  In arXiv:0912.1346, four quantum dilogarithm identities containing infinitely
many factors are proposed as wall-crossing formula for refined BPS invariant.
We give algebraic proof of these identities using the formula for universal
R-matrix of quantum affine algebra developed by K. Ito, which yields various
product presentation of universal R-matrix by choosing various convex orders on
affine root system. By the uniqueness of universal R-matrix and appropriate
degeneration, we can construct various quantum dilogarithm identities including
the ones proposed in arXiv:0912.1346, which turn out to correspond to convex
orders of multiple row type.

Quantum Algebra