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42
From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories
 J. Pure Appl. Alg
, 2003
"... We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground f ..."
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Cited by 108 (9 self)
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We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A = FVect, where F is a field. An object X ∈ A with twosided dual X gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A ≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple Hopf algebras, for which we prove H − mod ≈ ˆH − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205. 1
From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
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HOMYANGBAXTER EQUATION, HOMLIE ALGEBRAS, AND QUASITRIANGULAR BIALGEBRAS
, 2009
"... We study a twisted version of the YangBaxter Equation, called the HomYangBaxter Equation (HYBE), which is motivated by HomLie algebras. Three classes of solutions of the HYBE are constructed, one from HomLie algebras and the others from Drinfeld’s (dual) quasitriangular bialgebras. Each solutio ..."
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Cited by 42 (18 self)
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We study a twisted version of the YangBaxter Equation, called the HomYangBaxter Equation (HYBE), which is motivated by HomLie algebras. Three classes of solutions of the HYBE are constructed, one from HomLie algebras and the others from Drinfeld’s (dual) quasitriangular bialgebras. Each solution of the HYBE can be extended to operators that satisfy the braid relations. Assuming an invertibility condition, these operators give a representation of the braid group.
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 30 (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.
The HomYangBaxter equation and HomLie algebras
, 2009
"... Abstract. Motivated by recent work on HomLie algebras, a twisted version of the YangBaxter equation, called the HomYangBaxter equation (HYBE), was introduced by the author in [62]. In this paper, several more classes of solutions of the HYBE are constructed. Some of these solutions of the HYBE a ..."
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Cited by 25 (13 self)
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Abstract. Motivated by recent work on HomLie algebras, a twisted version of the YangBaxter equation, called the HomYangBaxter equation (HYBE), was introduced by the author in [62]. In this paper, several more classes of solutions of the HYBE are constructed. Some of these solutions of the HYBE are closely related to the quantum enveloping algebra of sl(2), the JonesConway polynomial, and YetterDrinfel’d modules. We also construct a new infinite sequence of solutions of the HYBE from a given one. Along the way, we compute all the Lie algebra endomorphisms on the (1 + 1)Poincaré algebra and sl(2). 1.
Homquantum groups I: quasitriangular Hombialgebras
"... Abstract. We introduce a Homtype generalization of quantum groups, called quasitriangular Hombialgebras. They are nonassociative and noncoassociative analogues of Drinfel’d’s quasitriangular bialgebras, in which the non(co)associativity is controlled by a twisting map. A family of quasitriang ..."
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Cited by 21 (10 self)
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Abstract. We introduce a Homtype generalization of quantum groups, called quasitriangular Hombialgebras. They are nonassociative and noncoassociative analogues of Drinfel’d’s quasitriangular bialgebras, in which the non(co)associativity is controlled by a twisting map. A family of quasitriangular Hombialgebras can be constructed from any quasitriangular bialgebra, such as Drinfel’d’s quantum enveloping algebras. Each quasitriangular Hombialgebra comes with a solution of the quantum HomYangBaxter equation, which is a nonassociative version of the quantum YangBaxter equation. Solutions of the HomYangBaxter equation can be obtained from modules of suitable quasitriangular Hombialgebras. 1.
*Products on Quantum Spaces
, 2001
"... In this paper we present explicit formulas for the ∗product on quantum spaces which are of particular importance in physics, i.e., the qdeformed Minkowski space and the qdeformed Euclidean space in 3 and 4 dimensions, respectively. Our formulas are complete and formulated using the deformation pa ..."
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Cited by 16 (13 self)
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In this paper we present explicit formulas for the ∗product on quantum spaces which are of particular importance in physics, i.e., the qdeformed Minkowski space and the qdeformed Euclidean space in 3 and 4 dimensions, respectively. Our formulas are complete and formulated using the deformation parameter q. In addition, we worked out an expansion in powers of h = ln q up to second order, for all considered cases.
Conformal Field Theory and DoplicherRoberts Reconstruction
 In
"... Abstract. After a brief review of recent rigorous results concerning the representation theory of rational chiral conformal field theories (RCQFTs) we focus on pairs (A, F) of conformal field theories, where F has a finite group G of global symmetries and A is the fixpoint theory. The comparison of ..."
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Cited by 16 (5 self)
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Abstract. After a brief review of recent rigorous results concerning the representation theory of rational chiral conformal field theories (RCQFTs) we focus on pairs (A, F) of conformal field theories, where F has a finite group G of global symmetries and A is the fixpoint theory. The comparison of the representation categories of A and F is strongly intertwined with various issues related to braided tensor categories. We
Homquantum groups II:cobraided Hombialgebras and Homquantum geometry
 UNIVERSITÉ DE HAUTEALSACE, LABORATOIRE DE MATHÉMATIQUES, INFORMATIQUE ET APPLICATIONS, 4 RUE DES FRÈRES LUMIÈRE, 68093
, 2009
"... A class of nonassociative and noncoassociative generalizations of cobraided bialgebras, called cobraided Hombialgebras, is introduced. The non(co)associativity in a cobraided Hombialgebra is controlled by a twisting map. Several methods for constructing cobraided Hombialgebras are given. In p ..."
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Cited by 16 (9 self)
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A class of nonassociative and noncoassociative generalizations of cobraided bialgebras, called cobraided Hombialgebras, is introduced. The non(co)associativity in a cobraided Hombialgebra is controlled by a twisting map. Several methods for constructing cobraided Hombialgebras are given. In particular, Homtype generalizations of FRT quantum groups, including quantum matrices and related quantum groups, are obtained. Each cobraided Hombialgebra comes with solutions of the operator quantum HomYangBaxter equations, which are twisted analogues of the operator form of the quantum YangBaxter equation. Solutions of the HomYangBaxter equation can be obtained from comodules of suitable cobraided Hombialgebras. Homtype generalizations of the usual quantum matrices coactions on the quantum planes give rise to nonassociative and noncoassociative analogues of quantum geometry.