University of Cambridge > > Isaac Newton Institute Seminar Series > Poisson geometry of directed networks and integrable lattices

Poisson geometry of directed networks and integrable lattices

Add to your list(s) Download to your calendar using vCal

If you have a question about this talk, please contact Mustapha Amrani.

Discrete Integrable Systems

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. Next, we generalize Postnikov’s construction by de ning 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 Coexeter-Toda flows in GL(n). This is a joint work with M. Shapiro and A. Vainshtein.

This talk is part of the Isaac Newton Institute Seminar Series series.

Tell a friend about this talk:

This talk is included in these lists:

Note that ex-directory lists are not shown.


© 2006-2021, University of Cambridge. Contact Us | Help and Documentation | Privacy and Publicity