From the Drinfeld realization to the Drinfeld-Jimbo presentation of affine quantum algebras: the injectivity

TL;DR

This paper proves the injectivity of the homomorphism from the Drinfeld realization to the Drinfeld-Jimbo presentation of affine quantum algebras, establishing a key structural property and providing a new presentation for affine Kac-Moody algebras.

Contribution

It demonstrates the injectivity of the homomorphism and extends the triangular decomposition to twisted affine quantum algebras, offering a new Drinfeld generator presentation.

Findings

Injectivity of the homomorphism established

Triangular decomposition extended to twisted cases

New presentation of affine Kac-Moody algebras provided

Abstract

In this paper the surjective homomorphism from the Drinfeld realization to the Drinfeld and Jimbo presentation of affine quantum algebras is proved to be injective. A consequence of the arguments used in the paper is the triangular decomposition of the Drinfeld realization of affine quantum algebras also in the twisted case. A presentation of the affine Kac-Moody algebras in terms of the "Drinfeld generators" is also provided.