Larson–Sweedler Theorem, Grouplike Elements, Invertible Modules and the Order of the Antipode in Weak Hopf Algebras
نویسنده
چکیده
We extend the Larson–Sweedler theorem for 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 establish the autonomous monoidal category of the modules of a weak Hopf algebra A and show the semisimplicity of the unit and the invertible modules of A. We also reveal the connection of these modules to left/right grouplike elements in the dual weak Hopf algebra Â. Defining distinguished left/right grouplike elements we derive the Radford formula 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 A of the underlying weak Hopf algebra A. E-mail: [email protected] Supported by the Hungarian Research Fund, OTKA – T 034 512
منابع مشابه
Larson–Sweedler Theorem and the Role of Grouplike Elements in Weak Hopf Algebras
We extend the Larson–Sweedler theorem [10] 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 grouplik...
متن کاملLarson–Sweedler Theorem, Grouplike Elements and Invertible Modules in Weak Hopf Algebras
We extend the Larson–Sweedler theorem for 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 establish the autonomous monoidal category of the modules of a weak Hopf algebra A and show the semisimplicity of the unit and the invertible modules of A. We also reveal the connection of these modules to lef...
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