PREFACE xiii

We prove that the exchange graph of the cluster algebra with principal coeﬃcients

covers the exchange graph of any other cluster algebra with the same exchange

matrix, see Theorem 7.1. In Section 7.2 we consider vertices and edges of an

exchange graph and prove that distinct seeds have distinct clusters for cluster al-

gebras of geometric type and for cluster algebras with arbitrary coeﬃcients and

a non-degenerate exchange matrix, see Theorem 7.4. Besides, we prove that if a

cluster algebra has the above cluster-defines-seed property, then adjacent vertices

of its exchange graph are clusters that differ only in one variable, see Theorem 7.5.

Finally, in Section 7.3 we prove that the exchange graph of a cluster algebra with a

non-degenerate exchange matrix does not depend on coeﬃcients, see Theorem 7.7.

In the remaining three chapters we develop an approach to the interaction of

Poisson and cluster structures based on the study of perfect networks—directed

networks on surfaces with boundary having trivalent interior vertices and univalent

boundary vertices. Perfect planar networks in the disk are treated in Chapter 8.

Main definitions, including weights and boundary measurements, are given in Sec-

tion 8.1. We prove that each boundary measurement is a rational function of the

edge weights, see Proposition 8.3, and define the boundary measurement map from

the space of edge weights of the network to the space of k × m matrices, where k

and m are the numbers of sources and sinks. Section 8.2 is central to this chapter;

it treats Poisson structures on the space of edge weights of the network and Poisson

structures on the space of k × m matrices induced by the boundary measurement

map. The former are defined axiomatically as satisfying certain natural conditions,

including an analog of the Poisson–Lie property for groups. It is proved that such

structures form a 6-parametric family, see Proposition 8.5. For fixed sets of sources

and sinks, the induced Poisson structures on the space of k × m matrices form a

2-parametric family that does not depend on the internal structure of the network,

see Theorem 8.6. Explicit expressions for this 2-parametric family are provided

in Theorem 8.7. In case k = m and separated sources and sinks, we recover on

k × k matrices the Sklyanin bracket corresponding to a 2-parametric family of clas-

sical R-matrices including the standard R-matrix, see Theorem 8.10. In Section 8.3

we extend the boundary measurement map to the Grassmannian Gk(k + m) and

prove that the obtained Grassmannian boundary measurement map induces a 2-

parametric family of Poisson structures on Gk(k + m) that does not depend on the

particular choice of k sources and m sinks; moreover, for any such choice, the fam-

ily of Poisson structures on k × m matrices representing the corresponding cell of

Gk(k+m) coincides with the family described in Theorem 8.7 (see Theorem 8.12 for

details). Next, we give an interpretation of the natural GLk+m action on Gk(k+m)

in terms of networks and establish that every member of the above 2-parametric

family of Poisson structures on Gk(k + m) makes it into a Poisson homogeneous

space of GLk+m equipped with the Sklyanin R-matrix bracket, see Theorem 8.17.

Finally, in Section 8.4 we prove that each bracket in this family is compatible with

the cluster algebra constructed in Chapter 4, see Theorem 8.20. An important

ingredient of the proof is the use of face weights instead of edge weights.

In Chapter 9 we extend the constructions of the previous chapter to perfect

networks in an annulus. Section 9.1.1 is parallel to Section 8.1; the main difference

is that in order to define boundary measurements we have to introduce an auxiliary

parameter λ that counts intersections of paths in the network with a cut whose end-

points belong to distinct boundary circles. We prove that boundary measurements