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Cluster algebras and derived categories. (English) Zbl 1299.13027

Kawamata, Yujiro (ed.), Derived categories in algebraic geometry. Proceedings of a conference held at the University of Tokyo, Japan in January 2011. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-115-6/hbk). EMS Series of Congress Reports, 123-183 (2012).
This beautiful survey is about cluster algebras, the quantum version and their (additive) categorifications. In the first five sections, the author presents the basic examples, main conjectures and some (not all) developments of cluster algebras. In section 6, the quantum cluster algebras (and their mutations) are introduced, together with the link to Donaldson-Thomas theory. A theorem about quantum dilogarithm identities is presented. In Section 7, the author describes an additive categorification of cluster algebras via quivers with potential. The author shows how the derived categories of Ginzburg dg algebras associated to quivers with potential categorify cluster algebras. As an application, the sketch of the proof of the previous theorem is given.
For the entire collection see [Zbl 1256.14001].
Reviewer: Yu Qiu (Trondheim)

MSC:

13F60 Cluster algebras
18E30 Derived categories, triangulated categories (MSC2010)
16G20 Representations of quivers and partially ordered sets
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