LarsonSweedler Theorem and the Role of Grouplike Elements in Weak Hopf Algebras
Abstract
We extend the LarsonSweedler theorem to weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a nondegenerate 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 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^T of the underlying weak Hopf algebra A.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 2001
 arXiv:
 arXiv:math/0111045
 Bibcode:
 2001math.....11045V
 Keywords:

 Mathematics  Quantum Algebra
 EPrint:
 version appeared in J.Algebra, 45 pages, plain TeX, extended introduction, shortened proofs