Results 1 
8 of
8
Extensions of locally compact quantum groups and the bicrossed product construction
, 2001
"... ..."
The Classification of Semisimple Hopf Algebras of dimension 16
 J. of Algebra
"... Abstract. In this paper we completely classify nontrivial semisimple Hopf algebras of dimension 16. We also compute all the possible structures of the Grothendieck ring of semisimple noncommutative Hopf algebras of dimension 16. Moreover, we prove that noncommutative semisimple Hopf algebras of di ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
Abstract. In this paper we completely classify nontrivial semisimple Hopf algebras of dimension 16. We also compute all the possible structures of the Grothendieck ring of semisimple noncommutative Hopf algebras of dimension 16. Moreover, we prove that noncommutative semisimple Hopf algebras of dimension p n, p is prime, cannot have a cyclic group of grouplikes. 1. Introduction. Recently various classification results were obtained for finitedimensional semisimple Hopf algebras over an algebraically closed field of characteristic 0. The smallest dimension, for which the question was still open, was 16. In this paper we completely classify all nontrivial (i.e. noncommutative and noncocommutative) Hopf algebras of dimension 16. Moreover, we consider all
Hopf Algebra Extensions and Cohomology
"... Abstract. This is an expository paper on ‘abelian ’ extensions of (quasi) Hopf algebras, which can be managed by the abelian cohomology, with emphasis on the author’s recent results which are motivated by an exact sequence due to George Kac. The cohomology plays here an important role in constructi ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
Abstract. This is an expository paper on ‘abelian ’ extensions of (quasi) Hopf algebras, which can be managed by the abelian cohomology, with emphasis on the author’s recent results which are motivated by an exact sequence due to George Kac. The cohomology plays here an important role in constructing and classifying those extensions, and even their cocycle deformations. We see also a strong connection of Hopf algebra extensions arising from a (matched) pair of Lie algebras with Lie bialgebra extensions.
Topological Quantum Double
, 1993
"... : Following a preceding paper showing how the introduction of a t.v.s. topology on quantum groups leads to a remarkable unification and rigidification of the different definitions, we adapt here, in the same way, the definition of quantum double. This topological double is dualizable and reflexive ( ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
: Following a preceding paper showing how the introduction of a t.v.s. topology on quantum groups leads to a remarkable unification and rigidification of the different definitions, we adapt here, in the same way, the definition of quantum double. This topological double is dualizable and reflexive (even for infinite dimensional algebras). In a simple case we show, considering the double as the "zero class" of an extension theory, the uniqueness of the double structure as a quasiHopf algebra. R'esum'e : A la suite d'un pr'ec'edent article montrant comment l'introduction d'une topologie d'e.v.t. sur les groupes quantiques permet une unification et une rigidification remarquables des diff'erentes d'efinitions, on adapte ici de la meme mani`ere la d'efinition du double quantique. Ce double topologique est alors dualisable et reflexif (meme pour des alg`ebres de dimension infinie). Dans un cas simple on montre, en consid'erant le double comme la "classe z'ero" d'une th'eorie d'extension...
COHOMOLOGY OF ABELIAN MATCHED PAIRS AND THE KAC SEQUENCE
, 2002
"... Abstract. The purpose of this paper is to introduce a cohomology theory for abelian matched pairs of Hopf algebras and to explore its relationship to Sweedler cohomology, to Singer cohomology and to extension theory. An exact sequence connecting these cohomology theories is obtained for a general ab ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
Abstract. The purpose of this paper is to introduce a cohomology theory for abelian matched pairs of Hopf algebras and to explore its relationship to Sweedler cohomology, to Singer cohomology and to extension theory. An exact sequence connecting these cohomology theories is obtained for a general abelian matched pair of Hopf algebras, generalizing those of Kac and Masuoka for matched pairs of finite groups and finite dimensional Lie algebras. The morphisms in the low degree part of this sequence are given explicitly, enabling concrete computations. In this paper we discuss various cohomology theories for Hopf algebras and their relation to extension theory. It is natural to think of building new algebraic objects from simpler structures, or to get information about the structure of complicated objects by
DAMTP98117 Cross Product Bialgebras
, 1998
"... This is the central part of a series of three articles on cross product bialgebras. We present a universal theory of cross product bialgebras with cocycles and dual cocycles. The construction provides an equivalent (co)modular cocyclic formulation. All known constructions as for instance bi or sm ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
This is the central part of a series of three articles on cross product bialgebras. We present a universal theory of cross product bialgebras with cocycles and dual cocycles. The construction provides an equivalent (co)modular cocyclic formulation. All known constructions as for instance bi or smash, doublecross and bicross product bialgebras as well as double biproduct bialgebras and bicrossed or cocycle bicross product bialgebras are now united within a single theory. Furthermore our construction bears various novel types of cross product bialgebras.
Hopf Algebra Extensions and Monoidal Categories
, 2002
"... Tannaka reconstruction provides a close link between monoidal categories and (quasi)Hopf algebras. We discuss some applications of the ideas of Tannaka reconstruction to the theory of Hopf algebra extensions, based on the following construction: For certain inclusions of a Hopf algebra into a coq ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Tannaka reconstruction provides a close link between monoidal categories and (quasi)Hopf algebras. We discuss some applications of the ideas of Tannaka reconstruction to the theory of Hopf algebra extensions, based on the following construction: For certain inclusions of a Hopf algebra into a coquasibialgebra one can consider a natural monoidal category consisting of Hopf modules, and one can reconstruct a new coquasibialgebra from that monoidal category.
CROSS PRODUCT QUANTISATION, NONABELIAN COHOMOLOGY AND TWISTING OF HOPF ALGEBRAS 1
, 1993
"... ABSTRACT This is an introduction to work on the generalisation to quantum groups of Mackey’s approach to quantisation on homogeneous spaces. We recall the bicrossproduct models of the author, which generalise the quantum double. We describe the general extension theory of Hopf algebras and the nonAb ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
ABSTRACT This is an introduction to work on the generalisation to quantum groups of Mackey’s approach to quantisation on homogeneous spaces. We recall the bicrossproduct models of the author, which generalise the quantum double. We describe the general extension theory of Hopf algebras and the nonAbelian cohomology spaces H 2 (H, A) which classify them. They form a new kind of topological quantum number in physics which is visible only in the quantum world. These same cross product quantisations can also be viewed as trivial quantum principal bundles in quantum group gauge theory. We also relate this nonAbelian cohomology H 2 (H, C) to Drinfeld’s theory of twisting.