Title: Finite-dimensional Hopf algebras, coends, and modular invariant Frobenius algebras

Lecturer: Juergen Fuchs, Karlstad University, Sweden

Time:
9:00 -- 10:30 on June 9
14:00 -- 15:30 on June 10
9:00 -- 11:00 on June 23
14:00 -- 16:00 on June 24

Place:
Room 1328 at BICMR, Resource Plaza, Peking University on June 9 and June 23.
Room 1213 at BICMR, Resource Plaza, Peking University on June 10 and June 24.

The aim of the lectures is to provide sufficient information for being able to appreciate results about certain Frobenius algebras in braided monoidal categories that come that finite-dimensional Hopf algebras. The study of these Frobenius algebras is motivated by conformal field theory.

• 1) Reminder about groups, algebras and representations. Graphical calculus. Motivation of Hopf algebras from representation theory. Weak Hopf algebras. Symmetric algebras. Frobenius algebras.
• 2) Characters. Grothendieck groups. Morita equivalence. Integrals and cointegrals of Hopf algebras. The Frobenius map.
• 3) Monoidal categories. Braiding, duality and twist. Quasitriangular, ribbon and factorizable Hopf algebras. The Drinfeld map and the quantum Fourier transform. H-mod and H-Bimod as ribbon categories.
• 4) Dinatural transformations and coends. Deligne product of abelian categories. Hopf pairing. Lyubashenko's coend as a Hopf algebra. Coends in H-Bimod which are Frobenius algebras.
• 5) Three-manifold invariants and mapping class group actions from coends. Modular invariance for coend Frobenius algebras.