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Pentagon relation for the quantum dilogarithm and quantized \(\mathcal M^{\text{cyc}}_{0,5}\). (English) Zbl 1139.81055

Kapranov, Mikhail (ed.) et al., Geometry and dynamics of groups and spaces. In memory of Alexander Reznikov. Partly based on the international conference on geometry and dynamics of groups and spaces in memory of Alexander Reznikov, Bonn, Germany, September 22–29, 2006. Basel: Birkhäuser (ISBN 978-3-7643-8607-8/hbk). Progress in Mathematics 265, 415-428 (2008).
Summary: We give a proof of the pentagon relation for the quantum dilogarithm by using functional analysis methods. We introduce a related Schwartz space and prove that it is preserved by the intertwiner operator defined using the quantum dilogarithm. Using this we can define a representation of the quantized moduli space of configurations of 5 points on the projective line.
For the entire collection see [Zbl 1130.00014].

MSC:

33E20 Other functions defined by series and integrals
33B30 Higher logarithm functions
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
81T99 Quantum field theory; related classical field theories
82B23 Exactly solvable models; Bethe ansatz
14D20 Algebraic moduli problems, moduli of vector bundles
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