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The periodicity conjecture for pairs of Dynkin diagrams. (English) Zbl 1320.17007

This important paper proves a periodicity conjecture first formulated by Al. B. Zamolodchikov [“On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories.” Phys. Lett. B 253, No. 3–4, 391–394 (1991), doi:10.1016/0370-2693(91)91737-G]. Let \(\Delta\), \(\Delta'\) be Dynkin diagrams. Then associated to the pair \((\Delta,\Delta')\) is the \(Y\)-system of algebraic equations: a system of recurrence relations in variables indexed by triples \(i,i',t\), where \(i\) is a vertex of \(\Delta\), \(i'\) is a vertex of \(\Delta'\) and \(t\) is an integer. The periodicity conjecture was originally stated for the pair \((\Delta,A_1)\), where \(\Delta\) is simply-laced.
The proof uses the theory of cluster algebras [S. Fomin and A. Zelevinsky, J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017)], in particular the \(Y\)-seeds from S. Fomin and A. Zelevinsky [Compos. Math. 143, No. 1, 112–164 (2007; Zbl 1127.16023)]. The periodicity is rephrased in terms of an generalized cluster category, introduced in [C. Amiot, Ann. Inst. Fourier 59, No. 6, 2525–2590 (2009; Zbl 1239.16011)], generalizing the cluster categories introduced in [A. B. Buan et al., Adv. Math. 204, No. 2, 572–618 (2006; Zbl 1127.16011)] and in [P. Caldero et al., Trans. Am. Math. Soc. 358, No. 3, 1347–1364 (2006; Zbl 1137.16020)] (for type \(A\)). The cluster category gives a categorification of the cluster algebra structure and the periodicity of a certain autoequivalence yields the conjecture.

MSC:

17B20 Simple, semisimple, reductive (super)algebras
13F60 Cluster algebras
18E30 Derived categories, triangulated categories (MSC2010)
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References:

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