Bicovariant Differential Calculi on a Weak Hopf Algebra

Abstract

Let H be a weak Hopf algebra with bijective antipode. In this paper we follow Woronowicz's fundamental method to characterize bicovariant differential calculi on H. We show that there exists a 1-1 correspondence between bicovariant differential calculi and some right ideals of H contained in kerε s such that these ideals are right H-comodules with coadjoint maps, where ε s is the source map of H. This is a generalization of well-known Woronowicz's theorem about bicovariant differential calculi on quantum groups.

Bicovariant differential calculi for finite global quotients

“…However, the same basic definition also works if A is replaced by a weak Hopf algebra. The problem of bicovariant differential calculi over weak Hopf algebras was originally studied in [6], especially in regard to differential calculi over weak smash products, and later again in [24]. However, the classification problem for bicovariant differential calculi over the weak Hopf algebra k(G⋉M ) has yet to be studied.…”