Representations of $$U_{q,p}({ \mathfrak {sl}}_2)$$

Abstract

In this chapter we introduce a notion of dynamical representation as representation of the dynamical quantum group \(U_{q,p}(\widehat {\mathfrak {sl}}_2)\). We then discuss both finite and infinite dimensional representations of \(U_{q,p}(\widehat {\mathfrak {sl}}_2)\). As examples, we present the evaluation representation associated with the vector representation (Sect. 4.2) and the level-1 highest weight representations (Sect. 4.3). Most of them can be extended to \(U_{q,p}(\mathfrak {g})\) for arbitrary untwisted affine Lie algebra \({\mathfrak {g}}\) (Jimbo et al., Comm. Math. Phys. 199, 605–647 (1999); Kojima and Konno, Comm. Math. Phys. 239, 405–447 (2003); Konno, SIGMA, 2, Paper 091, 25 pages (2006); Farghly et al., Algebr. Represent. Theory 18, 103–135 (2014)).