Bosonic representations of Yangian Double $\cD Y_{\hbar}(\gtg)$ with $\gtg=\gtgl_N,\gtsl_N$

TL;DR

This paper constructs bosonic representations of the Yangian double for general linear and special linear Lie algebras using the RTT formalism, providing explicit examples of finite and infinite-dimensional representations as deformations of affine cases.

Contribution

It introduces the quantum double of the Yangian for  and  with a central extension and constructs bosonic representations via Gauss decomposition.

Findings

Explicit bosonic representations of the Yangian double are constructed.

Finite and infinite-dimensional examples are provided as deformations of affine algebras.

The formalism extends the understanding of Yangian doubles and their representations.

Abstract

On the basis of `$RTT=TTR$' formalism, we introduce the quantum double of the Yangian $Y_{\hbar}(\gtg)$ for $\gtg=\gtgl_N,\gtsl_N$ with a central extension. The Gauss decomposition of T-matrices gives us the so-called Drinfel'd generators. Using these generators, we present some examples of both finite and infinite dimensional representations that are quite natural deformations of the corresponding affine counterpart.