Graduate courses

We will present the quantum algebra U_q(g) attached by Drinfeld and Jimbo to a simple Lie algebra g. We will try to motivate its definition by the problem of constructing solutions of the Yang-Baxter equation, and by the relationship with the Hall-Ringel algebra of a quiver.

We will start with the basics of the theory of Hopf algebras, and we will end with some features of the theory of compact quantum groups introduced by Woronowicz in the late 1980's in the framework of operator algebras, recalling the needed facts about operator algebras along the way. Besides compact and discrete groups, the main examples to be considered will be the so-called universal compact quantum groups introduced by Wang in 1995.

We will start with the basic notions related to quantum groups and their duality on the Hopf algebra level. Then a number of concrete examples of quantum groups will be discussed (some of these examples, like U_q(sl_2) and its dual, will be discussed in detail in other courses). In the center of our attention will be the representation category of a quantum group, including braided, ribbon and modular categories, and an action of a quantum group on an algebra. Finally, two important applications of the quantum group theory will be given: 1) to V. Jones' subfactor theory; 2) to topological invariants via modular categories (V. Turaev - N. Reshetikhin's approach).

In order to understand the interplay between noncommutative geometry and quantum groups, it is important to construct equivariant spectral triples on quantum groups and on their homogeneous spaces having the desired axioms of noncommutative geometry. We will discuss some of the equivariant spectral triples constructed on SU_q(2) and on the odd dimensional spheres from the point of view of local index formula. We show that the local index formula is applicable to these spectral triples. We prove by using a construction called the quantum double suspension of a spectral triple.