The subject of Quantum Groups is a rapidly diversifying field of mathematics and mathematical physics, originally launched by developments in theoretical physics and statistical mechanics involving quantum analogues of Lie algebras and coordinate rings of algebraic groups. The study of these objects and their representation theory has opened up important new directions in non-commutative algebra.
The aim of the course is to provide young researchers with the necessary tools to tackle open problems in the subject area, giving them the opportunity to learn the most recent results on the structure and representation theory of quantized coordinate rings and quantized enveloping algebras. The label "quantized coordinate ring" is used in the literature to refer to various non-communicative algebras which are, informally expressed, deformations of the classical coordinate rings of algebraic varieties or algebraic groups; the adjective "quantized" usually indicates that some solution to the quantum Yang-Baxter equation is involved in the construction and/or the representation theory of the algebra.
The known algebras that, by general agreement, carry the label "quantized coordinate rings" do share a substantial number of common features, which will be developed in the lectures. Similarly, quantum enveloping algebras of semi-simple, Lie algebras (or of affine Kac-Moody algebras). This class of algebras is somewhat more tightly defined; in that generators and relations are given by a standard process applied to the Serre relations for classical enveloping algebras. As in the classical setting, there is a duality between these algebras and the quantized coordinate rings of the corresponding semi-simple algebraic groups.