- The student is familar with classes of quantized universal enveloping algebras including its structures and classes of representations.
- The student is familiar with related Hopf algebraic structures, including quasitriangular Hopf algebras, Drinfeld double construction, Hopf algebras in duality.
- The student is familiar with some applications of quantum groups to harmonic analysis.
- The student is able to communicate orally and in writing on the topics of the course.
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Quantum groups came into being in the 1980s, and have been studied from various perspectives. Quantum groups., in one form or another, arose in the study of the school of Faddeev on mathematical physics in Russia, Jimbo in Japan, Woronowicz in Poland and Drinfeld (Fields Medal 1990) in Russia. The theory of quantum groups has seen a rapid development from the mid 1980s with far-reaching connections to various branches of mathematics, including knot theory, representation theory, operator algebra, and mathematical physics, notably integrable systems. While the term quantum group itself has no precise definition, it is used to denote a number of related constructions, in particular quantized universal enveloping algebras of semisimple Lie groups, and dually, deformations of the algebras of polynomial functions on the corresponding algebraic groups.
The course will focus at understanding quantized universal enveloping algebras as deformations of the universal enveloping algebra of a semisimple Lie algebra, of which some essential features will be recalled. The construction will involve the notion of Hopf algebras, and various constructions and properties involving Hopf algebras will be discussed. The basic example of the quantized universal enveloping algebra for sl(2) will be studied in detail. The representation theory will be studied and applications to harmonic analysis will be given. Possibly some other applications and extensions will be discussed.
There are many books and overview papers on quantum groups, and we will use a variety of sources for the course, including some lecture notes. This will be clearly indicated.
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