# zbMATH — the first resource for mathematics

Quantum Teichmüller space from the quantum plane. (English) Zbl 1271.30020
Summary: We derive the quantum Teichmüller space, previously constructed by R. M. Kashaev [Lett. Math. Phys. 43, No. 2, 105–115 (1998; Zbl 0897.57014)] and by L. O. Chekhov and V. V. Fock [Theor. Math. Phys. 120, No. 3, 1245–1259 (1999); translation from Teor. Mat. Fiz. 120, No. 3, 511–528 (1999; Zbl 0986.32007)], from tensor products of a single canonical representation of the modular double of the quantum plane. We show that the quantum dilogarithm function appears naturally in the decomposition of the tensor square, the quantum mutation operator arises from the tensor cube, the pentagon identity from the tensor fourth power of the canonical representation, and an operator of order three from isomorphisms between canonical representation and its left and right duals. We also show that the quantum universal Teichmüller space is realized in the infinite tensor power of the canonical representation naturally indexed by rational numbers including infinity. This suggests a relation to the same index set in the classification of projective modules over the quantum torus, the unitary counterpart of the quantum plane, and points to a new quantization of the universal Teichmüller space.

##### MSC:
 30F60 Teichmüller theory for Riemann surfaces 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 16T05 Hopf algebras and their applications 17B37 Quantum groups (quantized enveloping algebras) and related deformations 20G42 Quantum groups (quantized function algebras) and their representations
##### Citations:
Zbl 0897.57014; Zbl 0986.32007
Full Text:
##### References:
 [1] L. V. Ahlfors, Some remarks on Teichmüller’s space of Riemann surfaces , Ann. of Math. (2) 74 (1961), 171-191. · Zbl 0146.30602 [2] H. Bai, Quantum Teichmüller spaces and Kashaev’s 6 j-symbols , Algebr. Geom. Topol. 7 (2007), 1541-1560. · Zbl 1131.17005 [3] E. W. Barnes, Theory of the double gamma function , Phil. Trans. Roy. Soc. A 196 (1901), 265-388. · JFM 32.0442.02 [4] A. G. Bytsko and J. Teschner, R-operator, co-product and Haar-measure for the modular double of $$U_{q}(\mathfrak{sl}(2,\mathbb{R}))$$ , Comm. Math. Phys. 240 (2003), 171-196. · Zbl 1078.17007 [5] L. Chekhov and V. V. Fock, A quantum Teichmüller space , Theor. Math. Phys. 120 (1999), 1245-1259. · Zbl 0986.32007 [6] A. Connes, C*-algèbres et géométrie différentielle , C. R. Acad. Sci. Paris 290 (1980), 599-604. · Zbl 0433.46057 [7] L. D. Faddeev, Discrete Heisenberg-Weyl group and modular group , Lett. Math. Phys. 34 (1995), 249-254. · Zbl 0836.47012 [8] L. D. Faddeev, “Modular double of a quantum group” in Conférence Moshé Flato 1999, Vol. I (Dijon, France) , Math. Phys. Stud. 21 , Kluwer, Dordrecht, 2000, 149-156. · Zbl 1071.81533 [9] L. D. Faddeev and R. M. Kashaev, Quantum dilogarithm , Modern Phys. Lett. A 9 (1994), 427-434. · Zbl 0866.17010 [10] L. D. Faddeev, R. M. Kashaev, and A. Y. Volkov, Strongly coupled quantum discrete Liouville theory, I: Algebraic approach and duality , Comm. Math. Phys. 219 (2001), 199-219. · Zbl 0981.81052 [11] V. V. Fock, Dual Teichmüller spaces , preprint, [12] V. V. Fock and A. B. Goncharov, The quantum dilogarithm and representations of the quantum cluster varieties , Invent. Math. 175 (2009), 223-286. · Zbl 1183.14037 [13] A. B. Goncharov, “Pentagon relation for the quantum dilogarithm and quantized $$\mathcal{M}_{0,5}^{\mathrm{cyc}}$$” in Geometry and Dynamics of Groups and Spaces , Progr. Math. 265 , Birkhäuser, Basel, 2008, 415-428. · Zbl 1139.81055 [14] R. Guo and X. Liu, Quantum Teichmüller space and Kashaev algebra , Algebr. Geom. Topol. 