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

TL;DR

This paper reveals a universal structure in PBW bases of the positive part of quantized universal enveloping algebras and their relation to the quantized algebra of functions, connecting representation theory and solutions to the 3D reflection equation.

Contribution

It establishes that the transition matrix between PBW bases coincides with intertwiners of modules over the quantized algebra of functions, generalizing previous results and providing new insights into their structure.

Findings

Transition matrix matches intertwiners between modules.

Generalizes Sergeev's result from $A_2$ to general Lie algebras.

Provides a new interpretation of the 3D reflection equation solution.

Abstract

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 $A_q(\mathfrak{g})$.