We also show how one can interpret generalized acklund–Darboux transforma-
tions via equivalent transformations of the corresponding perfect networks in an
annulus. In conclusion, we explain that classical Darboux transformations can be
also interpreted via cluster transformations, see Proposition 10.39.
The process of writing spread over several years. We would like to thank all
individuals and institutions that supported us in this undertaking. The project
was started in Summer 2005 during our visit to Mathematisches Forschunginstitut
Oberwolfach in the framework of Research in Pairs program. Since then we con-
tinued to work on the book during our joint visits to Institut des Hautes Etudes
Scientifiques (Bures-sur-Yvette), Institut Mittag-Leffler (Djursholm), Max-Planck-
Institut fur Mathematik (Bonn), University of Stockholm, Royal School of Tech-
nology (Stockholm), University of Haifa, Michigan State University, and University
of Michigan. We thank all these institutions for warm hospitality and excellent
working conditions. During the process of writing, we had numerous discussions
with A. Berenstein, L. Chekhov, V. Fock, S. Fomin, R. Kulkarni, A. Postnikov,
N. Reshetikhin, J. Scott, B. Shapiro, M. Yakimov, A. Zelevinsky and others that
helped us to understand many different facets of cluster algebras. It is our pleasure
to thank all of them for their help.
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