Date: January, 2007
Type: mini-course
Speaker(s): Florian Koch

This mini-course covers the following six topics: 

1. Motivation: From Quantum Mechanics to Quantum Groups
The notion of 'quantization' commonly used in textbooks of quantum mechanics has to be specified in order to turn it into a defined mathematical operation. We discuss that on the trails of Weyl's phase space deformation, i.e. we introduce the Weyl-Moyal starproduct and the deformation of Poisson-manifolds.
Generalizing from this, we understand, why Hopf-algebras are the most genuine way to apply 'quantization'
to various other algebraic objects - and why this has direct physical applications.

2. Hopf-Algebras and their Representations
In order to consolidate the above motivation, we have to introduce Hopf-algebras on a mathematical footing. We define Hopf-algebras, discuss duality and especially we will have a closer look at the question why coproducts induce a multiplication on the dual algebra but not the other why around. With these preparations we close this unit by the discussion of representations and corepresentations - and how these are related for dual Hopf-algebras.

3. Universal Enveloping Algebras and dual Algebras of Functions
The two most relevant types of Hopf-algebras for applications in physics are discussed in this unit. Most central notion will be their duality and representation.

4. Quantization: quasitriangular Structures, dual R-matrices and twists
This is the central unit of the course - we quantize universal enveloping algebras and their duals. Central discussion is the fact that for the first type of Hopf-algebras the deformation of the coproduct is sufficient and for the second type it is the dual multiplication. This motivates the way quantization is performed in particular and how this gives rise for noncommutativity for the module and comodule spaces that are so interesting for physics. Currently most popular way to quantize universal enveloping algebras is the twisting according to Drinfeld. We discuss how and why this is such a good concept.

5. Quantum Groups in Physics
With the gained background we want to review known quantum groups that became relevant in physics. Especially q-Deformation, kappa-Poincare- and theta-Poincare-Algebras are discussed.

6. Cohomology
The notion of cohomology applied to Hopf-algebras is a generalized version of the Hochschild-cohomology.
We require this final discussion to classify quantizations of Hopf- algebras. In particular we will see that twists as well are quite suitable to discuss this issue.