For more information about this meeting, contact Luen-Chau Li.
|Title:||Poisson geometry of directed networks and integrable lattices|
|Speaker:||Michael Gekhtman, Notre Dame|
|Recently, Postnikov used weighted directed planar graphs in a disk to parametrize cells in Grassmannians. We investigate Poisson properties of Postnikov's map from the space of edge weights of a planar directed network into the Grassmannian. We show that this map is Poisson if the space of edge weights is equipped with a representative of a 6-parameter family of universal quadratic Poisson brackets and the Grasmannian is viewed as a Poisson homogeneous space GL(n) equipped with an appropriately chosen R-matrix Poisson-Lie structure. We generalize Postnikov's construction by defining a map from the space of edge weights of a directed network in an annulus into a space of loops in the Grassmannian. This family includes, for a particular kind of networks, the Poisson bracket associated with the trigonometric R-matrix. We use a special kind of directed networks in an annulus to study a cluster algebra structure on a certain space of rational functions and show that sequences of cluster transformations connecting two distinguished clusters are closely associated with Backlund-Darboux transformations between Coxeter-Toda flows in GL(n).|
Room Reservation Information
|Date:||10 / 23 / 2009|
|Time:||02:30pm - 03:30pm|