Pentagon relation in quantum cluster scattering diagrams
arXiv:2202.01588
Abstract
We formulate the pentagon relation for quantum dilogarithm elements in the structure group of a quantum cluster scattering diagram (QCSD). As an application, we show the nonpositivity of a certain class of nonskew-symmetric QCSDs. Also, we explicitly present various consistency relations for QCSDs of rank 2 completely or up to some degree, many of which are new in the literature.
v1: 30 pages; v2: normalization of generators in (2.5) and related formulas changed; v3: 32 pages, Appendix A is added, minor changes; v4: Lem. 4.2, Prop. 4.3, Prop. 5.3 modified, Ex. 5.4 added; v5: Ex. 5.4 modified; v6: Sec. 4.3, Sec. 5.1 revised; v7:typo in Lemma 4.2 corrected