Axioms for Weak Bialgebras

TL;DR

This paper introduces new axioms for weak bialgebras that ensure their representation categories are monoidal and rigid, providing a clearer framework for understanding weak Hopf algebras and their duals.

Contribution

It proposes a novel set of axioms for weak bialgebras and antipodes, establishing conditions for monoidality and rigidity of their representation categories.

Findings

A new set of counit axioms guarantees monoidality of Rep A.

Conditions for the antipode S to make Rep A rigid are established.

Dual weak bialgebras are monoidal iff S is a bialgebra anti-homomorphism.

Abstract

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.