Semisimple Hopf algebras and their categories of representations

Department of pure Mathematics, Faculty of Science of the University of Porto
Wednesday, 28 November, 2007

We shall review some results in the theory of Hopf algebras and their tensor categories of representations, putting special emphasis in the classification problem in the semisimple case. We shall discuss the construction of extensions and semisimple Hopf algebras arising from matched pairs of finite groups as well as the more recent construction of group theoretical Hopf algebras. We shall also discuss normal Hopf subalgebras and simplicity and its relation with the twisting construction.

Contents:
I. (Semisimple) Hopf algebras. Tensor categories and the twisting construction. Character algebra and the Class Equation. Applications: Masuoka's pn-Theorem.

II. Extensions. Normal Hopf subalgebras. Abelian extensions. The Kac exact sequence. Group-theoretical Hopf algebras.

III. Examples in low dimension. Simple Hopf algebras obtained by twisting.

Some references:

Part I.
H.-J. Schneider, Lectures on Hopf algebras, Trabajos de Matemática 31/95, FaMAF (1995). Available at www.mate.uncor.edu/natale

Part II.
- A. Masuoka, Extensions of Hopf algebras, Trabajos de Matemática 41/99, FaMAF (1999). Available at www.mate.uncor.edu/natale
- A. Masuoka, Hopf algebra extensions and cohomology, Math. Sci. Res. Inst. Publ. 43 (2002), 167-209.
- P. Etingof, D. Nikshych and V. Ostrik, On fusion categories, Ann. Math. (2) 162, 581-642 (2005).

Part III.
- C. Galindo and S. Natale, Simple Hopf algebras and deformations of finite groups, to appear in Math. Res. Lett. Preprint arXiv:math/0608734v2.
- C. Galindo and S. Natale, Normal Hopf subalgebras in cocycle deformations of finite groups. Preprint arXiv:math/0708.3407.
- S. Natale, Semisolvability of semisimple Hopf algebras of low dimension, Memoirs Amer. Math. Soc. 186, 123 pp. (2007).

Timetable:
Wednesday, November 28: 17.30-19.00
Thursday, November 29: 15.30-17.00
Friday, November 30: 14.30-16.00

Venue:
Edifício dos Departamentos de Matemática, Room 1.26
Faculdade de Ciências da Universidade do Porto
Rua do Campo Alegre, 687
4169-007 Porto

Speaker: 

Sonia Natale (Facultad de Matemática, Astronomía y Física Universidad Nacional de Córdoba, Argentina)