIE ALGEBRAS

1999 ◽  
Vol 02 (03) ◽  
pp. 441-460 ◽  
Author(s):  
H. NENCKA ◽  
R. F. STREATER
Keyword(s):  
Lie Algebras ◽  
Discrete Series ◽  
Complex Structure ◽  
Solvable Model ◽  
Coadjoint Orbits ◽  
Exactly Solvable Model ◽  
Metaplectic Representation ◽  
Canonical Algebra ◽  
Model Dynamics ◽  
Exactly Solvable

For certain unitary representations of a Lie algebra [Formula: see text] we define the statistical manifold ℳ of states as the convex cone of [Formula: see text] for which the partition function Z= Tr exp {-X} is finite. The Hessian of Ψ= log Z defines a Riemannian metric g on [Formula: see text], (the Bogoliubov–Kubo–Mori metric); [Formula: see text] foliates into the union of coadjoint orbits, each of which can be given a complex structure (that of Kostant). The program is carried out for so(3), and for sl(2,R) in the discrete series. We show that ℳ=R+× CP 1 and R+×H respectively. We show that for the metaplectic representation of the quadratic canonical algebra, ℳ=R+× CP 2/Z2. Exactly solvable model dynamics is constructed in each case.

scholarly journals On the cluster category of a marked surface without punctures

2011 ◽  
Vol 5 (4) ◽  
pp. 529-566 ◽  
Author(s):  
Thomas Brüstle ◽  
Jie Zhang
Keyword(s):  
Cluster Category ◽  
Marked Surface

scholarly journals Algebras of acyclic cluster type: Tree type and type Ã

2013 ◽  
Vol 211 ◽  
pp. 1-50 ◽  
Author(s):  
Claire Amiot ◽  
Steffen Oppermann
Keyword(s):  
Equivalence Class ◽  
Complete Classification ◽  
Global Dimension ◽  
Path Algebra ◽  
Derived Category ◽  
Derived Equivalence ◽  
Cluster Category ◽  
Cluster Type ◽  
Tree Type

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 describe the Auslander-Reiten quiver of its derived category. Together, these results in particular provide a complete classification of algebras which are cluster equivalent to tame acyclic quivers.

scholarly journals ELECTRIC-MAGNETIC DUALITY AND THE “LOOP REPRESENTATION” IN ABELIAN GAUGE THEORIES

1996 ◽  
Vol 11 (13) ◽  
pp. 1107-1114 ◽  
Author(s):  
LORENZO LEAL
Keyword(s):  
Phase Space ◽  
Gauge Theories ◽  
Topological Algebra ◽  
Geometric Representation ◽  
Abelian Gauge ◽  
Dual Algebra ◽  
Nonlocal Operators ◽  
Canonical Algebra

Abelian gauge theories are quantized in a geometric representation that generalizes the loop representation and treats electric and magnetic operators on the same footing. The usual canonical algebra is turned into a topological algebra of nonlocal operators that resembles the order-disorder dual algebra of ’t Hooft. These dual operators provide a complete description of the physical phase space of the theories.

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.

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