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

SIGMA 11 (2015), 007, 13 pages      arXiv:1205.2946
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

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.

pdf (413 kb)   tex (17 kb)


  1. Benkart G., Terwilliger P., Irreducible modules for the quantum affine algebra $U_q(\widehat{sl}_2)$ and its Borel subalgebra, J. Algebra 282 (2004), 172-194, math.QA/0311152.
  2. Chari V., Pressley A., Quantum affine algebras, Comm. Math. Phys. 142 (1991), 261-283.
  3. Ito T., Tanabe K., Terwilliger P., Some algebra related to $P$- and $Q$-polynomial association schemes, in Codes and Association Schemes (Piscataway, NJ, 1999), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., Vol. 56, Amer. Math. Soc., Providence, RI, 2001, 167-192, math.CO/0406556.
  4. Ito T., Terwilliger P., The augmented tridiagonal algebra, Kyushu J. Math. 64 (2010), 81-144, arXiv:0904.2889.
  5. Tolstoy V.N., Khoroshkin S.M., Universal $R$-matrix for quantized nontwisted affine Lie algebras, Funct. Anal. Appl. 26 (1992), 69-71.

Previous article  Next article   Contents of Volume 11 (2015)