Cluster algebras II: Finite type classification

Details Cluster algebras II: Finite type classification Cluster algebras II: Finite type classification

2002-08-29

arxiv.org/abs/math/0208229v2

Mathematics

Rings and Algebras

50 pages, 18 figures. Version 2: new introduction; final version, to appear in Invent. Math

Scientific paper

This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many clusters. This classification turns out to be identical to the Cartan-Killing classification of semisimple Lie algebras and finite root systems, which is intriguing since in most cases, the symmetry exhibited by the Cartan-Killing type of a cluster algebra is not at all apparent from its geometric origin. The combinatorial structure behind a cluster algebra of finite type is captured by its cluster complex. We identify this complex as the normal fan of a generalized associahedron introduced and studied in hep-th/0111053 and math.CO/0202004. Another essential combinatorial ingredient of our arguments is a new characterization of the Dynkin diagrams.

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