65-405:  Hopf algebras, quantum groups and topological field theory
Lecturer: Christoph Schweigert
Contents: 1. Hopf algebras and their representation categories
2. Finite-dimensional Hopf algebras
3. First application: topological field theories of Turaev-Viro type and applications to quantum codes
4. Quasi-triangular Hopf algebras and braided categories
5. Second application: topological field theories of Reshetikhin-Turaev 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 finite-dimensional Hopf algebras: their structure theory, examples and their representation categories. As an application, we present two constructions of topological field theories:
  • the Turaev-Viro construction with applications in the theory of quantum codes
  • the Reshetikhin-Turaev construction (which describes generalizations of Chern-Simons theories) with applications in the construction of invariants for knots and three-dimensional manifolds.
    For more information refer to:
  • 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 18-22. Please contact me, if you want to take an exam.
    • S. Dascalescu, C. Nastasescu, S. Raianu, Hopf Algebras. An Introduction. Monographs and Textbooks in Pure and Applied Mathematics 235, Marcel-Dekker, New-York, 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.
    • Hans-Jürgen 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 Yetter-Drinfeld 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, 10:15-11:45, Monday in Geom H4, Thursday in Geom H3. Start on Monday, October 15, 2012.
    Tutorials by Alexander Barvels: Monday, 12:15-13:45, in Geom 432. First meeting: Monday, October 22, 2012.