Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 145, 25 pages      arXiv:2008.07847      https://doi.org/10.3842/SIGMA.2020.145
Contribution to the Special Issue on Representation Theory and Integrable Systems in honor of Vitaly Tarasov on the 60th birthday and Alexander Varchenko on the 70th birthday

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

Naihuan Jing a, Ming Liu b and Alexander Molev c
a) Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
b) School of Mathematics, South China University of Technology, Guangzhou, 510640, China
c) School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

Received August 19, 2020, in final form December 25, 2020; Published online December 28, 2020

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.

Key words: $R$-matrix presentation; Drinfeld polynomials; highest weight representation; Gauss decomposition.

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