Cluster algebras

By Bernard Leclerc, Lauren K. Williams. Proceedings of the National Academy of Sciences of the United States of America, 2014

689 citations recorded by the source corpus.

Cluster algebras were conceived by Fomin and Zelevinsky (1) in the spring of 2000 as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. However, the theory of cluster algebras has since taken on a life of its own, as connections and applications have been discovered in diverse areas of mathematics, including representation theory of quivers and finite dimensional algebras, cf., for example, refs. 2⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–15; Poisson geometry (16⇓⇓–19); Teichmuller theory (20⇓⇓⇓–24); string theory (25⇓⇓⇓⇓⇓–31); discrete dynamical systems and integrability (6, 32⇓⇓⇓⇓⇓–38); and combinatorics (39⇓⇓⇓⇓⇓⇓⇓–47). Quite remarkably, cluster algebras provide a unifying algebraic and combinatorial framework for a wide variety of phenomena in these and other settings. We refer the reader to the survey papers (36, 48⇓⇓⇓⇓–53) and to the cluster algebras portal (www.math.lsa.umich.edu/~fomin/cluster.html) for various introductions to cluster algebras and their links with other subjects in mathematics (and physics). In brief, a cluster algebra A of rank k is a subring of an ambient field ℱ of rational functions in k variables, say x 1, …, x k . Unlike most commutative rings, a cluster algebra is not presented at the outset via a complete set of generators and relations. Instead, from the data of the initial seed — which includes the k initial cluster variables x 1, …, x k , plus an exchange matrix — one uses an iterative procedure called “mutation” to produce the rest of the cluster variables. In particular, each new cluster … [↵][1]1To whom correspondence should be addressed. Email: williams{at}math.berkeley.edu. [1]: #xref-corresp-1-1

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7d390be9d4d90d47e658f7391ab6dbfd1a41b4b0

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Citations in the corpus

Monoidal categorification of cluster algebras (2014, background): “Fomin and Zelevinsky proved that every cluster variable is a Laurent polynomial of the initial cluster {Xi}1≤i≤r and they conjectured that this Laurent polynomial has positive coefficients ([5]).”
$\tau $-tilting theory (2013, neutral): “例外列 (exceptional sequence)の変異 [7] と団傾対象 (cluster tilting object)の変異 [6, 10]の2種類が広く知られているが, いずれもある種の三角圏の構造を解析するものであり, 特に後者は高次元Auslander-Reiten理論 [9]や団代数 (cluster algebra)の圏論化 [12]にも応用される重要なものである.”
Infinitely many Lagrangian fillings (2020, methodology): “In terms of cluster algebras, the quotient Gr(3, 9)/(C∗)9 has no frozen cluster variables, whereas the Grassmannian Gr(3, 9) [26, 45] has the cyclically consecutive Plücker coordinates as frozen cluster variables.”
Regularity theorem for totally nonnegative flag varieties (2019, background): “Totally positive spaces have attracted a lot of interest due to their appearances in other contexts such as cluster algebras [9] and the physics of scattering amplitudes [1].”
The wall-crossing formula and Lagrangian mutations (2017, neutral): “LG seeds and their mutations are an instance of the general notions of cluster algebra [15].”
Dimer models and cluster categories of Grassmannians (2013, background): “The effect on Q(D) of applying the geometric exchange is to carry out the first two steps of Fomin-Zelevinsky quiver mutation [13] at the vertex I, i.”
INTEGRABLE CLUSTER DYNAMICS OF DIRECTED NETWORKS AND PENTAGRAM MAPS (2014, neutral): “This relation was discovered and explored by Glick [16] who proved that the pentagram map, acting on the quotient space Pn/R (the action of R∗ commutes with the map and is given by the formula Xi → tXi, Yi → t−1Yi), is described by coefficient dynamics [10] – also known as τ -transformations, see Ch”