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2 produkter
2 produkter
Del 45 - Mathematical Society Of Japan Memoirs
Cluster Algebras And Dilogarithm Identities
Häftad, Engelska, 2026
529 kr
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This is a reasonably self-contained exposition of the fascinating interplay between cluster algebras and the dilogarithm in the past two decades. The dilogarithm has a long and rich history since it was studied by Euler. The most intriguing property of the function is that it satisfies various functional relations, which we call dilogarithm identities (DIs). In the 1990s, various DIs were conjectured in the study of integrable models, but most of them were left unsolved. On the other hand, cluster algebras are a class of commutative algebras introduced by Fomin and Zelevinsky around 2000. In this text, we explain how the above DIs are proved using the techniques and results of cluster algebras. Also, we employ the DI associated with each period in a cluster pattern of cluster algebra as the leitmotif and present several proofs, variations, and generalizations of them with various methods and techniques. The quantum DIs are also treated from a unified point of view compared to the classical ones.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets
Del 41 - Mathematical Society Of Japan Memoirs
Cluster Algebras And Scattering Diagrams
Häftad, Engelska, 2023
779 kr
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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.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets