Algebraic & Geometric Topology

Quantum Teichmüller space and Kashaev algebra

Ren Guo and Xiaobo Liu

Full-text: Open access

Abstract

Kashaev algebra associated to a surface is a noncommutative deformation of the algebra of rational functions of Kashaev coordinates. For two arbitrary complex numbers, there is a generalized Kashaev algebra. The relationship between the shear coordinates and Kashaev coordinates induces a natural relationship between the quantum Teichmüller space and the generalized Kashaev algebra.

Article information

Source
Algebr. Geom. Topol., Volume 9, Number 3 (2009), 1791-1824.

Dates
Received: 6 May 2009
Accepted: 20 August 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513797044

Digital Object Identifier
doi:10.2140/agt.2009.9.1791

Mathematical Reviews number (MathSciNet)
MR2550095

Zentralblatt MATH identifier
1181.57034

Subjects
Primary: 57R56: Topological quantum field theories
Secondary: 57M50: Geometric structures on low-dimensional manifolds 20G42: Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50]

Keywords
Teichmüller space quantization Kashaev coordinates noncommutative algebra

Citation

Guo, Ren; Liu, Xiaobo. Quantum Teichmüller space and Kashaev algebra. Algebr. Geom. Topol. 9 (2009), no. 3, 1791--1824. doi:10.2140/agt.2009.9.1791. https://projecteuclid.org/euclid.agt/1513797044


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References

  • H Bai, A uniqueness property for the quantization of Teichmüller spaces, Geom. Dedicata 128 (2007) 1–16
  • H Bai, F Bonahon, X Liu, Local representations of the quantum Teichmüller space
  • F Bonahon, Shearing hyperbolic surfaces, bending pleated surfaces and Thurston's symplectic form, Ann. Fac. Sci. Toulouse Math. $(6)$ 5 (1996) 233–297
  • F Bonahon, Quantum Teichmüller theory and representations of the pure braid group, Commun. Contemp. Math. 10 (2008) 913–925
  • F Bonahon, X Liu, Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms, Geom. Topol. 11 (2007) 889–937
  • L O Chekhov, V V Fock, Observables in 3D gravity and geodesic algebras, Czechoslovak J. Phys. 50 (2000) 1201–1208 Quantum groups and integrable systems (Prague, 2000)
  • V V Fock, Dual Teichmüller spaces
  • V V Fok, L O Chekhov, Quantum Teichmüller spaces, Teoret. Mat. Fiz. 120 (1999) 511–528
  • R M Kashaev, Quantization of Teichmüller spaces and the quantum dilogarithm, Lett. Math. Phys. 43 (1998) 105–115
  • R M Kashaev, The Liouville central charge in quantum Teichmüller theory, Tr. Mat. Inst. Steklova 226 (1999) 72–81
  • R M Kashaev, On the spectrum of Dehn twists in quantum Teichmüller theory, from: “Physics and combinatorics, 2000 (Nagoya)”, World Sci. Publ., River Edge, NJ (2001) 63–81
  • R M Kashaev, On quantum moduli space of flat $\mathrm{PSL}_2(\mathbb{R})$–connections on a punctured surface, from: “Handbook of Teichmüller theory Vol I”, IRMA Lect. Math. Theor. Phys. 11, Eur. Math. Soc., Zürich (2007) 761–782
  • X Liu, Gromov–Witten invariants and moduli spaces of curves, from: “International Congress of Mathematicians Vol II”, Eur. Math. Soc., Zürich (2006) 791–812
  • X Liu, The quantum Teichmüller space as a non-commutative algebraic object, J. Knot Theory Ramifications 18 (2009) 705–726
  • R C Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987) 299–339
  • J Teschner, An analog of a modular functor from quantized Teichmüller theory, from: “Handbook of Teichmüller theory Vol I”, IRMA Lect. Math. Theor. Phys. 11, Eur. Math. Soc., Zürich (2007) 685–760
  • W P Thurston, The topology and geometry of 3–manifolds, lecture notes, Princeton University (1976–1979) Available at \setbox0\makeatletter\@url http://msri.org/communications/books/gt3m {\unhbox0