Cluster Algebras Reading Group
- The reading group meets every Wednesday at 4pm in MSS/N6
Introduction
Cluster algebras were introduced by Fomin and Zelevinsky in their study of double Bruhat cells (the intersection of Bruhat cells coming from two opposite Borel subgroups). See paper. These algebras arise naturally in Lie theory as the coordinate rings of varieties attached to algebraic groups.
This beautiful theory has many applications in different areas, such as the Thermodynamic Bethe Ansatz in physics and the description of Teichmüller spaces in geometry. Actually many algebras arising naturally representation theory have the structure of cluster algebra.
The quantum version obtained by Berenstein-Zelevinsky is conjectured to be related to the quantization of the decorated Teichmüller space obtained independently by Kashaev and Chekhov-Fock
The aim of this reading group is to explore this theory from many diffeent points of view (algebraic, topological, dynamical systems, integrable systems, geometrical and numeric).
Timetable
- G.Walker
- Wednesday 7th February 2007
- Total positivity of Bruhat cells and cluster algebras: We define TP-bases in G and describe how these can be derived from reducued words in the standard presentation of the Weyl group W=S(n). We show (for n=3) how this leads to a cluster algebra structure in the coordinate ring of G(w_0,w_0).
- G.Walker
- Wednesday 31st Janaury 2007
- Double Bruhat cells:We introduce Bruhat cells and double Bruhat cells in G=GL(n,C) or SL(n,C), and show (for n=3) that a matrix in G is TP if it is in TNN and lies in the open double Bruhat cell G(w_0,w_0).
- G.Walker
- Wednesday 17th January 2007
- Total positivity: We introduced the notion of totally positive (TP) matrix and totally non-negative (TNN) matrix giving several examples. We discussed the coordinate ring of SL (n,C) and related it to the cluster algebra structure
- M.Mazzocco
- Wednesday 13th December 2006
- Cluster algebras of geometric type: We started this talk by showing a simple example of the so called Laurent phenomenon. We then discussed the cluster algebra structure associated to triangulations of polygons..
Further information
For Further information please contact Marta Mazzocco (Marta.Mazzocco[at]manchester.ac.uk)
