The periodicity conjecture for pairs of Dynkin diagrams

By B. Keller. 2010

138 citations recorded by the source corpus.

We prove the periodicity conjecture for pairs of Dynkin diagrams using Fomin-Zelevinsky's cluster algebras and their (additive) categorification via triangulated categories.

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

View Paper: https://www.semanticscholar.org/paper/c9efe949e53256d3f21b052626cb34f8ede8a290

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Representations of Algebras and Related Topics (2011, background): “commutative: Db(Λ) − L ⊗ΛM // π Db(E) C2(Λ) F //Db(E)/per(E) ≃E. Remark 3.26. A very powerful application of cluster categories associated with algebras of global dimension at most 2 is made in [69], and allows to prove the periodicity conjecture for pairs of Dynkin diagrams (see also last section”
Classification of Complete N = 2 Supersymmetric Theories in 4 Dimensions (2011, methodology): “arity associated to the given Dynkin diagram G. The quiver of the (G,G′) model is given by the square tensor product of the Dynkin graphs of G and G′, G G′ (for the product orientation rule see refs. [2,55]). The quiver Am An is represented in ﬁgure 11. Up to mutation–equivalence one has the followi”
R-Twisting and 4d/2d Correspondences (2010, methodology): “The resulting quiver is called the square tensor product of the original quivers G and G [10], written G G.”
Coxeter's frieze patterns at the crossroads of algebra, geometry and combinatorics (2015, background): “It was proved in the Dynkin case in [38] and in a more general case in [53]; see also [52].”
Selfinjective quivers with potential and 2-representation-finite algebras (2010, background): “ientation (i.e., each vertex is either a sink or a source). Write Qi 1 = Xi ` Yi where Xi consists of sources and Yi of sinks, i.e, for each arrow x →a y ∈Qi 1 holds that x ∈X iand y ∈Y . Recall from [K3], that the square product Q1 Q2 is the quiver obtained from Q1 ⊗Q2 by replacing each arrow a sat”
Framed BPS states (2010, unknown): “For some references and background see [77] and the talk [78].”
Cluster algebras and derived categories (2012, unknown): “…in many other subjects, for example in • Poisson geometry [60] [61] [62] [63] [10] . . . ; • discrete dynamical systems [27] [47] [74] [79] [84] [81] [91] . . . ; • higher Teichmüller spaces [35] [36] [37] [38] [39] . . . ; • combinatorics and in particular the study of polyhedra like the…”
2-frieze patterns and the cluster structure of the space of polygons (2010, unknown): “Taking into account the symmetry of part (iii), the 2-frieze is determined by a fragment of size ( n − 2) × n . is closed of width 2.”