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From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
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QuasiQuantum Groups Related To Orbifold Models
 TALK PRESENTED BY V.PASQUIER AT THE INTERNATIONAL COLLOQUIUM ON MODERN QUANTUM FIELD THEORY, TATA INSTITUTE OF FUNDAMENTAL RESEARCH, 814 JANUARY 1990
, 1990
"... We construct nontrivial quasiquantum groups associated to any finite group G and any element of H³(G, U(1)). We analyze the set of representations of these algebras and show that we recover the data of particular 3dimensional topological field theories. We also give the Rmatrix structure in non ..."
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We construct nontrivial quasiquantum groups associated to any finite group G and any element of H³(G, U(1)). We analyze the set of representations of these algebras and show that we recover the data of particular 3dimensional topological field theories. We also give the Rmatrix structure in non abelian RCFT orbifold models.
On a universal solution to reflection equation
 Lett. Math. Phys
"... For a given quasitriangular Hopf algebra H we study relations between the braided group ˜ H ∗ and Drinfeld’s twist. We show that the braided bialgebra structure of ˜ H ∗ is naturally described by means of twisted tensor powers of H and their module algebras. We introduce universal solution to the re ..."
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For a given quasitriangular Hopf algebra H we study relations between the braided group ˜ H ∗ and Drinfeld’s twist. We show that the braided bialgebra structure of ˜ H ∗ is naturally described by means of twisted tensor powers of H and their module algebras. We introduce universal solution to the reflection equation (RE) and deduce a fusion prescription for REmatrices. Key words: reflection equation, twist, fusion procedure. AMS classification codes: 17B37, 16W30. 1
Derivatives and the role of the Drinfeld twist in Noncommutative string theory,” hepth/0003234
"... We consider the derivatives which appear in the context of noncommutative string theory. First, we identify the correct derivations to use when the underlying structure of the theory is a quasitriangular Hopf algebra. Then we show that this is a specific case of a more general structure utilising th ..."
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We consider the derivatives which appear in the context of noncommutative string theory. First, we identify the correct derivations to use when the underlying structure of the theory is a quasitriangular Hopf algebra. Then we show that this is a specific case of a more general structure utilising the Drinfel’d twist. We go on to present reasons as to why we feel that the lowenergy effective action, when written in terms of the original commuting coordinates, should explicitly exhibit this twisting.
Tensor categories: A selective guided tour
, 2008
"... These are the – only lightly edited – lecture notes for a short course on tensor categories. The coverage in these notes is relatively nontechnical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something int ..."
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These are the – only lightly edited – lecture notes for a short course on tensor categories. The coverage in these notes is relatively nontechnical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something interesting in them. Once the basic definitions are given, the focus is mainly on klinear categories with finite dimensional homspaces. Connections with quantum groups and low dimensional topology are pointed out, but these notes have no pretension to cover the latter subjects at any depth. Essentially, these notes should be considered as annotations to the extensive bibliography. We also recommend the recent review [33], which covers less ground in a deeper way.
P.N.Pyatov, GLq(N)Covariant Quantum Algebras and Covariant
 Differential Calculus, Dubna Prep. JINR
, 1992
"... We consider GLq(N)covariant quantum algebras with generators satisfying quadratic polynomial relations. We show that, up to some inessential arbitrariness, there are only two kinds of such quantum algebras, namely, the algebras with qdeformed commutation and qdeformed anticommutation relations. T ..."
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We consider GLq(N)covariant quantum algebras with generators satisfying quadratic polynomial relations. We show that, up to some inessential arbitrariness, there are only two kinds of such quantum algebras, namely, the algebras with qdeformed commutation and qdeformed anticommutation relations. The connection with the bicovariant differential calculus on the linear quantum groups is disscussed.
Connected components of compact matrix quantum groups and finiteness conditions
, 2012
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Existence of Solutions of the Classical YangBaxter Equation for a Real Lie Algebra
, 2000
"... We characterize finitedimensional Lie algebras over the real numbers for which the classical YangBaxter equation has a nontrivial skewsymmetric solution (resp. a nontrivial solution with invariant symmetric part). Equivalently, we obtain a characterization of those finitedimensional real Lie a ..."
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We characterize finitedimensional Lie algebras over the real numbers for which the classical YangBaxter equation has a nontrivial skewsymmetric solution (resp. a nontrivial solution with invariant symmetric part). Equivalently, we obtain a characterization of those finitedimensional real Lie algebras which admit a nontrivial (quasi)triangular Lie bialgebra structure.
Existence of triangular Lie bialgebra structures
 J. Pure Appl. Algebra
, 1999
"... Dedicated to the memory of my father We characterize finitedimensional Lie algebras over an arbitrary field of characteristic zero which admit a nontrivial (quasi) triangular Lie bialgebra structure. 2000 Mathematics Subject Classification: 17B62, 81R05, 81R50 In general, a complete classificatio ..."
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Dedicated to the memory of my father We characterize finitedimensional Lie algebras over an arbitrary field of characteristic zero which admit a nontrivial (quasi) triangular Lie bialgebra structure. 2000 Mathematics Subject Classification: 17B62, 81R05, 81R50 In general, a complete classification of all triangular Lie bialgebra structures is very difficult. Nevertheless, Belavin and Drinfel’d succeeded in [2] to obtain such a classification for every finitedimensional simple Lie algebra over the complex numbers. The aim of this paper is much more modest in asking when there exist nontrivial (quasi) triangular Lie bialgebra structures. Michaelis