Displaying similar documents to “Braided monoidal categories and Doi-Hopf modules for monoidal Hom-Hopf algebras”

A Maschke type theorem for relative Hom-Hopf modules

Shuangjian Guo, Xiu-Li Chen (2014)

Czechoslovak Mathematical Journal

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Let ( H , α ) be a monoidal Hom-Hopf algebra and ( A , β ) a right ( H , α ) -Hom-comodule algebra. We first introduce the notion of a relative Hom-Hopf module and prove that the functor F from the category of relative Hom-Hopf modules to the category of right ( A , β ) -Hom-modules has a right adjoint. Furthermore, we prove a Maschke type theorem for the category of relative Hom-Hopf modules. In fact, we give necessary and sufficient conditions for the functor that forgets the ( H , α ) -coaction to be separable. This leads...

The fundamental theorem and Maschke's theorem in the category of relative Hom-Hopf modules

Yuanyuan Chen, Zhongwei Wang, Liangyun Zhang (2016)

Colloquium Mathematicae

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We introduce the concept of relative Hom-Hopf modules and investigate their structure in a monoidal category ̃ ( ℳ k ) . More particularly, the fundamental theorem for relative Hom-Hopf modules is proved under the assumption that the Hom-comodule algebra is cleft. Moreover, Maschke’s theorem for relative Hom-Hopf modules is established when there is a multiplicative total Hom-integral.

Serre Theorem for involutory Hopf algebras

Gigel Militaru (2010)

Open Mathematics

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We call a monoidal category C a Serre category if for any C, D ∈ C such that C ⊗ D is semisimple, C and D are semisimple objects in C. Let H be an involutory Hopf algebra, M, N two H-(co)modules such that M ⊗ N is (co)semisimple as a H-(co)module. If N (resp. M) is a finitely generated projective k-module with invertible Hattory-Stallings rank in k then M (resp. N) is (co)semisimple as a H-(co)module. In particular, the full subcategory of all finite dimensional modules, comodules or...

Smash (co)products and skew pairings.

José N. Alonso Alvarez, José Manuel Fernández Vilaboa, Ramón González Rodríguez (2001)

Publicacions Matemàtiques

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Let τ be an invertible skew pairing on (B,H) where B and H are Hopf algebras in a symmetric monoidal category C with (co)equalizers. Assume that H is quasitriangular. Then we obtain a new algebra structure such that B is a Hopf algebra in the braided category γD and there exists a Hopf algebra isomorphism w: B ∞ H → B [×] H in C, where B ∞ H is a Hopf algebra with (co)algebra structure the smash (co)product and B [×] H is the Hopf algebra defined by Doi and Takeuchi. ...

Bicovariant differential calculi and cross products on braided Hopf algebras

Yuri Bespalov, Bernhard Drabant (1997)

Banach Center Publications

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In a braided monoidal category C we consider Hopf bimodules and crossed modules over a braided Hopf algebra H. We show that both categories are equivalent. It is discussed that the category of Hopf bimodule bialgebras coincides up to isomorphism with the category of bialgebra projections over H. Using these results we generalize the Radford-Majid criterion and show that bialgebra cross products over the Hopf algebra H are precisely described by H-crossed module bialgebras. In specific...

Squared Hopf algebras and reconstruction theorems

Volodymyr Lyubashenko (1997)

Banach Center Publications

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Given an abelian 𝑉-linear rigid monoidal category 𝑉, where 𝑉 is a perfect field, we define squared coalgebras as objects of cocompleted 𝑉 ⨂ 𝑉 (Deligne's tensor product of categories) equipped with the appropriate notion of comultiplication. Based on this, (squared) bialgebras and Hopf algebras are defined without use of braiding. If 𝑉 is the category of 𝑉-vector spaces, squared (co)algebras coincide with conventional ones. If 𝑉 is braided, a braided Hopf algebra can be obtained...