A Common Structure in PBW Bases of the Nilpotent Subalgebra of U q (g)

For a finite-dimensional simple Lie algebra g, let U + q (g) be the positive part of the quantized universal enveloping algebra, and A q (g) be the quantized algebra of functions. We show that the transition matrix of the PBW bases of U + q (g) coincides with the intertwiner between the irreducible A q (g)-modules labeled by two different reduced expressions of the longest element of the Weyl group of 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 (g) in a quotient ring of A q (g).