For a finite-dimensional simple Lie algebra $\mathfrak{ g}$, let $U^+_q(\mathfrak{ g})$ be the positive part of the quantized universal enveloping algebra, and $A_q(\mathfrak{ g})$ be the quantized algebra of functions. We show that the transition matrix of the PBW bases of $U^+_q(\mathfrak{ g})$ coincides with the intertwiner between the irreducible $A_q(\mathfrak{ g})$-modules labeled by two different reduced expressions of the longest element of the Weyl group of $\mathfrak{ g}$. This generalizes the earlier result by Sergeev on $A_2$ related to the tetrahedron equation and endows a new representation theoretical interpretation with the recent solution to the 3D reflection equation for $C_2$. Our proof is based on a realization of $U^+_q(\mathfrak{ g})$ in a quotient ring of $A_q(\mathfrak{ g})$.
Categories: math.QA math-ph math.MP nlin.SI