Cluster algebras I: Foundations

Abstract

In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.

33 pages

Cited by in corpus (31)

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• Periodicities in cluster algebras and dilogarithm identities
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• Dimers and cluster integrable systems
• Network, Cluster coordinates and N=2 theory I
• Donaldson-Thomas trasnsformations of moduli spaces of G-local systems
• Quantum hypermultiplet moduli spaces in N=2 string vacua: a review
• A categorification of acyclic principal coefficient cluster algebras
• Q-system Cluster Algebras, Paths and Total Positivity
• Augmentations, Fillings, and Clusters
• T-systems, networks and dimers
• Factoriality and class groups of cluster algebras
• Cluster Variables and Perfect Matchings of Subgraphs of the $d{P}_3$ Lattice
• The solution of the $A_r$ T-system for arbitrary boundary
• Donaldson-Thomas Transformation of Double Bruhat Cells in Semisimple Lie Groups
• Cluster Structures on Double Bott-Samelson Cells
• Decorated marked surfaces II: Intersection numbers and dimensions of Homs
• Dual canonical bases and quantum cluster algebras
• Exact WKB analysis and cluster algebras
• An inhomogeneous Lambda-determinant
• Three dimensional Seiberg-like duality and tropical cluster algebra
• A formula for $F$-Polynomials in terms of $C$-Vectors and Stabilization of $F$-Polynomials
• Cluster Algebras and Scattering Diagrams, Part II. Cluster Patterns and Scattering Diagrams
• Stable Cluster Variables
• Exchange graphs for mutation-finite non-integer quivers of rank 3
• Truncated determinants and the refined enumeration of Alternating Sign Matrices and Descending Plane Partitions
• From frieze patterns to cluster categories
• The formula for the permutation of mutation sequences in $A_n$ straight orientation
• Braiding Operator via Quantum Cluster Algebra
• Cluster algebras and their bases
• Cluster Algebras and Scattering Diagrams, Part III. Cluster Scattering Diagrams