Lecture course: Lie algebras and their representations (Thorsten Heidersdorf)

Advanced Topics in Algebra: Hopf algebras, quantum groups and Nichols algebras


Thorsten Heidersdorf
Universität Bonn SS 2021
Tuesday, 10.15 - 12.00 am, Friday 10.15 - 12.00 am
First lecture: April 16 2021


The course will be run via Zoom. Register in advance for this course: Registration link

After registering, you will receive a confirmation email containing information about joining the meeting. The passcode can be found in BASIS.

Lecture Notes: Sciebo Link The password was announced in the lecture, starts with Q and is one of the words from the title of the lecture.

Content: Hopf algebras are certain bialgebras that occur naturally in many fields such as algebraic topology, representation theory and combinatorics. Some important examples are:

  • The universal enveloping algebra $\mathcal{U}(\mathfrak{g})$ of a Lie algebra (or a Lie superalgebra);
  • The coordinate ring $k[G]$ of an affine (super) group scheme $G$;
  • The Hopf algebra of symmetric functions;
  • The Hopf algebra of rooted trees
  • and many more.
We will be in particular interested in the class of quasitriangular and braided Hopf algebras. For these the category of modules is a braided monoidal category, and these play a major role for example in the theory of quantum link invariants.

The most important examples in this course are Drinfeld's quantum groups, certain deformations of the universal enveloping algebra of a Lie algebra (note that there are many non-equivalent definitions of quantum groups in the literature). We will define and study the so-called small and big quantum group attached to a semisimple Lie algebra $\mathfrak{g}$. These quantum groups are quasitriangular Hopf algebras and are of fundamental importance in many mathematical areas.

Another source of quasitriangular Hopf algebras is obtained from the Drinfeld double of a finite dimensional Hopf algebra. We will study modules over this Drinfeld double and identify them with so-called Yetter-Drinfeld modules over the original Hopf algebra.

In the last part of the course we will introduce the notion of a Nichols algebra, braided Hopf algebras in some category of Yetter-Drinfeld modules. They play a major role in classification problems. In particular one can see the universal enveloping enveloping algebras of Lie algebras and Lie superalgebras as well as their quantized versions as special cases of the general theory of Nichols algebras. While we will not be able to prove any major theorems in this area, the aim is to motivate the constructions and give an overview about some major theorems and developments.

Prerequisites:
  • Knowledge of Lie algebras and their representations as in my course on Advanced Algebra 1 in the winter term (see e.g. Humphreys: Introduction to Lie algebras and representation theory). Mostly we will need the part on the universal enveloping algebra and the classification of simple Lie algebras.
  • Basic category theory (categories and functors, natural transformations, (co)limits,...)


Literature: The literature on this topic is vast and more may be provided during the course. Some sources are
  • Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V.: Algebras, rings and modules. Lie algebras and Hopf algebras. zbmath page
  • Kassel, Christian: Quantum groups. zbmath page
  • Chari, Vyjayanthi; Pressley, Andrew: A guide to quantum groups. zbmath page
  • Heckenberger, István; Schneider, Hans-Jürgen: Hopf algebras and root systems. zbmath page