A matrix $A$ over the field of real numbers is totally positive (resp. totally nonnegative) if every minor (= determinant of any submatrix) is a positive (resp. nonnegative) real number. Total positivity implies a number of remarkable properties; for example all eigenvalues are distinct and positive.
George Lusztig discovered that total positivity is closely related to some phenomena in the theory of Lie groups and quantum groups. Later, S. Fomin and A. Zelevinsky studied the canonical bases for quantum groups and discovered the combinatorics of simple transformations and defined associated classical and quantum cluster algebras to such situations. In particular, Stasheff associahedra are associated to these cluster algebras. Remarkably, they found an unusual algebraic geometry related to cluster algebras, possessing new, and at the beginning mysterious, Laurent phenomenon. Later, the cluster algebras appeared also in the connection to the representations of quivers, tilting theory and the wall crossing phenomenon, with the applications in representation theory and the study of triangulated categories.
A cluster algebra of rank $n$ comes equipped with some subsets of size $n$ called clusters. Some of these clusters are related by sequences of operations called mutations.
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Sergey Fomin, Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497–529, math.RT/0104151; Cluster algebras. II. Finite type classifications. Invent. Math. 154 (2003), no. 1, 63–121 – classification of cluster algebras of finite type via root systems math.RA/0208229, Cluster algebras, IV: Coefficients, Compos. Math. 143 (2007), 112–164, MR2295199 doi
Arkady Berenstein, Andrei Zelevinsky, Quantum cluster algebras, math.QA/0404446
K. Goodearl, M. Yakimov, Quantum cluster algebras and quantum nilpotent algebras Proc Natl Acad Sci USA 111(27):9696-703 (2014) doi
B. Keller, Cluster algebras, quiver representations and triangulated categories, arXiv:0807.1960 MR2681708
Lots of links at Cluster algebras portal including to the Fomin’s course slides.
description and conference info “Cluster Algebras and Lusztig’s Semicanonical Basis”, Oregon, June 2011, html
Maxim Kontsevich, Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435
Yuji Kodama, Lauren Williams, KP solitons, total positivity, and cluster algebras, Proc. Natl. Acad. Sci. arxiv/1105.4170, doi
Bernard Leclerc, Cluster algebras and representation theory, arxiv/1009.4552
Christof Geiss, Bernard Leclerc, Jan Schröer, Preprojective algebras and cluster algebras, arxiv/0804.3168; Kac-Moody groups and cluster algebras, arxiv/1001.3545; Cluster structures on quantum coordinate rings, arxiv/1104.0531
Tomoki Nakanishi, Dilogarithm identities for conformal field theories and cluster algebras: Simply laced case, Nagoya Math. J. 202 (2011), 23-43, MR2804544 doi
The connections to the exact WKB method a la Voros are studied in
Generalized cluster algebras are studied in