Abstract:
We introduce a Poisson variety compatible with a cluster algebra structure and a compatible toric action on this variety. We study Poisson and topological properties of the union of generic orbits of this toric action. In particular, we compute the number of connected components of the union of generic toric orbits for cluster algebras over real numbers. As a corollary we compute the number of connected components of refined open Bruhat cells in Grassmanians $G(k,n)$ over $\mathbb R$.
Key words and phrases:Cluster algebras, Poisson brackets, toric action, symplectic leaves, real Grassmannians, Sklyanin bracket.