
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, 11, Nonoe, Suzu, Ishikawa 9271213, 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), 81144], it is discussed that a certain subalgebra of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_2)$ controls the second kind TDalgebra 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 finitedimensional irreducible representations of $U'_q(\widehat{\mathfrak{sl}}_2)$. Intertwiners are also determined.
Key words:
degenerate $q$Onsager algebra; quantum affine algebra; TDalgebra; augmented TDalgebra; TDpair.
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