Duality of Lusztig and RTT integral forms of $U_v(L\mathfrak{sl}_n)$

TL;DR

This paper establishes a duality between Lusztig and RTT integral forms of the quantum loop algebra of type A using shuffle algebra and PBWD bases, revealing a new pairing relationship.

Contribution

It demonstrates the duality of Lusztig and RTT integral forms via a new Drinfeld pairing, connecting shuffle algebra and PBWD bases in type A quantum loop algebras.

Findings

Lusztig integral form is dual to RTT integral form under the new pairing

Utilizes shuffle algebra realization for Lusztig form

Employs PBWD bases for RTT form

Abstract

We show that the Lusztig integral form is dual to the RTT integral form of the type A quantum loop algebra with respect to the new Drinfeld pairing, by utilizing the shuffle algebra realization of the former and the PBWD bases of the latter obtained in arXiv:1808.09536.