Cluster mutation-periodic quivers and associated Laurent sequences.(English)Zbl 1272.13020

Let $$Q$$ be a quiver. The mutation of $$Q$$ at node $$k$$ is a new quiver $$\mu_{k}Q$$ obtained from $$Q$$ via first reversing all arrows originating or terminating at $$k$$; then adding $$pq$$ arrows from node $$i$$ to node $$j$$ in addition to these arrows from node $$i$$ to node $$j$$ existing in $$Q$$, if there are $$p$$ arrows from node $$i$$ to node $$k$$ and $$q$$ arrows from node $$k$$ to node $$j$$ in the quiver $$Q$$; and finally removing any two-cycles created in the previous steps. Let us order the nodes of $$Q$$ as $$1, 2, \dots, N$$ and define a permutation of the nodes of $$Q$$ as $$\rho: (1, 2, \dots, N)\rightarrow (N, 1, \dots, N-1)$$. Let us denote by $$Q(1)$$ the quiver $$\mu_{1}Q$$ and by $$Q(k+1)$$ the quiver $$\mu_{k}Q(k)$$. A quiver $$Q$$ is said to have period $$m$$ if $$Q(m+1)=\rho^{m}Q(1)$$. A quiver $$Q$$ is said to be a period $$m$$ sink-type quiver if it is of period $$m$$ and for $$1\leq i\leq m$$, node $$i$$ of $$Q(i)$$ is a sink.
In this paper, the authors study some periodic quivers. First of all, the authors classify all periodic quivers of sink-type; and the results are stated in Sections $$3, 4$$ and $$5$$. Then, they completely classify all period $$1$$ quivers and construct some specific examples in Section $$6$$. The authors also classify all quivers of period $$2$$ with at most five nodes in Section $$7$$. In the rest of the paper, the authors study the recurrences which can be associated to periods $$1$$ and $$2$$ quivers. In particular, they prove that the recurrences associated to period $$1$$ primitive quivers can be linearized. They also extend the construction of mutation periodic quivers to quivers involving frozen cluster variables and thus introduce parameters into recurrences. Moreover, they study the connections with supersymmetric quiver gauge theories.

MSC:

 13F60 Cluster algebras 16G20 Representations of quivers and partially ordered sets 81T60 Supersymmetric field theories in quantum mechanics

