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Quantum affine algebras and their representations. (English) Zbl 0855.17009
Allison, Bruce N. (ed.) et al., Representations of groups. Canadian Mathematical Society annual seminar, June 15-24, 1994, Banff, Alberta, Canada. Providence, RI: American Mathematical Society. CMS Conf. Proc. 16, 59-78 (1995).
Let $${\mathfrak g}$$ denote a finite-dimensional complex simple Lie algebra, and let $$\widehat {\mathfrak g}$$ denote the corresponding untwisted affine Lie algebra. The finite-dimensional irreducible representations of the Hopf algebra $$U_q (\widehat {\mathfrak g})$$, the quantum deformation of the universal enveloping algebra of $$\widehat {\mathfrak g}$$, are considered with respect to the representations of $$U_q ({\mathfrak g})$$. Starting with an introduction that motivates the interest in finite-dimensional representations of quantum groups by several applications in physics, some facts of the structure theory and of representation theory of $$U_q ({\mathfrak g})$$ and $$U_q (\widehat {\mathfrak g})$$ are summarized. The essential part is theorem 3.3 stating a parametrization of the finite-dimensional irreducible representations of $$U_q (\widehat {\mathfrak g})$$ by a generalized “highest weight technology”. There is a separate direct proof of this main theorem in the special case of $$sl_2$$ and the authors claim, that the same method can be carried through in the general case. Nevertheless the proof of the general case given in point 5 is more sophisticated. In the last point the notion of affinization of a representation of $$U_q ({\mathfrak g})$$ is introduced and some results on minimal affinizations in the case $${\mathfrak g}= sl_{n+ 1} (\mathbb{C})$$ are stated (see also the following review).
For the entire collection see [Zbl 0829.00018].

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 16W30 Hopf algebras (associative rings and algebras) (MSC2000)