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42
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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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
, 2000
"... 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.
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 15 (4 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
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 8 (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.
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
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.
The Dirichlet Hopf algebra of arithmetics
 JOURNAL OF KNOT THEORY AND ITS RAMIFICATIUONS
, 2006
"... Many constructs in mathematical physics entail notational complexities, deriving from the manipulation of various types of index sets which often can be reduced to labelling by various multisets of integers. In this work, we develop systematically the “Dirichlet Hopf algebra of arithmetics ” by dual ..."
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Cited by 4 (3 self)
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Many constructs in mathematical physics entail notational complexities, deriving from the manipulation of various types of index sets which often can be reduced to labelling by various multisets of integers. In this work, we develop systematically the “Dirichlet Hopf algebra of arithmetics ” by dualizing the addition and multiplication maps. Then we study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra structures, but form only a weaker Hopf gebra obeying a weakened homomorphism axiom. A careful identification of the algebraic structures involved is done featuring subtraction, division and derivations derived from coproducts and chochains using branching operators. The consequences of the weakened structure of a Hopf gebra on cohomology are explored, showing this has major impact on number theory. This features multiplicativity versus complete multiplicativity of number theoretic arithmetic functions. The deficiency of not being a Hopf algebra is then cured by introducing an ‘unrenormalized’ coproduct and an ‘unrenormalized ’ pairing. It is then argued that exactly the failure of the homomorphism property (complete multiplicativity) for noncoprime integers is a blueprint for the problems in quantum field theory (QFT) leading to the need for renormalization. Renormalization turns out to be the morphism from the algebraically sound Hopf algebra to the physical and number
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.