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A quantum generalization of Rogers’ five-term, or “pentagon” dilogarithm identity is suggested. It is shown that the classical limit gives the usual Rogers’ identity. The case where the quantum identity is realized in finite-dimensional space is also considered and the quantum dilogarithm is constructed as a function on Fermat curve, while the identity itself is equivalent to the restricted star-triangle relation introduced by Bazhanov and Baxter.
Faddeev et al. (Sun,) studied this question.