The octagon equation of the non-compact quantum dilogarithm

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The octagon equation of the non-compact quantum dilogarithm

   Let $P$, $Q$, and $V$ be as in Lemma cited in the source. The octagon equation.

   $$\Phi^{3\hbar}(3P)\Phi^\hbar(Q)=\Phi^\hbar(Q)\Phi^{3\hbar}(3P+3Q)\Phi^\hbar(3P+2Q)\Phi^{3\hbar}(6P+3Q)\Phi^\hbar(3P+Q)\Phi^{3\hbar}(3P).$$

   This identity is the operator relation corresponding to Fock–Goncharov's $G_2$-type identity; the supplied text presents it as a suggested relation in a heuristic discussion, without establishing its resolution here.

   source: Hyun Kyu Kim, “Phase constants in the Fock-Goncharov quantization of cluster varieties: long version”, arXiv:1602.00361 (2016).