Cluster algebras: Notes for the CDM-03 conference

By Sergey Fomin, Andrei Zelevinsky. 2003

144 citations recorded by the source corpus.

This is an expanded version of the notes of our lectures given at the conference Current Developments in Mathematics 2003 held at Harvard University on November 21–22, 2003. We present an overview of the main definitions, results and applications of the theory of cluster algebras.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/fab1bd7507cb63da229d5ee13853365f40c203e6

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Citations in the corpus

Total positivity and cluster algebras (2010, background): “…minors play an important role in classical mechanics (theory of small oscillations), probability (one-dimensional diffusion processes), discrete potential theory (planar resistor networks), asymptotic representation theory (the Edrei-Thoma theorem), algebraic and enumerative…”
Tensor diagrams and cluster algebras (2012, background): “[13]) as a monomial in the elements of any given extended cluster.”
Acyclic Calabi–Yau categories (2006, background): “It was shown [BMRRT06, BMR06, BMR08, BMRT05, CC06, CK05, CK06] that this category fully determines the combinatorics of the cluster algebra associated with Q and carries considerably more information.”
Cluster algebras IV: Coefficients (2006, unknown): “In Theorem 3.7 below we give a formula that expresses a cluster variable in an arbitrary cluster algebra with a given initial exchange matrix B0 = Bt0 in terms of similar formulas for the corresponding cluster algebra with principal coefficients.”
Cluster algebras and triangulated surfaces. Part I: Cluster complexes (2006, background): “ch of the main construction studied in this paper, a class of cluster algebras associated with 2-dimensional surfaces, aimed at a reader familiar with the basics of cluster algebra theory (see, e.g., [16]). Let S be a connected oriented (2- dimensional) Riemann surface with boundary. Fix a ﬁnite set”
Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants (2010, methodology): “ectic double torus. Using the relation of symplectomorphisms and mutations as in [34] one obtains the following result, which we formulate using the standard terminology of cluster algebras (see e.g. [20],[19]). Theorem 8. Let Qbe an acyclic quiver which is obtained by mutations from a Dynkin quiver”
A Quantum Cluster Algebra of Kronecker Type and the Dual Canonical Basis (2010, unknown): “We refer to [9] for deﬁnitions and to [12] for good survey about cluster algebras.”
Dual Teichmuller and lamination spaces (2005, methodology): “The closely related objects, cluster algebras, were introduced by Sergey Fomin and Andrei Zelevinsky [10], and studied by them and Arkady Berenstein, Mikhail Gekhtman, Mikhail Shapiro, Alek Vainshtein ([ 11 ] and references therein).”
CLUSTER MUTATION VIA QUIVER REPRESENTATIONS (2004, background): “See also [ FZ2 ] for an overview of the theory of cluster algebras.”
On the properties of the exchange graph of a cluster algebra (2007, background): “In the present paper we prove some conjectures on the general structure of cluster algebras formulated by Fomin and Zelevinsky in [ FZ3 ].”
Cluster-tilted algebras are Gorenstein and stably Calabi–Yau (2005, background): “We refer to [23] for more background on cluster algebras and to [1,2,10,11,14–18,28,33,34,46,47] for some recent developments in the study of their links with representation theory.”
Kac–Moody groups and cluster algebras (2010, unknown): “We refer to [FZ4] for an excellent survey on cluster algebras.”
R-Twisting and 4d/2d Correspondences (2010, methodology): “[45, 48, 49] for reviews), which we will momentarily define.”
A cluster algebra approach to $q$-characters of Kirillov–Reshetikhin modules (2013, unknown): “2 Cluster algebras We refer the reader to [FZ2] and [GSV] for an introduction to cluster algebras, and for any undefined terminology.”