Separable Functors for the Category of Doi–Hopf Modules, Applications

Doi-hopf modules over weak hopf algebras

“…In this paper we generalize the definition of Doi-Hopf modules to the case when H is a Weak Bialgebra (WBA). Our definitions are supported by the fact that many results of [10,5,6] remain valid in this case.…”

“…It is the category of the modules over the algebra A which are also comodules over the coalgebra C and satisfy certain compatibility condition involving H. The study of C M(H) A turned out to be very useful: It was shown in [7,4] that many categories investigated independently before -such as the module and comodule categories over bialgebras, the Hopf modules category [15], and the Yetter-Drinfeld category [16,14] -are special cases of C M(H) A . Using this observation many results known for module categories over bialgebras or Hopf algebras were generalized to this more general setting [5,6].…”

“…The paper is organized as follows: we define and examine the structures such as the weak Doi-Hopf datum (generalizing the Doi-Hopf datum of [7]) the weak Doi-Hopf module (generalizing the Doi-Hopf module of [7]) the weak smash product (generalizing the analogous notion of [10]) and the weak Doi-Hopf integral (generalizing definitions of [6,8]). We illustrate these notions on the same four examples generalizing some classical examples of [7,4].…”

Separable functors for the category of Doi-Hopf π-modules

“…For all α ∈ π , we consider the functor F (α) from the category of Doi-Hopf π -modules to the category of left A α -modules, and we give necessary and sufficient conditions for this functor to be separable. In particular, if π is a trivial group, the results in [3] are recovered as special cases.…”

Heisenberg Double, Pentagon Equation, Structure and Classification of Finite-Dimensional Hopf Algebras

“…The Heisenberg double H(L) = L#L * is in fact a particular case (for the Doi-Koppinen datum (H, A, C) = (L, L, L)) of the general smash product A#C * introduced in [6] in the right-right version. The right-left version of it and the above description of the Heisenberg double are given in [4]. The canonical isomorphism of vector spaces…”