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Quantum groups. (English) Zbl 0795.17005
Zampieri, G (ed.) et al., D-modules, representation theory, and quantum groups. Lectures given at the 2nd session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Venezia, Italy, June 12-20, 1992. Berlin: Springer-Verlag. Lect. Notes Math. 1565, 31-140 (1993).
Given a finite-dimensional complex simple Lie algebra \({\mathfrak g}\), one can define a Hopf algebra \(U_ q ({\mathfrak g})\) over the ring \(\mathbb{C}(q)\) of rational functions of an indeterminate \(q\); \(U_ q ({\mathfrak g})\) is called a quantum group because if \(B\) given by generators and relations which become, in the limit \(q \to 1\), those of the universal enveloping algebra \(U({\mathfrak g})\). If \(\varepsilon \in \mathbb{C}^ \times\) is not a root of unity of low order, the defining relations of \(U_ q ({\mathfrak g})\) make sense when \(q\) is replaced by \(\varepsilon\), and the corresponding Hopf algebra \(U_{\varepsilon} ({\mathfrak g})\) over \(\mathbb{C}\) is called the non-restricted specialization of \(U_ q ({\mathfrak g})\). (There is also a restricted specialization \(U_ \varepsilon^{\text{res}} ({\mathfrak g})\), whose definition is slightly more complicated. If \(\varepsilon\) is not a root of unity, \(U_ \varepsilon^{\text{res}} ({\mathfrak g}) \cong U_ \varepsilon ({\mathfrak g})\), but if \(\varepsilon\) is a root of unity, \(U_ \varepsilon^{\text{res}} ({\mathfrak g})\) and \(U_ \varepsilon ({\mathfrak g})\) have quite different representation theories.)
This paper surveys the main results in the representation theory of \(U_ \varepsilon ({\mathfrak g})\), which are due mainly to V. G. Kac and the authors. (The restricted theory at a root of unity is not addressed in this paper.) It is divided into six sections. The first collects some necessary results about Hopf algebras, and the second discusses the representation theory of some associative algebras, notably twisted polynomial algebras. The next four sections are essentially the union of the two papers [C. De Concini and V. G. Kac Prog. Math. 92, 471-506 (1990; Zbl 0738.17008)]; C. De Concini, V. G. Kac and C. Procesi, J. Am. Math. Soc. 5, 151-190 (1992; Zbl 0747.17018)], although there are a number of technical improvements over these papers.
For a detailed discussion of the results, we refer the reader to the reviews of these two papers.
For the entire collection see [Zbl 0782.00093].

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W30 Hopf algebras (associative rings and algebras) (MSC2000)