nLab
cluster algebra

Contents

Idea

A matrix AA 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 nn comes equipped with some subsets of size nn called clusters. Some of these clusters are related by sequences of operations called mutations.

Definition

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The connections to the exact WKB method a la Voros are studied in

Generalized cluster algebras are studied in