x PREFACE

positivity. It is still an open question whether Laurent positivity holds for all dis-

tinguished elements for an arbitrary cluster algebra. All Laurent positive elements

of a cluster algebra form a cone. Conjecturally, for cluster algebras arising from

reductive semisimple Lie groups extremal rays of this cone form a basis closely

connected to the dual canonical basis mentioned above.

Since then, the theory of cluster algebras has witnessed a spectacular growth,

first and foremost due to the many links that have been discovered with a wide

range of subjects including

• representation theory of quivers and finite-dimensional algebras and cat-

egorification;

• discrete dynamical systems based on rational recurrences, in particular,

Y -systems in the thermodynamic Bethe Ansatz;

• Teichm¨ uller and higher Teichm¨ uller spaces;

• combinatorics and the study of combinatorial polyhedra, such as the

Stasheff associahedron and its generalizations;

• commutative and non-commutative algebraic geometry, in particular,

– Grassmannians, projective configurations and their tropical analogues,

– the study of stability conditions in the sense of Bridgeland,

– Calabi-Yau algebras,

– Donaldson-Thomas invariants,

– moduli space of (stable) quiver representations.

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.

In Chapter 1 we introduce necessary notions and notation. Section 1.1 provides

a very concise introduction to flag varieties, Grassmannians and Pl¨ ucker coordi-

nates. Section 1.2 treats simple Lie algebras and groups. Here we remind to the

reader the standard objects and constructions used in Lie theory, including the

adjoint action, the Killing form, Cartan subalgebras, root systems, and Dynkin dia-

grams. We discuss in some detail Bruhat decompositions of a simple Lie group, and

double Bruhat cells, which feature prominently in the next Chapter. Poisson–Lie

groups are introduced in Section 1.3. We start with providing basic definitions of

Poisson geometry, and proceed to define main objects in Poisson–Lie theory, in-

cluding the classical R-matrix. Sklyanin brackets are then defined as a particular

example of an R-matrix Poisson bracket. Finally, we treat in some detail the case

of the standard Poisson–Lie structure on a simple Lie group, which will play an

important role in subsequent chapters.