Axioms for Weak Bialgebras
  Let A be a finite dimensional unital associative algebra over a field K, which is also equipped with a coassociative counital coalgebra structure (\Delta,\eps). A is called a Weak Bialgebra if the coproduct \Delta is multiplicative. We do not require \Delta(1) = 1 \otimes 1 nor multiplicativity of the counit \eps. Instead, we propose a new set of counit axioms, which are modelled so as to guarantee that \Rep\A becomes a monoidal category with unit object given by the cyclic A-submodule \E := (A --> \eps) \subset \hat A (\hat A denoting the dual weak bialgebra). Under these monoidality axioms \E and \bar\E := (\eps <-- A) become commuting unital subalgebras of \hat A which are trivial if and only if the counit \eps is multiplicative. We also propose axioms for an antipode S such that the category \Rep\A becomes rigid. S is uniquely determined, provided it exists. If a monoidal weak bialgebra A has an antipode S, then its dual \hat A is monoidal if and only if S is a bialgebra anti-homomorphism, in which case S is also invertible. In this way we obtain a definition of weak Hopf algebras which in Appendix A will be shown to be equivalent to the one given independently by G. B\"ohm and K. Szlach\'anyi. Special examples are given by the face algebras of T. Hayashi and the generalised Kac algebras of T. Yamanouchi. 
Florian Nill