Triangulated Categories: Cluster algebras, quiver representations and triangulated categories

By B. Keller. 2008

204 citations recorded by the source corpus.

This is an introduction to some aspects of Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/18d9d7edb2941360276d2a41d72660247cb049c1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. David E. Speyer
2. Fan Qin
3. C. Vafa
4. S. Cecotti
5. Sophie Morier-Genoud
6. Andrew Neitzke
7. Philipp Lampe
8. Sarah Scherotzke

In Geometry and Topology

1. David E. Speyer
2. S. Cecotti
3. C. Vafa
4. B. Keller
5. Andrew Neitzke
6. Fan Qin
7. Nathan Reading
8. Sophie Morier-Genoud

In Mathematical Physics

1. Fan Qin
2. Nathan Reading
3. David E. Speyer
4. B. Keller
5. Philipp Lampe
6. C. Vafa
7. S. Cecotti
8. Andrew Neitzke

Citations in the corpus

• Coxeter's frieze patterns at the crossroads of algebra, geometry and combinatorics (2015, background): “We define the main notions and give the main results that we need from quiver representations and from the theory of cluster algebras; details can be found in classical textbooks or surveys on the subjects; see, for example, [7, 9, 41, 43, 52, 61, 73, 77].”
• Sortable elements in infinite Coxeter groups (2008, background): “Our work is motivated by connections to cluster algebras, as is much current research on quivers; the reader who wishes to know the best current results relating quiver representation theory to cluster algebras should consult [25].”
• The Hall algebra of a spherical object (2008, unknown): “[2, 14, 19, 20].”
• Triangular bases in quantum cluster algebras and monoidal categorification conjectures (2015, background): “ter algebras. 10 FANQIN 2.2. Quivers and cluster categories. We review the necessary notions on the additive categorification of a cluster algebra by cluster categories, more details could be found in [Kel08] [Pla11b]. Let Be be the m ×n matrix introduced before. Assume that its principal part B is s”
• A Quantum Cluster Algebra of Kronecker Type and the Dual Canonical Basis (2010, background): “Keller ([20]) proved that C Q is indeed triangulated.”