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Quantum generalized cluster algebras and quantum dilogarithms of higher degrees. (English. Russian original) Zbl 1338.81241
Theor. Math. Phys. 185, No. 3, 1759-1768 (2015); translation from Teor. Mat. Fiz. 185, No. 3, 460-470 (2015).
Summary: We extend the notion of quantizing the coefficients of ordinary cluster algebras to the generalized cluster algebras of L. Chekhov and M. Shapiro [Int. Math. Res. Not. 2014, No. 10, 2746–2772 (2014; Zbl 1301.30042)]. In parallel to the ordinary case, it is tightly integrated with certain generalizations of the ordinary quantum dilogarithm, which we call the quantum dilogarithms of higher degrees. As an application, we derive the identities of these generalized quantum dilogarithms associated with any period of quantum \(Y\)-seeds.

MSC:
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
11G55 Polylogarithms and relations with \(K\)-theory
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[1] Chekhov, L.; Shapiro, M., No article title, Int. Math. Res. Notices, 2014, 2746-2772, (2014) · Zbl 1301.30042
[2] Fomin, S.; Zelevinsky, A., No article title, Invent. Math., 154, 63-121, (2003) · Zbl 1054.17024
[3] Gekhtman, M.; Shapiro, M.; Vainshtein, A., No article title, Moscow Math. J., 3, 899-934, (2003) · Zbl 1057.53064
[4] A. Gleitz, “Quantum affine algebras at roots of unity and generalized cluster algebras,” arXiv:1410.2446v1 [math.RT] (2014).
[5] K. Iwaki and T. Nakanishi, “Exact WKB analysis and cluster algebras II: Simple poles, orbifold points, and generalized cluster algebras,” arXiv:1409.4641v2 [math.CA] (2014). · Zbl 1311.81113
[6] Nakanishi, T., No article title, Pacific J. Math., 277, 201-218, (2015) · Zbl 1366.13017
[7] Berenstein, A.; Zelevinsky, A., No article title, Adv. Math., 195, 405-455, (2005) · Zbl 1124.20028
[8] Fock, V. V.; Goncharov, A. B., No article title, Ann. Sci. Éc. Norm. Supér., 42, 865-930, (2009) · Zbl 1180.53081
[9] Fock, V. V.; Goncharov, A. B., No article title, Invent. Math., 175, 223-286, (2009) · Zbl 1183.14037
[10] Faddeev, L. D.; Volkov, A. Yu., No article title, Phys. Lett., 315, 311-318, (1993) · Zbl 0864.17042
[11] Faddeev, L. D.; Kashaev, R. M., No article title, Modern Phys. Lett. A, 9, 427-434, (1994) · Zbl 0866.17010
[12] L. Lewin, Polylogarithms and Associated Functions, North-Holland, Amsterdam (1981). · Zbl 0465.33001
[13] Keller, B.; Skowroński, A. (ed.); Yamagata, K. (ed.), On cluster theory and quantum dilogarithm identities, 85-116, (2011) · Zbl 1307.13028
[14] Kashaev, R. M.; Nakanishi, T., No article title, SIGMA, 7, 102, (2011)
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