Representations of Quantum Affine Algebras in their $R$-Matrix Realization

TL;DR

This paper explores the structure of quantum affine algebras and Yangians in their $R$-matrix realizations, providing a detailed description of finite-dimensional irreducible representations through isomorphisms and Gauss decomposition.

Contribution

It establishes explicit isomorphisms between $R$-matrix and Drinfeld presentations for types $B$, $C$, and $D$, and applies Gauss decomposition to unify representation descriptions.

Findings

Describes finite-dimensional irreducible representations in $R$-matrix form.

Establishes isomorphisms between $R$-matrix and Drinfeld presentations.

Uses Gauss decomposition to relate Yangian representations.

Abstract

We use the isomorphisms between the $R$-matrix and Drinfeld presentations of the quantum affine algebras in types $B$, $C$ and $D$ produced in our previous work to describe finite-dimensional irreducible representations in the $R$-matrix realization. We also review the isomorphisms for the Yangians of these types and use Gauss decomposition to establish an equivalence of the descriptions of the representations in the $R$-matrix and Drinfeld presentations of the Yangians.