Larson–Sweedler theorem and the role of grouplike elements in weak Hopf algebras |
| |
Authors: | Peter Vecsernys |
| |
Institution: | Research Institute for Particle and Nuclear Physics, PO Box 49, H-1525, Budapest 114, Hungary |
| |
Abstract: | We extend the Larson–Sweedler theorem Amer. J. Math. 91 (1969) 75] to weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a non-degenerate left integral. We show that the category of modules over a weak Hopf algebra is autonomous monoidal with semisimple unit and invertible modules. We also reveal the connection of invertible modules to left and right grouplike elements in the dual weak Hopf algebra. Defining distinguished left and right grouplike elements, we derive the Radford formula Amer. J. Math. 98 (1976) 333] for the fourth power of the antipode in a weak Hopf algebra and prove that the order of the antipode is finite up to an inner automorphism by a grouplike element in the trivial subalgebra AT of the underlying weak Hopf algebra A. |
| |
Keywords: | |
本文献已被 ScienceDirect 等数据库收录! |
|