A matrix 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.
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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
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
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
Generalized cluster algebras are studied in