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34
Quantum Double for Quasi–Hopf Algebras
"... Abstract We introduce a quantum double quasitriangular quasiHopf algebra D(H) associated to any quasiHopf algebra H. The algebra structure is a cocycle double cross product. We use categorical reconstruction methods. As an example, we recover the quasiHopf algebra of Dijkgraaf, Pasquier and Roche ..."
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Cited by 20 (2 self)
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Abstract We introduce a quantum double quasitriangular quasiHopf algebra D(H) associated to any quasiHopf algebra H. The algebra structure is a cocycle double cross product. We use categorical reconstruction methods. As an example, we recover the quasiHopf algebra of Dijkgraaf, Pasquier and Roche as the quantum double D φ (G) associated to a finite group G and group 3cocycle φ. We also discuss D φ (Ug) associate to a Lie algebra g and Drinfeld’s cocycle φ obtained from a solution of the KZ equation.
Vertex operator algebras, generalized doubles and dual pairs
 Math. Z
"... Let V be a simple vertex operator algebra and G a finite automorphism group. Then there is a natural right Gaction on the set of all inequivalent irreducible Vmodules. Let S be a finite set of inequivalent irreducible Vmodules which is closed under the action of G. There is a finite dimensional s ..."
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Cited by 17 (4 self)
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Let V be a simple vertex operator algebra and G a finite automorphism group. Then there is a natural right Gaction on the set of all inequivalent irreducible Vmodules. Let S be a finite set of inequivalent irreducible Vmodules which is closed under the action of G. There is a finite dimensional semisimple associative algebra Aα(G, S) for a suitable 2cocycle naturally determined by the Gaction on S such that Aα(G, S) and the vertex operator algebra V G form a dual pair on the sum of Vmodules in S in the sense of Howe. In particular, every irreducible Vmodule is completely reducible V Gmodule. 1
Lazy cohomology: an analogue of the Schur multiplier for arbitrary Hopf algebra
 J. Pure Appl. Algebra
"... We propose a detailed systematic study of a group H2 L (A) associated, by elementary means of lazy 2cocycles, to any Hopf algebra A. This group was introduced by Schauenburg in order to generalize G.I. Kac’s exact sequence. We study the various interplays of lazy cohomology in Hopf algebra theory: ..."
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Cited by 13 (3 self)
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We propose a detailed systematic study of a group H2 L (A) associated, by elementary means of lazy 2cocycles, to any Hopf algebra A. This group was introduced by Schauenburg in order to generalize G.I. Kac’s exact sequence. We study the various interplays of lazy cohomology in Hopf algebra theory: Galois and biGalois objects, Brauer groups and projective representations. We obtain a KacSchauenburgtype sequence for double crossed products of possibly infinitedimensional Hopf algebras. Finally the explicit computation of H2 L (A) for monomial Hopf algebras and for a class of cotriangular Hopf algebras is performed. Key words: Hopf 2cocycle, Galois objects, biGalois objects.
Universal Deformation Formulae, Symplectic Lie groups and Symmetric Spaces
, 2003
"... We apply methods from strict quantization of solvable symmetric spaces to obtain universal deformation formulae for actions of a class of solvable Lie groups. We also study compatible coproducts by generalizing the notion of smash product in the context of Hopf algebras. 1 ..."
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Cited by 9 (4 self)
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We apply methods from strict quantization of solvable symmetric spaces to obtain universal deformation formulae for actions of a class of solvable Lie groups. We also study compatible coproducts by generalizing the notion of smash product in the context of Hopf algebras. 1
Universal Deformation Formulae for ThreeDimensional Solvable Lie groups
, 2003
"... We apply methods from strict quantization of solvable symmetric spaces to obtain universal deformation formulae for actions of every threedimensional solvable Lie group. We also study compatible coproducts by generalizing the notion of smash product in the context of Hopf algebras. We investigate ..."
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Cited by 7 (1 self)
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We apply methods from strict quantization of solvable symmetric spaces to obtain universal deformation formulae for actions of every threedimensional solvable Lie group. We also study compatible coproducts by generalizing the notion of smash product in the context of Hopf algebras. We investigate in particular the dressing action of the ‘book ’ group on SU(2). 1
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 ..."
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Cited by 7 (0 self)
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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.
CROSSED PRODUCTS BY A COALGEBRA
, 1996
"... We introduce the notion of a crossed product of an algebra by a coalgebra C, which generalises the notion of a crossed product by a bialgebra wellstudied in the theory of Hopf algebras. The result of such a crossed product is an algebra which is also a right Ccomodule. We find the necessary and s ..."
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Cited by 5 (0 self)
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We introduce the notion of a crossed product of an algebra by a coalgebra C, which generalises the notion of a crossed product by a bialgebra wellstudied in the theory of Hopf algebras. The result of such a crossed product is an algebra which is also a right Ccomodule. We find the necessary and sufficient conditions for two coalgebra crossed products be equivalent. We show that the twodimensional quantum Euclidean group is a coalgebra crossed product. The paper is completed with an appendix describing the dualisation of construction of coalgebra crossed products.
Topological Hopf algebras, quantum groups and deformation quantization, in ”Hopf algebras in noncommutative geometry and physics
 55–70, Lecture
"... After a presentation of the context and a brief reminder of deformation quantization, we indicate how the introduction of natural topological vector space topologies on Hopf algebras associated with Poisson Lie groups, Lie bialgebras and their doubles explains their dualities and provides a comprehe ..."
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Cited by 4 (1 self)
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After a presentation of the context and a brief reminder of deformation quantization, we indicate how the introduction of natural topological vector space topologies on Hopf algebras associated with Poisson Lie groups, Lie bialgebras and their doubles explains their dualities and provides a comprehensive framework. Relations with deformation quantization and applications to the deformation quantization of symmetric spaces are described.
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 ( ..."
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Cited by 3 (0 self)
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: 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...