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The category of partial Doi-Hopf modules and functors. (English) Zbl 1284.16037

Let \(H\) be a Hopf algebra, \(A\) an algebra and \(C\) a coalgebra, all over a field. The usual notions of \(A\) being a right \(H\)-comodule algebra and \(C\) being a right \(H\)-module coalgebra are weakened by replacing properties of unit elements of \(A\) and \(H\) in the first case, and of \(H\) in the second case, by more general technical properties. This gives the notions of partial coactions and partial actions. They generalize notions of partial group actions. Their definitions were introduced by S. Caenepeel and K. Janssen using notions of partial entwining structures [Commun. Algebra 36, No. 8, 2923-2946 (2008; Zbl 1168.16021)]. A partial Doi-Hopf datum is a triple \((H,A,C)\), \(A\) a partial right \(H\)-comodule algebra, \(C\) a partial right \(A\)-module coalgebra. A partial Doi-Hopf module \(M\) is a right \(A\)-module with a map of \(M\) to \(M\otimes C\) satisfying certain conditions. This generalizes the usual notion of Doi-Hopf module given by Y. Doi [J. Algebra 153, No. 2, 373-385 (1992; Zbl 0782.16025)].
The paper under review generalizes a result of S. Caenepeel and S. Raianu for induced functors on Doi-Hopf modules to the case of partial Doi-Hopf modules [Abelian groups and modules. Proceedings of the Padova conference 1994. Math. Appl., Dordr. 343, 73-94 (1995; Zbl 0843.16035)]. Let \((\mathcal M(H)^C)_A\) be the category of partial Doi-Hopf modules for the partial Doi-Hopf datum \((H,A,C)\). Let \((H',A',C')\) be another partial Doi-Hopf datum. Given morphisms from \(H\) to \(H'\), \(A\) to \(A'\) and \(C\) to \(C'\), the authors define an induction functor from \((\mathcal M(H)^C)_A\) to \((\mathcal M(H')^{C'})_A'\), and prove that it has a right adjoint. They define a normalized \(A\)-integral as a map from \(C\otimes C\) to \(A\) satisfying certain conditions, and show that there is one if and only if the forgetful functor from \((\mathcal M(H)^C)_A\) to \(\mathcal M(H)_A\) is separable. Finally, a Maschke-type theorem for partial Doi-Hopf modules is proved.

MSC:

16T05 Hopf algebras and their applications
16T15 Coalgebras and comodules; corings
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[1] S. CAENEPEEL - K. JANSSEN, Partial (Co)Actions of hopf algebras and Partial Hopf-Galois Theory, Comm. Algebra, 36 (2008), pp. 2923-2946. · Zbl 1168.16021 · doi:10.1080/00927870802110334
[2] S. CAENEPEEL - G. MILITARU - B. ION, Separable functors for the category of Doi-Hopf modules, Applications, Adv. Mathematics, 145 (1999), pp. 239-290. · Zbl 0943.18007 · doi:10.1006/aima.1998.1817
[3] S. CAENEPEEL - S. RAIANU, Induction functors for the Doi-Koppinen unified Hopf modules, in Abelian Groups and Modules, pp. 73-94, Kluwer Acad. Publ., Dordrecht, 1995. · Zbl 0843.16035
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