Poisson Geometry of Directed Networks in a Disk

TL;DR

This paper explores the Poisson geometric structure of Postnikov's map from planar directed networks to the Grassmannian, revealing compatibility with cluster algebra structures and specific Poisson brackets.

Contribution

It establishes that Postnikov's map is Poisson with respect to a family of quadratic brackets and compatible with cluster algebra structures on the Grassmannian.

Findings

Poisson property of Postnikov's map proven

Compatibility with cluster algebra structure shown

Identification of a 6-parameter family of Poisson brackets

Abstract

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 of the general linear group equipped with an appropriately chosen R-matrix Poisson-Lie structure. We also prove that Poisson brackets on the Grassmannian arising in this way are compatible with the natural cluster algebra structure.