Geodesic
Quantum generalized cluster algebras and quantum dilogarithms of higher degrees
Teoretičeskaâ i matematičeskaâ fizika, Tome 185 (2015) no. 3, pp. 460-470 Cet article a éte moissonné depuis la source Math-Net.Ru

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We extend the notion of quantizing the coefficients of ordinary cluster algebras to the generalized cluster algebras of Chekhov and Shapiro. 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.
Keywords: cluster algebra, quantum dilogarithm. 
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T. Nakanishi. Quantum generalized cluster algebras and quantum dilogarithms of higher degrees. Teoretičeskaâ i matematičeskaâ fizika, Tome 185 (2015) no. 3, pp. 460-470. http://geodesic.mathdoc.fr/item/TMF_2015_185_3_a4/

[1] L. Chekhov, M. Shapiro, Int. Math. Res. Notices, 2014:10 (2014), 2746–2772, arXiv: 1111.3963 | DOI | MR | Zbl

[2] S. Fomin, A. Zelevinsky, Invent. Math., 154:1 (2003), 63–121, arXiv: math/0208229 | DOI | MR | Zbl

[3] M. Gekhtman, M. Shapiro, A. Vainshtein, Moscow Math. J., 3:3 (2003), 899–934, arXiv: math/0208033 | MR | Zbl

[4] A. Gleitz, Quantum affine algebras at roots of unity and generalized cluster algebras, arXiv: 1410.2446

[5] K. Iwaki, T. Nakanishi, Exact WKB analysis and cluster algebras II: simple poles, orbifold points, and generalized cluster algebras, arXiv: 1409.4641 | MR

[6] T. Nakanishi, Pacific J. Math., 277:1 (2015), 201–218, arXiv: 1409.5967 | DOI | MR | Zbl

[7] A. Berenstein, A. Zelevinsky, Adv. Math., 195:2 (2005), 405–455, arXiv: math/0404446 | DOI | MR | Zbl

[8] V. V. Fock, A. B. Goncharov, Ann. Sci. Éc. Norm. Supér., 42:6 (2009), 865–930, arXiv: math/0311245 | DOI | MR | Zbl

[9] V. V. Fock, A. B. Goncharov, Invent. Math., 175:2 (2009), 223–286, arXiv: math/0702397 | DOI | MR | Zbl

[10] L. D. Faddeev, A. Yu. Volkov, Phys. Lett., 315:3–4 (1993), 311–318, arXiv: hep-th/9307048 | DOI | MR | Zbl

[11] L. D. Faddeev, R. M. Kashaev, Modern Phys. Lett. A, 9:5 (1994), 427–434, arXiv: hep-th/9310070 | DOI | MR | Zbl

[12] L. Lewin, Polylogarithms and Associated Functions, North-Holland, Amsterdam, 1981 | MR | Zbl

[13] B. Keller, “On cluster theory and quantum dilogarithm identities”, Representations of Algebras and Related Topics (Tokyo, Japan, August 6–15, 2010), EMS Series of Congress Reports, European Mathematical Society, eds. A. Skowroński, K. Yamagata, Eur. Math. Soc., Zürich, 2011, 85–116, arXiv: 1102.4148 | MR | Zbl

[14] R. M. Kashaev, T. Nakanishi, SIGMA, 7 (2011), 102, 29 pp., arXiv: 1104.4630 | DOI | MR | Zbl