Center of the Yangian double in type A

TL;DR

This paper establishes the isomorphism between R-matrix and Drinfeld presentations of the Yangian double in type A, constructs its central elements, and relates their images to eigenvalues in Wakimoto modules.

Contribution

It proves the isomorphism of two key presentations of the Yangian double in type A and computes the images of central elements via a Harish-Chandra homomorphism.

Findings

R-matrix and Drinfeld presentations are isomorphic

Constructed central elements at the critical level

Calculated images of central elements matching Wakimoto module eigenvalues

Abstract

We prove the R-matrix and Drinfeld presentations of the Yangian double in type A are isomorphic. The central elements of the completed Yangian double in type A at the critical level are constructed. The images of these elements under a Harish-Chandra-type homomorphism are calculated by applying a version of the Poincar\'e-Birkhoff-Witt theorem for the R-matrix presentation. These images coincide with the eigenvalues of the central elements in the Wakimoto modules.