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SUMMARY:Poisson geometry of directed networks and integrable lattices - Ge
 khtman\, M (Notre Dame)
DTSTART:20090514T140000Z
DTEND:20090514T150000Z
UID:TALK18414@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Recently\, Postnikov used weighted directed planar graphs in a
  disk to parametrize cells in Grassmannians. We investigate Poisson proper
 ties of Postnikov's map from the space of edge weights of a planar directe
 d network into the Grassmannian. We show that this map is Poisson if the s
 pace of edge weights is equipped with a representative of a 6-parameter fa
 mily of universal quadratic Poisson brackets and the Grasmannian is viewed
  as a Poisson homogeneous space GL(n) equipped with an appropriately chose
 n R-matrix Poisson-Lie structure. Next\, we generalize Postnikov's constru
 ction by de ning a map from the space of edge weights of a directed networ
 k in an annulus into a space of loops in the Grassmannian. This family inc
 ludes\, for a particular kind of networks\, the Poisson bracket associated
  with the trigonometric R-matrix. We use a special kind of directed networ
 ks in an annulus to study a cluster algebra structure on a certain space o
 f rational functions and show that sequences of cluster transformations co
 nnecting two distinguished clusters are closely associated with Backlund-D
 arboux transformations between Coexeter-Toda flows in GL(n). This is a joi
 nt work with M. Shapiro and A. Vainshtein.
LOCATION:Satellite
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