Cluster algebras and scattering diagrams
Tomoki. Nakanishi
Bok · Engelsk · 2023
| Omfang | xiv, 279 pages : : illustrations ;
|
|---|---|
| Opplysninger | Basics in Cluster Algebras: -- Introduction to Part I -- Getting Started -- Basic Notions -- Fundamental Results -- Separation Formulas -- Principal Coefficients -- Upper Cluster Algebras -- Generalized Cluster Algebras -- Cluster Patterns and Scattering Diagrams: -- Introduction to Part II -- Cluster Patterns and Separation Formulas -- More about C- and G-Matrices -- Scattering Diagrams and C and G-Matrices -- More about F-Polynomials -- Scattering Diagrams -- Scattering Diagrams and F-Polynomials -- Cluster Scattering Diagrams: -- Introduction to Part III -- Underlying Algebraic Structure -- Consistent Scattering Diagrams -- Cluster Scattering Diagrams -- Principal Extension Method -- Positive Realization of CSD and Pentagon Relation -- Mutation Invariance of CSDs -- Broken Lines and Theta Functions.. - "The theme of this monograph is the relation between cluster algebras and scattering diagrams. Cluster algebras were introduced by Fomin and Zelevinsky around 2000 as an algebraic and combinatorial structure originated in Lie theory. Recently, Gross, Hacking, Keel, and Kontsevich solved several important conjectures in cluster algebra theory by the scattering diagram method introduced in the homological mirror symmetry. This monograph is the first comprehensive exposition of this important development. The text consists of three parts. Part I is a first step guide to the theory of cluster algebras for readers without any knowledge on cluster algebras. Part II is the main part of the monograph, where we focus on the column sign-coherence of C-matrices and the Laurent positivity for cluster patterns, both of which were conjectured by Fomin and Zelevinsky and proved by Gross, Hacking, Keel, and Kontsevich based on the scattering diagram method. Part III is a self-contained exposition of several fundamental properties of cluster scattering diagrams with emphasis on the roles of the dilogarithm elements and the pentagon relation. As a specific feature of this monograph, each part is written without explicitly relying on the other parts. Thus, readers can start reading from any part depending on their interest and knowledge."- publisher.
|
| Emner | |
| ISBN | 9784864971058
|