The octagon equation of the non-compact quantum dilogarithm

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Let $P$ , $Q$ , and $V$ be as in Lem.. The octagon equation of the non-compact quantum dilogarithm.

\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 presented as the equation corresponding to Fock–Goncharov's $G_2$ -type operator identity and is introduced after a heuristic discussion of related quantum dilogarithm identities. The supplied text does not establish whether the equation has been proved in full generality.

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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).

References

Primary source

Hyun Kyu Kim, “Phase constants in the Fock-Goncharov quantum cluster varieties”, arXiv:1602.00797 (2019).

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