Seminar of June 6th

7/6/2018, Room 27, 2:30 pm, FaMAF

From Hopf Algebras to tensor categories


Speaker: Héctor Peña Pollastri

This talk follows the exposition given in [1]. We’ll show a systematic way to construct tensor categories from categories of representations of certain Hopf Algebras taking the quotient by objects of zero quantum-trace (a process which was introduced in [2]).
In particular, we’ll require the Hopf algebra to be spherical, notion which we’ll explain and also show some conditions that guarantee that a given Hopf algebra is of this type. After this, we’ll discuss how to obtain fusion subcategories from the tensor categories constructed before, in particular, we’ll explain the method of Tilting modules for quasi hereditary Algebras [3]. Finally, we’ll discuss the special case of quantum groups with q a foot of unity, where the Tilting modules are obtained using good filtrations and Weyl filtrations.

[1] “From Hopf Algebras to Tensor Categories”. N. Andruskiewitch, I. Angiono, A. García Iglesias, B. Torrecillas and C. Vay. From ‘Conformal Field Theories and Tensor Categories’, Mathematical lectures from Peking University, DOI 10.10007/978-3-642-39383-9. Springer- Velag Berlin Heindelberg 2014.
[2] ”Spherical Categories” John W. Barrett and Bruce W. Westbury. Advances in Mathematics, Volume 143, Number 2, 1999. Pág 357.
[3] “The category of modules with good filtrations over quasi-hereditary algebra has almost split sequences” C.M. Ringel. Mathematische Zeitschrift Band 208, Heft 2, 1991. Pág 209.
[4] “Tensor Products of Quantized Tilting Modules” H.H. Andersen. Comm. In Mathematical Physics, Volume 149, Number 1, 1992. Pág 149.
[5] “From Quantum Groups to Unitary Modular Tensor Categories”. Eric C. Rowell. Contemporary Mathematics 2005 (arXiv:math/0503226).