9 (2009), 1791-1824. · Zbl 1181.57034 [15] I. Ip, The classical limit of representation theory of the quantum plane , preprint, · Zbl 1323.81045 [16] R. M. Kashaev, Quantum dilogarithm as a 6 j-symbol , Modern Phys. Lett. A 9 (1994), 3757-3768. · Zbl 1015.17500 [17] R. M. Kashaev, Quantization of Teichmüller spaces and the quantum dilogarithm , Lett. Math. Phys. 43 (1998), 105-115. · Zbl 0897.57014 [18] R. M. Kashaev, The Liouville central charge in quantum Teichmüller theory , Proc. Steklov Inst. Math. 226 (1999), 63-71. · Zbl 0982.81047 [19] R. M. Kashaev, “On the spectrum of Dehn twists in quantum Teichmüller theory” in Physics and Combinatorics (Nagoya, 2000), World Sci., River Edge, N. J., 2001, 63-81. · Zbl 0996.53056 [20] A. A. Kirillov and D. V. Yuriev, Representations of the Virasoro algebra by the orbit method , J. Geom. Phys. 5 (1988), 351-363. · Zbl 0698.17015 [21] H.-H. Kuo, Gaussian measures in Banach spaces , Lecture Notes in Math. 463 , Springer, Berlin, 1975. · Zbl 0306.28010 [22] S. Nag and D. Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the H q 1/2 space on the circle , Osaka J. Math. 32 (1995), 1-34. · Zbl 0820.30027 [23] R. C. Penner, The decorated Teichmüller space of punctured surfaces , Comm. Math. Phys. 113 (1987), 299-339. · Zbl 0642.32012 [24] R. C. Penner, Universal constructions in Teichmüller theory , Adv. Math. 98 (1993), 143-215. · Zbl 0772.30040 [25] R. C. Penner, On Hilbert, Fourier and wavelet transforms , Comm. Pure Appl. Math. 55 (2002), 772-814. · Zbl 1020.42016 [26] R. C. Penner, Decorated Teichmüller space of bordered surfaces , Comm. Anal. Geom. 12 (2004), 793-820. · Zbl 1072.32008 [27] B. Ponsot and J. Teschner, Clebsh-Gordan and Racah-Wigner coefficients for a continuous series of representations of $$\mathcal{U}_{q}(\mathfrak{sl}(2,\mathbb{R}))$$ , Comm. Math. Phys. 224 (2001), 613-655. · Zbl 1010.33013 [28] B. Ponsot and J. Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group , preprint, · Zbl 0988.81068 [29] M. A. Rieffel, C*-algebras associated with irrational rotations , Pacific J. Math. 93 (1981), 415-429. · Zbl 0499.46039 [30] S. N. M. Ruijsenaars, First order analytic difference equations and integrable quantum systems , J. Math. Phys. 38 (1997), 1069-1146. · Zbl 0877.39002 [31] S. N. M. Ruijsenaars, A unitary joint eigenfunction transform for the A\Delta Os exp( ia \pm 1 d / dz ) + exp(2 \pi z / a \mp ), J. Nonlinear Math. Phys. 12 (2005), suppl. 2 253-294. · Zbl 1082.81082 [32] T. Shintani, On a Kronecker limit formula for real quadratic fields , J. Fac. Sci. Univ. Tokyo Sect. 1A 24 (1977), 167-199. · Zbl 0364.12012 [33] K. Schmüdgen, Operator Representations of \Bbb R 2 , Publ. Res. Inst. Math. Sci. 28 (1992), 1029-1061. [34] W. P. Thurston, The geometry and topology of 3- manifolds , lecture notes, Princeton University, 1980, available at [35] A. M. Vershik, Strange factor representations of type II 1 , and pairs of dual dynamical systems , Moscow Math. J. 3 (2003), 1441-1457. · Zbl 1059.46041 [36] A. Y. Volkov, Noncommutative hypergeometry , Comm. Math. Phys. 258 (2005), 257-273. · Zbl 1097.33012 [37] S. Wolpert, On the symplectic geometry of deformations of a hyperbolic surface , Ann. of Math. (2) 117 (1983), 207-234. · Zbl 0518.30040 [38] S. L. Woronowicz, Quantum exponential function , Rev. Math. Phys. 12 (2000), 873-920. · Zbl 0961.47013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.