PBW and duality theorems for quantum groups and quantum current algebras

TL;DR

This paper proves PBW and duality theorems for quantum Kac-Moody and current algebras using Lie bialgebra duality, and explores their classical limits related to toroidal algebras.

Contribution

It provides new proofs of fundamental theorems for quantum algebras and clarifies their classical limits, especially for affine Cartan matrices.

Findings

PBW and duality theorems established for quantum Kac-Moody algebras

Classical limits relate quantum current algebras to toroidal algebra quotients

Trivial quotients except in the $A_1^{(1)}$ case

Abstract

We give proofs of the PBW and duality theorems for the quantum Kac-Moody algebras and quantum current algebras, relying on Lie bialgebra duality. We also show that the classical limit of the quantum current algebras associated with an untwisted affine Cartan matrix is the enveloping algebra of a quotient of the corresponding toroidal algebra; this quotient is trivial in all cases except the $A_1^{(1)}$ case.