
65401: 
Hopf algebras, quantum groups and topological field theory

Lecturer: 
Christoph Schweigert

Contents: 
1. Hopf algebras and their representation categories
2. Finitedimensional Hopf algebras
3. Quasitriangular Hopf algebras and braided categories
4. Topological field theories and quantum codes 
Aim: 
We present an introduction to Hopf algebras over a field and their applications to topological
field theories. The study of Hopf algebras (sometimes also known as quantum groups) is a very
active field, relating algebra, representation theory and mathematical physics. Hopf algebras
and topological field theories have applications in
representation theory, topology, string theory, quantum gravity and
quantum information theory.
Special emphasis in this class will be on complex finitedimensional Hopf algebras: their
structure theory, examples and their representation categories. As an application, we present
two constructions of topological field theories:
the TuraevViro construction with applications in the theory of quantum codes
and
the ReshetikhinTuraev construction (which describes generalizations of ChernSimons
theories) with applications in the construction of invariants for knots and threedimensional
manifolds.
For more information refer to:
http://www.math.unihamburg.de/home/schweigert/ws19/hopf.html

Prerequisites: 
This lecture aims at students in the master programs of mathematics,
mathematical physics and physics. It is accessible to motivated bachelor students as well.
Prerequisites are a good knowledge of linear algebra (in particular vector spaces,
their duals, linear maps,
bilinear maps and tensor products). Some notions from algebra
(in particular about groups and algebras) or the theory of Lie algebras are helpful,
but not indispensable.

Exam: 
Depending on the number of exams, oral or written.

Literature: 
 S. Dascalescu, C. Nastasescu, S. Raianu, Hopf Algebras. An Introduction. Monographs
and Textbooks in Pure and Applied Mathematics 235, MarcelDekker, NewYork, 2001.
 C. Kassel, Quantum Groups, Graduate Texts in Mathematics 155, Springer, Berlin, 1995.
 C. Kassel, M. Rosso, Vl. Turaev: Quantum groups and knot invariants.
Panoramas et Synthèses, Soc. Math. de France, Paris, 1993

S. Montgomery, Hopf algebras and their actions on rings, CMBS Reg. Conf. Ser. In Math.
82, Am. Math. Soc., Providence, 1993.

HansJürgen Schneider,
Lectures on Hopf algebras, Notes by Sonia Natale. Trabajos de Matemática 31/95, FaMAF, 1995.

Lecture notes: 
as a
pdf file.
Figures for
Proposition 2.5.11, for
Theorem 3.1.5, for
Observation
3.1.10 and
2d TFT. Notes on
YetterDrinfeld modules,
traces
and twists and the
TuraevViro
construction.
Overview scheme by Ana Ros Camacho.

Problem sheets: 
will be provided on STiNE. Please register in STiNE.

Time and Place: 
See information in STiNE.
Lectures start on Monday, October 14, 2019, tutorials on Wednesday, October
23, 2019.


