On a Certain Subalgebra of $U_q(\widehat{\mathfrak{sl}}_2)$ Related to the Degenerate $q$-Onsager Algebra

TL;DR

This paper classifies finite-dimensional irreducible representations and intertwiners of a specific subalgebra of the quantum affine algebra $U_q(\\widehat{\rak{sl}}_2)$, which is related to the degenerate $q$-Onsager algebra, expanding understanding of its structure.

Contribution

It explicitly determines the finite-dimensional irreducible representations and intertwiners of the subalgebra $U'_q(\widehat{\frak{sl}}_2)$, a new insight into its representation theory.

Findings

Finite-dimensional irreducible representations are classified.

Intertwiners for the subalgebra are explicitly constructed.

The structure of the subalgebra related to the degenerate $q$-Onsager algebra is clarified.

Abstract

In [Kyushu J. Math. 64 (2010), 81-144, arXiv:0904.2889], it is discussed that a certain subalgebra of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_2)$ controls the second kind TD-algebra of type I (the degenerate $q$-Onsager algebra). The subalgebra, which we denote by $U'_q(\widehat{\mathfrak{sl}}_2)$, is generated by $e_0^+$, $e_1^\pm$, $k_i^{\pm1}$ $(i=0,1)$ with $e^-_0$ missing from the Chevalley generators $e_i^\pm$, $k_i^{\pm1}$ $(i=0,1)$ of $U_q(\widehat{\mathfrak{sl}}_2)$. In this paper, we determine the finite-dimensional irreducible representations of $U'_q(\widehat{\mathfrak{sl}}_2)$. Intertwiners are also determined.