## Braided Hopf algebras over non abelian finite groups

Venue: | Acad. Nac. Ciencias (Córdoba |

Citations: | 38 - 11 self |

### BibTeX

@ARTICLE{Andruskiewitsch_braidedhopf,

author = {Nicolás Andruskiewitsch and Matías Graña},

title = {Braided Hopf algebras over non abelian finite groups},

journal = {Acad. Nac. Ciencias (Córdoba},

year = {}

}

### OpenURL

### Abstract

Abstract. In the last years a new theory of Hopf algebras has begun to be developed: that of Hopf algebras in braided categories, or, briefly, braided Hopf algebras. This is a survey of general aspects of the theory with emphasis in H HYD, the Yetter–Drinfeld category over H, where H is the group algebra of a non abelian finite group Γ. We discuss a special class of braided graded Hopf algebras from different points of view following Lusztig, Nichols and Schauenburg. We present some finite dimensional examples arising in an unpublished work by Milinski and Schneider. Sinopsis. En los últimos años comenzó a ser desarrollada una nueva teoría de álgebras de Hopf en categorías trenzadas, o brevemente, álgebras de Hopf trenzadas. Presentamos aquí aspectos generales de la teoría con énfasis en H HYD, la categoría de Yetter–Drinfeld sobre H, donde H es el álgebra de grupo de un grupo finito no abeliano Γ. Discutimos una clase especial de álgebras de Hopf trenzadas graduadas desde diferentes puntos de vista, siguiendo a Lusztig, Nichols y Schauenburg. Presentamos algunos ejemplos de dimensión finita que aparecen en un trabajo inédito de Milinski y Schneider. 0. Introduction and notations 0.1. Introduction.

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Citation Context ...ion of the simple objects in terms of irreducible representations of some subgroups of Γ. This seems to be folklore; it can be found e.g. in [CR, Prop 3.3] in the language of Hopf bimodules (see also =-=[Ci]-=-). Thanks to 1.1.16, in order to give all the simple objects of H HYD it is enough to give a collection of mutually non isomorphic simple objects for which the sum of the squares of their dimensions i... |

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Citation Context ...a Hopf algebra of type one, in the sense of Nichols [N]. An important example of TOBA is the quantum analog of the enveloping algebra of the nilpotent part of a Borel algebra, see [Lz], [Scg3], [R1], =-=[R2]-=-. The article is organized as follows: In section 1 we define and give examples of braided categories, Hopf algebras in braided categories and review the bosonization construction. In section 2 we giv... |

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Citation Context ...8]), from where we borrow the name. The first one can be seen as a rewriting of that of Nichols in the language of braided categories, and is due to Schauenburg (see [Sch96], see also [Ros95, Ros92], =-=[BD97]-=-). The last one is inspired in the work of Lusztig [Lus93] and is stated for the category H HYD∞, where H = kΓ (Lusztig’s algebras f and ′ f are braided Hopf algebras in a category of comodules). The ... |

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Citation Context ... elements is isomorphic to V . The first two constructions (which work for a general Hopf algebra H) use quantum shuffles and universal properties, and are essentially contained in [N], [Scg1], [R1], =-=[Ro]-=-, [W]. The third construction, by means of a bilinear form, is done for H = kΓ and seems to be new. It is however inspired by [Lz], [Scg1], [R1], [Mü]. We finally discuss some explicit examples from [... |

3 |
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Citation Context ...e of primitive elements is isomorphic to V . The first two constructions (which work for a general Hopf algebra H) use quantum shuffles and universal properties, and are essentially contained in [N], =-=[Scg1]-=-, [R1], [Ro], [W]. The third construction, by means of a bilinear form, is done for H = kΓ and seems to be new. It is however inspired by [Lz], [Scg1], [R1], [Mü]. We finally discuss some explicit exa... |

2 |
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2 |
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2 |
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1 |
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1 |
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