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Braided Hopf algebras over non abelian finite groups
 Acad. Nac. Ciencias (Córdoba
"... 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 al ..."
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Cited by 56 (12 self)
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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 We present some finite dimensional examples arising in an unpublished work by Milinski and Schneider.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.
Braided Hopf Algebras
, 2005
"... The term “quantum group” was popularized by Drinfeld in his address to the International Congress of Mathematicians in Berkeley. However, the concepts of quantum groups and quasitriangular Hopf algebras are the same. Therefore, Hopf algebras have close connections with various areas of mathematics a ..."
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Cited by 18 (7 self)
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The term “quantum group” was popularized by Drinfeld in his address to the International Congress of Mathematicians in Berkeley. However, the concepts of quantum groups and quasitriangular Hopf algebras are the same. Therefore, Hopf algebras have close connections with various areas of mathematics and physics. The development of Hopf algebras can be divided into five stages. The first stage is that integral and Maschke’s theorem are found. Maschke’s theorem gives a nice criterion of semisimplicity, which is due to M.E. Sweedler. The second stage is that the Lagrange’s theorem is proved, which is due to W.D. Nichols and M.B. Zoeller. The third stage is the research of the actions of Hopf algebras, which unifies actions of groups and Lie algebras. S. Montgomery’s “Hopf algebras and their actions on rings ” contains the main results in this area. S. Montgomery and R.J. Blattner show the duality theorem. S. Montegomery, M. Cohen, H.J. Schneider, W. Chin, J.R. Fisher and the author study the relation between algebra R and the crossed product R#σH of the semisimplicity, semiprimeness, Morita equivalence and radical. S. Montgomery and M. Cohen ask whether R#σH is semiprime when R is semiprime for a finitedimensional semiprime Hopf algebra H. This is a famous semiprime problem. The
Algebras versus coalgebras
 Appl. Categorical Structures, DOI
, 2007
"... Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and computer scientists. Some of the ..."
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Cited by 17 (14 self)
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Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and computer scientists. Some of them have developed specialised parts of the theory and often reinvented constructions already known in a neighbouring area. One purpose of this survey is to show the connection between results from different fields and to trace a number of them back to some fundamental papers in category theory from the early 70’s. Another intention is to look at the interplay between algebraic and coalgebraic notions. Hopf algebras are one of the most interesting objects in this setting. While knowledge of algebras and coalgebras are folklore in general category theory, the notion of Hopf algebras is usually only considered for monoidal categories. In the course of the text we do suggest how to overcome this defect by defining a Hopf monad on an arbitrary category as a monad and comonad satisfying some compatibility conditions and inducing an equivalence between
PBW deformations of braided symmetric algebras and a MilnorMoore type theorem
"... Abstract. Braided bialgebras were defined by mimicking the definition of bialgebras in a braided category; see [Ta]. In this paper we are interested in those braided bialgebras that are connected as a coalgebra, and such that, up to multiplication by a certain scalar, their braiding restricted to th ..."
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Cited by 6 (4 self)
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Abstract. Braided bialgebras were defined by mimicking the definition of bialgebras in a braided category; see [Ta]. In this paper we are interested in those braided bialgebras that are connected as a coalgebra, and such that, up to multiplication by a certain scalar, their braiding restricted to the primitive part is a Hecke operator. To every braided bialgebra as above we associate a braided Lie algebra. Conversely, for each braided Lie algebra we construct a braided bialgebra, namely its enveloping algebra. Braided symmetric and exterior algebras are examples of enveloping algebras (they correspond to a trivial Lie bracket). We show many of the properties of ordinary enveloping algebras still hold in the braided case: the graded associated (with respect to the standard filtration) is a braided symmetric algebra; the coalgebra counterpart is isomorphic to the coalgebra structure of a braided symmetric algebra; the coradical filtration and the standard filtration are identical. As in the classical case, braided symmetric and exterior algebras are Koszul algebras. The proof of this fact is based on a new characterization of Koszul algebras. These properties are used to prove a MilnorMoore type theorem for infinitesimally cocommutative connected braided bialgebras (see Theorem 5.5). We apply our result to three different classes of braided bialgebras: connected bialgebras in the category of comodules over a coquasitriangular cosemisimple Hopf algebra (e.g. superbialgebras and enveloping algebras of
Differential calculus in braided Abelian categories
, 1997
"... Braided noncommutative differential geometry is studied. We investigate the theory of (bicovariant) differential calculi in braided abelian categories. Previous results on crossed modules and Hopf bimodules in braided categories are used to construct higher order bicovariant differential calculi ov ..."
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Cited by 5 (0 self)
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Braided noncommutative differential geometry is studied. We investigate the theory of (bicovariant) differential calculi in braided abelian categories. Previous results on crossed modules and Hopf bimodules in braided categories are used to construct higher order bicovariant differential calculi over braided Hopf algebras out of first order ones. These graded objects are shown to be braided differential Hopf algebras with universal bialgebra properties. The article extends Woronowicz’s results on (bicovariant) differential calculi to the braided noncommutative case.
Fusion in the entwined category of Yetter–Drinfeld modules of a rank1 Nichols algebra
, 1109
"... ar ..."
WEAK PROJECTIONS ONTO A BRAIDED HOPF ALGEBRA
, 2006
"... Abstract. We show that, under some mild conditions, a bialgebra in an abelian and coabelian braided monoidal category has a weak projection onto a formally smooth (as a coalgebra) subbialgebra with antipode; see Theorem 1.12. In the second part of the paper we prove that bialgebras with weak project ..."
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Cited by 3 (1 self)
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Abstract. We show that, under some mild conditions, a bialgebra in an abelian and coabelian braided monoidal category has a weak projection onto a formally smooth (as a coalgebra) subbialgebra with antipode; see Theorem 1.12. In the second part of the paper we prove that bialgebras with weak projections are cross product bialgebras; see Theorem 2.12. In the particular case when the bialgebra A is cocommutative and a certain cocycle associated to the weak projection is trivial we prove that A is a double cross product, or biproduct in Madjid’s terminology. The last result is based on a universal property of double cross products which, by Theorem 2.15, works in braided monoidal categories. We also investigate the situation when the right action of the associated matched pair is trivial.