Doi-Hopf Modules For Hopf π-Coalgebra And Braided T-Algebra

The notions of Hopf Ï€-coalgebras, where Ï€ is a discrete group, generalize those of Hopf algebras. Hopf Ï€-coalgebra was used by V. G. Turaev to construct Hennings-like and Kuperberg-like invariants of principal Ï€-bundles over link complements and over 3-manifolds. This paper mainly studies Maschke type theorem and Frobenius properties of Doi-Hopf modules for Hopf Ï€-algebras. Also we generalize such results to entwined Ï€-modules. V. G. Turaev introduced the notion of a modular crossed Ï€-category(or called T-category following M. Zunino[4]) and showed that such a category gives rise to a three-dimensional homotopy quantum field theory with target space K(Ï€, 1). Finally, we study the properties of T-algebra and get an example of T-category: the category of corepresentation of T-algebra.In section 1, we review the basic definitions, notions and examples used in this paper.In section 2, firstly, we introduce the concept of a normalized Ï€-integral. Using it, we prove a Maschke type theorem of Doi-Hopf modules for Hopf Ï€-coalgebras. That is theorem 2.7: Let (H, A, C) be a Doi-Hopf Ï€-datum, if there exists a normalized Ï€-integral {φ_α : C_α (?) A₁ → C^*_α-1 (?) A₁}_{α ∈π} of (H, A, C), then the following assertions hold.(1) A morphism u = {u_α : N_α → M_α}_{α ∈π} in ^CM_A^Ï€ has a retraction (resp. a section) in ^CM_A^Ï€, if the A₁-linear map u₁ : N_α → M_α has a retraction (resp. a section) in M_A1;(2) M ∈ ^CM_A^Ï€, if M₁ is semisimple as a right A₁-module, then M is semisimple as an object in ^CM_A^Ï€.As an application, we obtain a characterization for (H, A, C) Doi-Hopf 7r-modules to be projective as A-modules (Corollary 2.9). Meanwhile, we apply our results to character the separability of the forgetful functor F : ^CM_A^Ï€ → M_a1 (Theorem 2.12). At the end of this section we generalize this result to entwined Ï€-modules (Theorem 2.18).