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.
Based on a pairing of two regular multiplier Hopf algebras A and B, Heisenberg double H is the smash product A#B with respect to the left regular action of B on A. Let D = A ⊲⊳ B be the Drinfel'd double, then Heisenberg double H is a Yetter-Drinfel'd D-module algebra, and it is also braided commutative by the braiding of Yetter-Drinfel'd module, which generalizes the results in [10] to some infinite dimensional cases.
In this paper, we introduce a class of 2-cocycles on monoidal Hom–Hopf algebras, study their properties, and extend neat lazy 2-cocycles to a Radford [Formula: see text]-biproduct monoidal Hom–Hopf algebra.
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