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The affineness criterion for quantum Hom-Yetter-Drinfel'd modules

100%

Guo S., Wang S.

Colloquium Mathematicum

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2016

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tom 143

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nr 2

169-185

EN

Quantum integrals associated to quantum Hom-Yetter-Drinfel'd modules are defined, and the affineness criterion for quantum Hom-Yetter-Drinfel'd modules is proved in the following form. Let (H,α) be a monoidal Hom-Hopf algebra, (A,β) an (H,α)-Hom-bicomodule algebra and $B = A^{coH}$. Under the assumption that there exists a total quantum integral γ: H → Hom(H,A) and the canonical map $β: A ⊗_{B} A → A ⊗ H$, $a ⊗_{B} b↦ S^{-1}(b_{[1]})α(b_{[0][-1]}) ⊗ β^{-1}(a)β(b_{[0][0]})$, is surjective, we prove that the induction functor $A ⊗_{B}-: 𝓗̃ (𝓜 _{k})_{B} → ^{H}𝓗 𝓨 𝓓_{A}$ is an equivalence of categories.

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Separable functors for the category of Doi Hom-Hopf modules

100%

Guo S., Zhang X.

Colloquium Mathematicum

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2016

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tom 143

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nr 1

23-37

EN

Let $𝓗̃ (𝓜 _{k})(H)^{C}_{A}$ be the category of Doi Hom-Hopf modules, $𝓗̃ (𝓜 _{k})_{A}$ be the category of A-Hom-modules, and F be the forgetful functor from $𝓗̃ (𝓜 _{k})(H)^{C}_{A}$ to $𝓗̃ (𝓜 _{k})_{A}$. The aim of this paper is to give a necessary and suffcient condition for F to be separable. This leads to a generalized notion of integral. Finally, applications of our results are given. In particular, we prove a Maschke type theorem for Doi Hom-Hopf modules.

3

Braided monoidal categories and Doi-Hopf modules for monoidal Hom-Hopf algebras

81%

Guo S., Zhang X., Wang S.

Colloquium Mathematicum

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2016

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tom 143

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nr 1

79-103

EN

We continue our study of the category of Doi Hom-Hopf modules introduced in [Colloq. Math., to appear]. We find a sufficient condition for the category of Doi Hom-Hopf modules to be monoidal. We also obtain a condition for a monoidal Hom-algebra and monoidal Hom-coalgebra to be monoidal Hom-bialgebras. Moreover, we introduce morphisms between the underlying monoidal Hom-Hopf algebras, Hom-comodule algebras and Hom-module coalgebras, which give rise to functors between the category of Doi Hom-Hopf modules, and we study tensor identities for monodial categories of Doi Hom-Hopf modules. Furthermore, we construct a braiding on the category of Doi Hom-Hopf modules. Finally, as an application of our theory, we get a braiding on the category of Hom-modules, on the category of Hom-comodules, and on the category of Hom-Yetter-Drinfeld modules.

Ograniczanie wyników

3 Colloquium Mathematicum

3 Guo S.

2 Wang S.

2 Zhang X.

3 2016

The affineness criterion for quantum Hom-Yetter-Drinfel'd modules

Separable functors for the category of Doi Hom-Hopf modules

Braided monoidal categories and Doi-Hopf modules for monoidal Hom-Hopf algebras