Cluster algebras' Reading Sessions

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
-m-Clusters

See for instance [BMRRT], [F-R] and [T] for references

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