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28
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 52 (6 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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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.
*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 12 (3 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
Grassmann variables on quantum spaces
, 2005
"... In this article we consider quantum spaces which could be of particular importance in physics, i.e. the 2dimensional quantum plane, the qdeformed Euclidean space with 3 or 4 dimensions as well as the qdeformed Minkowski space. For each of these spaces we present some standard techniques for deali ..."
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Cited by 10 (10 self)
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In this article we consider quantum spaces which could be of particular importance in physics, i.e. the 2dimensional quantum plane, the qdeformed Euclidean space with 3 or 4 dimensions as well as the qdeformed Minkowski space. For each of these spaces we present some standard techniques for dealing with qdeformed Grassmann variables. Especially, we give formulae for multiplying two supernumbers and show how symmetry generators and fermionic derivatives act on antisymmetrized quantum spaces. Furthermore, we review for all types of quantum spaces their Hopf structures. From the corresponding formulae for the coproduct we are then able to read off a realization of the Lmatrices in terms of the symmetry generators. This means that the commutation relations between all types of quantum spaces are calculable as soon as the actions of the symmetry generators are known.
HOMYANGBAXTER EQUATION, HOMLIE ALGEBRAS, AND QUASITRIANGULAR BIALGEBRAS
, 903
"... Abstract. 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. Ea ..."
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Cited by 9 (5 self)
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Abstract. 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. 1.
Quantum groups and noncommutative geometry
 J. Math. Phys
"... Abstract Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalisation of symmetry groups for certain integrable systems, and on the other as part of a generalisation of geometry itself powerful enough to make sense in the quantum domain. Jus ..."
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Cited by 9 (0 self)
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Abstract Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalisation of symmetry groups for certain integrable systems, and on the other as part of a generalisation of geometry itself powerful enough to make sense in the quantum domain. Just as the last century saw the birth of classical geometry, so the present century sees at its end the birth of this quantum or noncommutative geometry, both as an elegant mathematical reality and in the form of the first theoretical predictions for Planckscale physics via ongoing astronomical measurements. Noncommutativity of spacetime, in particular, amounts to a postulated new force or physical effect called cogravity.
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 5 (3 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.
Analysis on qdeformed quantum spaces
, 604
"... A qdeformed version of classical analysis is given to quantum spaces of physical importance, i.e. Manin plane, qdeformed Euclidean space in three or four dimensions, and qdeformed Minkowski space. The subject is presented in a rather complete and selfcontained way. All relevant notions are introd ..."
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Cited by 5 (5 self)
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A qdeformed version of classical analysis is given to quantum spaces of physical importance, i.e. Manin plane, qdeformed Euclidean space in three or four dimensions, and qdeformed Minkowski space. The subject is presented in a rather complete and selfcontained way. All relevant notions are introduced and explained in detail. The different possibilities to realize the objects of qdeformed analysis are discussed and their elementary properties are studied. In this manner attention is focused on star products, qdeformed tensor products, qdeformed translations, qdeformed partial derivatives, dual pairings, qdeformed exponentials, and qdeformed integration. The main concern of this work is to show that these objects fit together in a consistent framework, which is suitable to formulate physical theories on quantum spaces.