TL;DR
This paper discusses the development and significance of cluster categories, which provide a categorical framework for understanding cluster algebras and their connections to representation theory and Calabi-Yau categories.
Contribution
It reviews the introduction, properties, and recent advancements of cluster categories, highlighting their role in categorifying cluster algebras and related algebraic structures.
Findings
Cluster categories are triangulated, as shown by Keller.
They establish a link between cluster algebras and representation theory.
Recent work extends to categories of Calabi-Yau dimension 2.
Abstract
Cluster algebras were introduced by Fomin-Zelevinsky in 2002 in order to give a combinatorial framework for phenomena occurring in the context of algebraic groups. Cluster algebras also have links to a wide range of other subjects, including the representation theory of finite dimensional algebras, as first discovered by Marsh- Reineke-Zelevinsky. Modifying module categories over hereditary algebras, cluster categories were introduced in work with Buan-Marsh-Reineke-Todorov in order to "categorify" the essential ingredients in the definition of cluster algebras in the acyclic case. They were shown to be triangulated by Keller. Related work was done by Geiss-Leclerc-Schr\"oer using preprojective algebras of Dynkin type. In work by many authors there have been further developments, leading to feedback to cluster algebras, new interesting classes of finite dimensional algebras, and the…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology · Finite Group Theory Research