Ebook: Cluster Algebras and Poisson Geometry

In this book, however, we deal only with one aspect of the cluster algebra the-
ory: its relations to Poisson geometry and theory of integrable systems. First of all,
we show that the cluster algebra structure, which is purely algebraic in its nature,
is closely related to certain Poisson (or, dually, pre-symplectic) structures. In the
cases of double Bruhat cells and Grassmannians discussed below, the corresponding
families of Poisson structures include, among others, standard R-matrix Poisson-
Lie structures (or their push-forwards). A large part of the book is devoted to the
interplay between cluster structures and Poisson/pre-symplectic structures. This
leads, in particular, to revealing of cluster structure related to integrable systems
called Toda lattices and to dynamical interpretation of cluster transformations, see
the last chapter. Vice versa, Poisson/pre-symplectic structures turned out to be
instrumental for the proof of purely algebraic results in the general theory of cluster
algebras.