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For a finite-dimensional simple Lie algebra $\mathfrak{g}$, let$U^+_q(\mathfrak{g})$ be the positive part of the quantized universalenveloping algebra, and $A_q(\mathfrak{g})$ be the quantized algebra offunctions. 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 thelongest element of the Weyl group of $\mathfrak{g}$. This generalizes theearlier result by Sergeev on $A_2$ related to the tetrahedron equation andendows a new representation theoretical interpretation with the recent solutionto the 3D reflection equation for $C_2$. Our proof is based on a realization of$U^+_q(\mathfrak{g})$ in a quotient ring of $A_q(\mathfrak{g})$.

A Common Structure in PBW Bases of the Nilpotent Subalgebra of $U_q(\mathfrak{g})$ and Quantized Algebra of Functions