Finite dimensional representations of quantum groups
The quantum algebra, a quantization of the universal enveloping algebra U(\({\mathfrak g})\) of a finite dimensional semisimple Lie algebra \({\mathfrak g}\) was found by Drinfel'd and Jimbo in their study of the quantum Yang- Baxter equation. Their formulations are slightly different. Drinfel'd's \(H_{\hslash}({\mathfrak g})\) is an algebra over the formal power series ring K\(\hslash\), while Jimbo's \(U_ q({\mathfrak g})\) is that over the field of rational functions \(k(q^{1/2})\), where k is a field with characteristic zero. The indeterminates \(\hslash\) and q are related by \(q^{1/2}=e^{\hslash /4}.\) The purpose of this paper is to give a counterpart for \(U_{\hslash}({\mathfrak g})\) of the results of \textit{G. Lusztig} [Adv. Math. 70, 237-249 (1988; Zbl 0651.17007)] and \textit{M. Rosso} [Commun. Math. Phys. 117, 581-593 (1988; Zbl 0651.17008)] concerning finite dimensional \(U_ q({\mathfrak g})\)-modules, by using the advantage that \(U_{\hslash}({\mathfrak g})\) is a topologically free k\(\hslash\)-module satisfying \(U_{\hslash}({\mathfrak g})|_{\hslash =0}\simeq U({\mathfrak g})\). Also proved is the irreducible decomposition as a two-sided \(U_{\hslash}({\mathfrak g})\)-module of the coordinate algebra A[G] of the quantum group G. The final section is devoted to a proof of a quantum analogue of the Borel-Weil-Bott theorem.
scientific article; zbMATH DE number 4204612