## Hopf (Bi-)Modules and Crossed Modules in Braided Monoidal Categories (1995)

Venue: | J. Pure Appl. Algebra |

Citations: | 15 - 1 self |

### BibTeX

@ARTICLE{Bespalov95hopf(bi-)modules,

author = {Yuri Bespalov and Bernhard Drabant},

title = {Hopf (Bi-)Modules and Crossed Modules in Braided Monoidal Categories},

journal = {J. Pure Appl. Algebra},

year = {1995},

volume = {123},

pages = {105--129}

}

### OpenURL

### Abstract

Hopf (bi-)modules and crossed modules over a bialgebra B in a braided monoidal category C are considered. The (braided) monoidal equivalence of both categories is proved provided B is a Hopf algebra (with invertible antipode). Bialgebra projections and Hopf bimodule bialgebras over a Hopf algebra in C are found to be isomorphic categories. A generalization of the Majid-Radford criterion for a braided Hopf algebra to be a cross product is obtained as an application of these results. Keywords: Braided category, Braided Hopf algebra, Crossed Module, Hopf (Bi-)Module Mathematical Subject Classification (1991): 16W30, 17B37, 18D10, 81R50 1 Introduction For bialgebras over a field k the smash product and the smash coproduct are investigated extensively in the literature [Rad, Mol]. Let H be a bialgebra, B be an H-right module algebra and an H-right comodule coalgebra. If the smash product algebra structure and the smash coproduct coalgebra structure on H\Omega B form a bialgebra then Ra...

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Citation Context ...on to knot theory and quantum groups was exhibited. Hopf bimodules are a special form of Hopf modules and appear as the basic notion in Woronowicz's approach to differential calculi on quantum groups =-=[Wor]-=- where they are called bicovariant bimodules. A connection of Hopf bimodules and crossed modules in terms of a choosen basis over k has been found in [Wor]. For a symmetric monoidal category which adm... |

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Citation Context ...lgebra, B be an H-right module algebra and an H-right comodule coalgebra. If the smash product algebra structure and the smash coproduct coalgebra structure on H\Omega B form a bialgebra then Radford =-=[Rad]-=- speaks of an admissible pair (H; B). It is noted by Majid [Ma1] that in the work [Rad] the notion of a crossed module implicitely occurs as one of the conditions for a pair (H; B) to be admissible. C... |

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Citation Context ... \Gamma1 \Gamma resp: \Delta := \Psi HH \Gamma1 ffi \Delta \Delta : (5) The opposite (co-)multiplication in (5) may not be confused with the opposite (co-)multiplication defined in [Ma4]. Lemma 2.2.1 =-=[Ma2]-=- H op and H op are Hopf algebras in C with antipode S \Gamma1 . 3 Hopf modules From [Lyu2] it is known that the Structure Theorem of Hopf modules [Swe] also holds for Hopf algebras in braided monoidal... |

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Citation Context ...modules over a particular Hopf algebra. In the present paper we generalize this result to arbitrary braided monoidal (base) categories C. The fundamental theorem of Hopf modules in braided categories =-=[Lyu2]-=- will be proven without the assumption of the existence of (co-)equalizers but with the help of the weaker condition that idempotents in C split. We recall the results of [B2] on crossed modules and d... |

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Citation Context ...oalgebra. If the smash product algebra structure and the smash coproduct coalgebra structure on H\Omega B form a bialgebra then Radford [Rad] speaks of an admissible pair (H; B). It is noted by Majid =-=[Ma1]-=- that in the work [Rad] the notion of a crossed module implicitely occurs as one of the conditions for a pair (H; B) to be admissible. Crossed modules have been explicitely investigated in [Yet, RT] w... |

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Citation Context ... m := m ffi \Psi HH \Gamma1 \Gamma resp: \Delta := \Psi HH \Gamma1 ffi \Delta \Delta : (5) The opposite (co-)multiplication in (5) may not be confused with the opposite (co-)multiplication defined in =-=[Ma4]-=-. Lemma 2.2.1 [Ma2] H op and H op are Hopf algebras in C with antipode S \Gamma1 . 3 Hopf modules From [Lyu2] it is known that the Structure Theorem of Hopf modules [Swe] also holds for Hopf algebras ... |

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Citation Context ... f ) := ft 2 C(X; Y )---f te = tg : The full embedding i is given through i(X) = X idX ; i(f) = f . Further information on the Karoubi enveloping category can be found in the appendix. It is noted in =-=[Lyu1]-=- that for C being a (braided) monoidal category the category b C can be equipped with a (braided) monoidal structure. 1 := (1; id 1 ) ; X e\Omega Y f := (X\Omega Y ) e\Omega f ; ff Xe ;Y f ;Zg := ((e\... |

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Citation Context ...ossed modules in terms of a choosen basis over k has been found in [Wor]. For a symmetric monoidal category which admits (co-)equalizers, a coordinate free version of Woronowicz's result was found in =-=[Sch]-=-. The main theorem in [Sch] states the equivalence of the category of Hopf bimodules and the category of crossed modules over a particular Hopf algebra. In the present paper we generalize this result ... |

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Citation Context ...imodule H leads to the Hcrossed modules H ad and H ad . This yields new solutions of the Yang-Baxter equation for H which arise through the braiding of these ojects in the category of crossed modules =-=[Ma6]-=-. Proposition 4.2.2 will now be used to construct the functor Hn(\Gamma) from the Hopf bimodules into the crossed modules. Proposition 4.2.3 Let H be a Hopf algebra in C. Then for an H-Hopf bimodule X... |

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Citation Context ...ty of the tensor product, ae andsare the natural isomorphisms for the right and left tensor multiplication with the unit object respectively, and \Psi is the braiding. By Mac Lane's coherence theorem =-=[Mac]-=-, C is equivalent to a strict monoidal category, i.e. a category where ff; ae andsare identity morphisms. This allows us to neglect the morphisms ff; ae andsin most of the calculations. We suppose tha... |

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Citation Context ... in braided categories [Lyu2] will be proven without the assumption of the existence of (co-)equalizers but with the help of the weaker condition that idempotents in C split. We recall the results of =-=[B2]-=- on crossed modules and define Hopf bimodules in the braided monoidal category C. Then we prove in particular the (pre-)braided monoidal This work was supported by the International Science Foundation... |

1 | On Braided FRT-construction
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Citation Context ...ough the generalization to braided categories (see below) it is possible to construct (bicovariant) differential calculi in such categories; this will be demonstrated in a forthcoming paper. See also =-=[B3] on the -=-construction of a first order differential calculus on the dual quantum braided matrix group. Definition 4.2.1 Let B be a bialgebra in C. An object (X; �� r ; �� l ;sr ;sl ) is called a B-Hopf... |

1 | Braided Supersymmetry and (Co-)Homology', preprint UVA-FWI - Drabant - 1994 |