Functors for long dimodules and Yetter-Drinfeld modules in a weak setting (Q6082489)

Summary: In this paper, for two weak Hopf monoids \(H\) and \(B\) with invertible antipode, we define a functor between the category of left-left \(H\otimes B\)-Yetter-Drinfeld modules and the category of \(H\)-\(B\)-Long dimodules. We also show that, if moreover \(H\) is quasitriangular and \(B\) is coquasitriangular, this functor is a retraction of the well-known injective functor between left-left \(H\)-\(B\)-Long dimodules and left-left \(H\otimes B\)-Yetter-Drinfeld modules.