2004/02/04 by Aslak Bakke Buan, Bethany Marsh, Buan, Aslak Bakke +7 · 44 citations
Mathematics · #16G20 #16G70 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Bounded function #Cluster (spacecraft) #Cluster algebra #Combinatorics #Computer science #Conjecture #Derived category #FOS: Mathematics #Field (mathematics) #Functor #Injective function #Link (geometry) #Mathematics #Physics #Pure mathematics #Quotient #Representation Theory (math.RT) #Representation theory #Rings and Algebras (math.RA) #Type (biology) #math.RA #math.RT #msc:16G20 #msc:16G70
paper · pdf · doi:10.48550/arxiv.math/0402054
published in arXiv (Cornell University) (Cornell University) · 36 pages, no separate figures
arxiv created 2004/02/04 · openalex publication_date 2004/02/04 · arxiv updated 2020/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply-laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin-Zelevinsky cluster algebra. In this model, the tilting modules correspond to the clusters of Fomin-Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of self-injective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.