65-401:  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. Quasi-triangular 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 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
    and
  • 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:
    http://www.math.uni-hamburg.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, 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 and 2d TFT. Notes on Yetter-Drinfeld modules, traces and twists and the Turaev-Viro 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.