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Remarks on the quantum dilogarithm. (English) Zbl 0876.17035

Summary: A quantum analogue of the dilogarithm function has been introduced recently by L. D. Faddeev and R. M. Kashaev [Mod. Phys. Lett. A 9, 427-434 (1994; Zbl 0866.17010)] in such a way that a certain identity in the Weyl algebra \(\overline W_q\) plays the role of the five-term dilogarithm identity. We study this identity in the limit when \(q\) approaches a root of unity and show that it then reduces to the ‘restricted star-triangle relation’ which has been used previously by V. V. Bazhanov and R. J. Baxter [J. Stat. Phys. 71, 839-864 (1993; Zbl 0876.17031)] as a local integrability condition of a class of three-dimensional solvable lattice models.

MSC:

17B81 Applications of Lie (super)algebras to physics, etc.
33B99 Elementary classical functions
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
82B23 Exactly solvable models; Bethe ansatz
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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