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

SIGMA 11 (2015), 007, 13 pages      arXiv:1205.2946      https://doi.org/10.3842/SIGMA.2015.007
Contribution to the Special Issue on New Directions in Lie Theory

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

Tomoya Hattai a and Tatsuro Ito b
a) Iida Highschool, 1-1, Nonoe, Suzu, Ishikawa 927-1213, Japan
b) School of Mathematical Sciences, Anhui University, 111 Jiulong Road, Hefei 230601, China

Received September 30, 2014, in final form January 15, 2015; Published online January 19, 2015

Abstract
In [Kyushu J. Math. 64 (2010), 81-144], 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.

Key words: degenerate $q$-Onsager algebra; quantum affine algebra; TD-algebra; augmented TD-algebra; TD-pair.

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References

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