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 [1] Assem, I., Simson, D., Skowroski, A.: Elements of the Representation Theory of Associative Algebras, vol. 1. Techniques of Representation Theory. London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge (2006) · Zbl 1092.16001 [2] Assem, I., Reutenauer, C., Smith, D.: Frises. Preprint (2009). arXiv:0906.2026v1 [math.RA] [3] Bellon, M.; Viallet, C.-M., Algebraic entropy, Commun. Math. Phys., 204, 425-437, (1999) · Zbl 0987.37007 [4] Derksen, H.; Weyman, J.; Zelevinsky, A., Quivers with potentials and their representations I: Mutations, Sel. Math., New Ser., 14, 59-119, (2008) · Zbl 1204.16008 [5] Feng, B.; Hanany, A.; He, Y.-H., D-brane gauge theories from toric singularities and toric duality, Nucl. Phys. B, 595, 165-200, (2001) · Zbl 0972.81138 [6] Feng, B.; Hanany, A.; He, Y.-H.; Uranga, A. M., Toric duality as Seiberg duality and brane diamonds, J. High Energy Phys., 12, (2001) [7] Fomin, S.; Zelevinsky, A., Cluster algebras I: foundations, J. Am. Math. Soc., 15, 497-529, (2002) · Zbl 1021.16017 [8] Fomin, S.; Zelevinsky, A., The Laurent phenomenon, Adv. Appl. Math., 28, 119-144, (2002) · Zbl 1012.05012 [9] Fordy, A.P.: Mutation-periodic quivers, integrable maps and associated Poisson algebras. Preprint (2010). arXiv:1003.3952v1 [nlin.SI] · Zbl 1219.17020 [10] Fordy, A.P., Hone, A.N.W.: Integrable maps and Poisson algebras derived from cluster algebras. In preparation · Zbl 1344.37076 [11] Franco, S.; Hanany, A.; Kennaway, K. D.; Vegh, D.; Wecht, B., Brane dimers and quiver gauge theories, J. High Energy Phys., 01, (2006) [12] Fu, C.; Keller, B., On cluster algebras with coefficients and 2-Calabi-Yau categories, Trans. Am. Math. Soc., 362, 859-895, (2010) · Zbl 1201.18007 [13] Gale, D., The strange and surprising saga of the Somos sequences, Math. Intell., 13, 40-42, (1991) [14] Gekhtman, M.; Shapiro, M.; Vainshtein, A., Cluster algebras and Poisson geometry, Mosc. Math. J., 3, 899-934, (2003) · Zbl 1057.53064 [15] Grammaticos, B.; Ramani, A.; Papageorgiou, V. G., Do integrable mappings have the Painlevé property?, Phys. Rev. Lett., 67, 1825-1828, (1991) · Zbl 0990.37518 [16] Halburd, R. G., Diophantine integrability, J. Phys. A, 38, l263-l269, (2005) · Zbl 1076.39013 [17] Heideman, P.; Hogan, E., A new family of Somos-like recurrences, Electron. J. Comb., 15, r54, (2008) · Zbl 1206.11016 [18] Hoggatt, V. E.; Bicknell-Johnson, M., A primer for the Fibonacci numbers XVII: generalized Fibonacci numbers satisfying $$u_{n+1}u_{n-1}-u_n^2=\pm1,$$ Fibonacci Q., 2, 130-137, (1978) · Zbl 0388.10008 [19] Hone, A. N.W., Laurent polynomials and superintegrable maps, SIGMA, 3, (2007) · Zbl 1165.37024 [20] Kashiwara, M., On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J., 63, 465-516, (1991) · Zbl 0739.17005 [21] Keller, B.: The periodicity conjecture for pairs of Dynkin diagrams. Preprint (2010). arXiv:1001.1531v3 [math.RT] · Zbl 1320.17007 [22] Keller, B., Scherotzke, S.: Linear recurrence relations for cluster variables of affine quivers. Preprint (2010). arXiv:1004.0613v2 [math.RT] · Zbl 1252.16012 [23] Lusztig, G., Canonical bases arising from quantized enveloping algebras, J. Am. Math. Soc., 3, 447-498, (1990) · Zbl 0703.17008 [24] Mukhopadhyay, S.; Ray, K., Seiberg duality as derived equivalence for some quiver gauge theories, J. High Energy Phys., 02, (2004) [25] Nakanishi, T.: Periodic cluster algebras and dilogarithm identities. Preprint (2010). arXiv:1006.0632v3 [math.QA] [26] Oota, T.; Yasui, Y., New example of infinite family of quiver gauge theories, Nucl. Phys. B, 762, 377-391, (2007) · Zbl 1116.81057 [27] Quispel, G. R.W.; Roberts, J. A.G.; Thompson, C. J., Integrable mappings and soliton equations, Phys. Lett. A, 126, 419-421, (1988) · Zbl 0679.58023 [28] Sloane, N.J.A.: The on-line encyclopedia of integer sequences. www.research.att.com/njas/sequences (2009) · Zbl 1044.11108 [29] Stienstra, J.; Yui, N. (ed.); Verrill, H. (ed.); Doran, C. F. (ed.), Hypergeometric systems in two variables, quivers, dimers and dessins d’enfants, No. 54, 125-161, (2008), Providence · Zbl 1166.33001 [30] Veselov, A. P., Integrable maps, Russ. Math. Surv., 46, 1-51, (1991) · Zbl 0785.58027 [31] Vitoria, J., Mutations vs. Seiberg duality, J. Algebra, 321, 816-828, (2009) · Zbl 1183.16015 [32] Zay, B., An application of the continued fractions for $$\sqrt{D}$$ in solving some types of Pell’s equations, Acta Acad. Paedagog. Agriensis Sect. Mat. (NS), 25, 3-14, (1998) · Zbl 0923.11019
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