Can the famous Abel's five terms relation satisfied by the dilogarithm be derived from (a particular case of) the theory of Yang-Baxter equations? If yes, how?
Thanks for any help.
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Quite a well-known paper Quantum Dilogarithm L. D. Faddeev, R. M. Kashaev discovers that in an appriate limit the q-exponential function degenerates to dilogarithm. (It might be suprising dilogorithm and q-exp were known for more that 100 years, but the relation between them have been uncovered in 1993).
Moreover the authors show that five-term dilogarithm identity is a limit of the identity for the q-exponential functions. The last identity in contrast to the classical one, includes non-commuting variables. It is related in certain sense to Yang-Baxter (star-triangle) equation, however it seems not quite direct relation, although it is clearly from the realm of Yang-Baxter quantum group theory.
I also heard that similar identities for q-exponential function appear in
the proof that certain construction of universal R-matrix indeed satisfies the Yang-Baxter equation.
Wikipedia article might be quite useful. The so-called "quantum dilogarithm" now appears in many papers on quantum Teichmuller theory (see papers by Fock,Rosly, Chekhov, et. al. on arXiv e.g. The quantum dilogarithm and representations quantum cluster varieties V. V. Fock, A. B. Goncharov)
answered Jul 5, 2014 at 15:32