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
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