Quantized coordinate rings, PBW-type bases and $q$-boson algebras

TL;DR

This paper provides a new proof connecting PBW-type bases transition matrices in quantized universal enveloping algebras with intertwiner matrix coefficients in quantized coordinate rings, using $q$-boson algebra representation theory.

Contribution

It introduces a novel proof of a known result by employing $q$-boson algebra representations and Drinfeld pairing techniques.

Findings

Established the equivalence between transition matrices and intertwiner matrix coefficients.

Utilized representation theory of $q$-boson algebra for proof.

Connected algebraic structures via Drinfeld pairing.

Abstract

Recently, Kuniba, Okado and Yamada proved that the transition matrix of PBW-type bases of the positive-half of a quantized universal enveloping algebra $U_q(\mathfrak{g})$ coincides with a matrix coefficients of the intertwiner between certain irreducible modules over the corresponding quantized coordinate ring $A_q(\mathfrak{g})$, introduced by Soibelman. In the present article, we give a new proof of their result, by using representation theory of the $q$-boson algebra, and the Drinfeld paring of $U_q(\mathfrak{g})$.