Reading Session on cluster algebras

We will take the time we need for each lecture.

Lecture 1: The Cartan-Killing classification (by Anouk Bergeron-Brlek)     
- Dynkin diagrams,
- Definite positive symmetric matrices,
- Crystallographic root systems
See for instance in [H] (Chapter 2) or in  [F-R] (Lectures 1 & 2)

Lecture 2: Cluster algebras (by Philippe Choquette)     
- Sees and clusters
- Cluster algebras
- Example: homogeneous coordiante ring of Gr(2,n+3)
(illustrated with  the associahedron)
See for instance in [F-R] (Lecture 3 & Lecture 4.1) or in [F-Z 1] and [F-Z 4]

Lecture 3: The classification of finite type cluster algebras (by Huilan Li)     
See for instance [F-R] (Lecture 4.2) or [F-Z 2]

Lecture 4: Cluster complex and generalized associahedra (by Ziting Zeng)     
See for instance  [F-R] (Lecture 4.3 & 4.4) or in [F-Z 3]

Lecture 5: Quivers: an introduction (by Andrew Douglas)     
- Representations of quivers
- quivers of finite type
See for instance [F-R] (end of lecture 2.3) or [C-C-S 1] and [C -C-S 2] for references

Lecture 6: The (m-)Cluster category
(by Hugh Thomas)     
-Derived category of representations of quivers
-Defining the cluster category
-Recovering the cluster algebra from the cluster category
See for instance [BMRRT], [F-R]  and [T] for references


[B-F-Z] A. Berenstein, S. Fomin and A. Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126(1)  (2005), 1-52.
[C-C-S 1] P. Caldero, F. Chapoton and R. Schiffler, Quivers with relations arising from clusters (A_n case), to appear in Trans. Amer. Math. Soc. (2004).
[C-C-S 2] P. Caldero, F. Chapoton and R. Schiffler, Quivers with relations and cluster tilted algebras (2004).
[F-R] S. Fomin and N. Reading, Root systems and generalized associahedra (Lectures Notes) (2005)
[F-Z 1] S. Fomin and A. Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15(2) (2002), 497- 529 (electronic).
[F-Z 2] S. Fomin and A. Zelevinsky, Cluster algebras. II. Finite type classification, Inventiones Mathematicae 154 (2003), 63-121.
[F-Z 3] S. Fomin and A. Zelevinsky, Y-Systems and generalized associahedra, Ann. of Math 158 (2003), 977-1018
[F-Z 4] S. Fomin and A. Zelevinsky, Cluster algebras: Notes for the CDM-03 conference (2004)
[H] J. E. Humphreys, Reflection groups and Coxeter groups, book, Cambridge university press
[BMRRT] A. B. Buan, R. Marsh, M. Reinecke, I. Reiten, and G. Todorov, Tilting theory and Cluster combinatorics (2004).
[T] H. Thomas, preprint in preparation

See also the web page  of the reading sessions on cluster algebras,  Lyon, France (in french)

You will find more on cluster algebras  on the ArXiv

Algebraic Combinatorics Seminar home