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 Grassmannian is viewed as a Poisson homogeneous space of the general linear group equipped with an appropriate R-matrix Poisson-Lie structure. We also prove that the Poisson brackets on the Grassmannian arising in this way are compatible with the natural cluster algebra structure.
Bibliographical noteFunding Information:
We wish to express our gratitude to A. Postnikov who explained to us the details of his construction and to S. Fomin for stimulating discussions. M. G. was supported in part by NSF Grant DMS #0400484. M. S. was supported in part by NSF Grants DMS #0401178 and PHY#0555346. The authors were also supported by the BSF Grant # 2002375. A. V. is grateful to Centre Interfacultaire Bernoulli at École Polytechnique Fédérale de Lausanne for hospitality during his Spring 2008 visit. We are also grateful to the Institute of Advanced Studies of the Hebrew University of Jerusalem for the invitation to the Midrasha Mathematicae in May 2008, during which this paper was completed.
- Poisson structure
- Postnikov's map,cluster algebra
ASJC Scopus subject areas
- Mathematics (all)
- Physics and Astronomy (all)