The Category of Partial Doi-Hopf Modules and Functors

  • Q.-C. Chen

    Yili Normal College, Yining, China
  • D.-G. Wang

    Qufu Normal University, Qufu, Shandong, China


Let (H,A,C)(H, A, C), (H,A,C)(H', A', C') be two partial Doi-Hopf datums consisting of a Hopf algebra HH, a partial right HH-comodule algebra AA and a partial right HH-module coalgebra. Given α:HH{\alpha}: H \rightarrow H ', β:AA{\beta}: A \rightarrow A ' and γ:CC{\gamma}: C \rightarrow C', we define an induction functor between the category M(H)AC{\cal M}(H)^{C}_{A} of all partial Doi-Hopf modules and the category M(H)AC{\cal M}(H')^{C'}_{A'}, and we prove that this functor has a right adjoint. Specially, we then give necessary and sufficient conditions for the functor F:M(H)ACM(H)AF\kern -1pt :\kern -1pt {\cal M}(H)^{C}_{A} \kern -1pt \rightarrow \kern -1pt {\cal M}(H)_{A} (exactly the category of right AA-modules). This leads to a generalized notion of integrals. Moreover, from these results, we deduce a version of Maschke-type Theorems for partial Doi-Hopf modules. The applications of our results are considered.

Cite this article

Q.-C. Chen, D.-G. Wang, The Category of Partial Doi-Hopf Modules and Functors. Rend. Sem. Mat. Univ. Padova 129 (2013), pp. 189–204

DOI 10.4171/RSMUP/129-11