
65405: 
Hopf algebras, quantum groups and topological field theory

Lecturer: 
Christoph Schweigert

Contents: 
1. Hopf algebras and their representation categories
2. Finitedimensional Hopf algebras
3. First application: topological field theories of TuraevViro type and applications to quantum codes
4. Quasitriangular Hopf algebras and braided categories
5. Second application: topological field theories of ReshetikhinTuraev type and applications to knots.

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 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/ws12/hopf.html

Prerequisites: 
This lecture aims at students in the master programs of mathematics, mathematical
physics and physics. It is accessible to advanced 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: 
Individual oral exam: either on Wednesday, February 6, 2013 or in the
week March 1822. Please contact me, if you want to take an exam.

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 for
Proposition 3.1.19 and
2d TFT. Notes on
YetterDrinfeld modules and
traces and twists.
Overview scheme by Ana Ros Camacho.

Problem sheets: 
Sheet 1: problems and
hints.
Sheet 2: problems and
hints.
Sheet 3: problems and
hints.
Sheet 4: problems and
hints.
Sheet 5: problems and
hints and handwritten
notes.
Sheet 6: problems and
hints.
Sheet 7: problems and
hints.
Sheet 8: problems and
hints and handwritten
notes.
Sheet 9: problems and
hints and handwritten
notes.
Sheet 10: problems and
hints and handwritten
notes.
Sheet 11: problems and
hints and handwritten
notes.
No sheet for the Christmas break. Question time during the tutorials
of January 7, 2013.

Time and Place: 
Lectures:
Monday and Thursday, 10:1511:45,
Monday in Geom H4, Thursday in Geom H3. Start on Monday, October 15, 2012.
Tutorials by Alexander Barvels: Monday, 12:1513:45, in Geom 432.
First meeting: Monday, October 22, 2012.


