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Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
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Cited by 218 (13 self)
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For a copy with the handdrawn figures please email
Module categories, weak Hopf algebras and modular invariants
 Transform. Groups
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On qanalog of McKay correspondence and ADE classification of sl (2) conformal field theories
 Adv. Math
"... Abstract. The goal of this paper is to classify “finite subgroups in Uq(sl2)” where q = e πi/l is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of Uq(sl2); we show that this definition is a natural generalization of the notion of a subgroup i ..."
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Cited by 84 (8 self)
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Abstract. The goal of this paper is to classify “finite subgroups in Uq(sl2)” where q = e πi/l is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of Uq(sl2); we show that this definition is a natural generalization of the notion of a subgroup in a reductive group, and that it is also related with extensions of the chiral (vertex operator) algebra corresponding to ̂ sl2 at level k = l − 2. We show that “finite subgroups in Uq(sl2) ” are classified by Dynkin diagrams of types An, D2n, E6, E8 with Coxeter number equal to l, give a description of this correspondence similar to the classical McKay correspondence, and discuss relation with modular invariants in ( ̂ sl2)k conformal field theory.
CATEGORY THEORY FOR CONFORMAL BOUNDARY CONDITIONS
, 2001
"... We study properties of the category of modules of an algebra object A in a tensor category C. We show that the module category inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a byproduct we obtain results about the FrobeniusSchur ind ..."
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Cited by 75 (18 self)
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We study properties of the category of modules of an algebra object A in a tensor category C. We show that the module category inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a byproduct we obtain results about the FrobeniusSchur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the description of boundary conditions in twodimensional conformal field theory and present illustrative examples. We show that module categories give rise to NIMreps of the fusion rules, and discuss a possible relation with the representation theory of vertex operator algebras.
Module categories over representations of SLq(2) and graphs
"... Abstract. We classify semisimple module categories over the tensor category of representations of quantum SL(2). 1. ..."
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Cited by 23 (4 self)
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Abstract. We classify semisimple module categories over the tensor category of representations of quantum SL(2). 1.
Skein Theory for the D2n Planar Algebras
, 2008
"... Abstract We give a combinatorial description of the “D2n planar algebra”, by giving generators and relations. This includes a direct proof that the relations are consistent, as well as a complete description of the associated tensor category and its principal graph. While we use the usual langauge o ..."
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Cited by 18 (4 self)
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Abstract We give a combinatorial description of the “D2n planar algebra”, by giving generators and relations. This includes a direct proof that the relations are consistent, as well as a complete description of the associated tensor category and its principal graph. While we use the usual langauge of subfactor planar algebras, the entire argument is ‘skeintheoretic’, and there is no functional analysis. AMS Classification 46L37; 57M15 46M99
Orders and dimensions for sl(2) or sl(3) module categories and Boundary Conformal Field Theories on a torus
 J MATH PHYS 48 P 043511
, 2006
"... After giving a short description, in terms of action of categories, of some of the structures associated with sl(2) and sl(3) boundary conformal field theories on a torus, we provide tables of dimensions describing the semisimple and cosemisimple blocks of the corresponding weak bialgebras (quantum ..."
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Cited by 11 (5 self)
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After giving a short description, in terms of action of categories, of some of the structures associated with sl(2) and sl(3) boundary conformal field theories on a torus, we provide tables of dimensions describing the semisimple and cosemisimple blocks of the corresponding weak bialgebras (quantum groupoids), tables of quantum dimensions and orders, and tables describing induction restriction. For reasons of size, the sl(3) tables of induction are only given for theories with selffusion (existence of a monoidal structure).
Topological lattice field theories from intertwiner dynamics,” arXiv:1311.1798 [grqc
"... We introduce a class of 2D lattice models that describe the dynamics of intertwiners, or, in a condensed matter interpretation, the fusion and splitting of anyons. We identify different families and instances of triangulation invariant, that is, topological, models inside this class. These models gi ..."
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Cited by 3 (3 self)
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We introduce a class of 2D lattice models that describe the dynamics of intertwiners, or, in a condensed matter interpretation, the fusion and splitting of anyons. We identify different families and instances of triangulation invariant, that is, topological, models inside this class. These models give examples for symmetry protected topologically ordered 1D quantum phases with quantum group symmetries. Furthermore the models provide realizations for anyon condensation into a new effective vacuum. We explain the relevance of our findings for the problem of identifying the continuum limit of spin foam and spin net models. 1
CATEGORY THEORY FOR CONFORMAL Abstract BOUNDARY CONDITIONS
, 2001
"... We study properties of the category of modules of an algebra object A in a tensor category C. We show that the module category inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a byproduct we obtain results about the FrobeniusSchur ind ..."
Abstract
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We study properties of the category of modules of an algebra object A in a tensor category C. We show that the module category inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a byproduct we obtain results about the FrobeniusSchur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the description of boundary conditions in twodimensional conformal field theory and present illustrative examples. We show that module categories give rise to NIMreps of the fusion rules, and discuss a possible relation with the representation theory of vertex operator algebras. 1 1 CFT boundary conditions Boundary conditions in conformal field theory have various physical applications, ranging from the study of defects in condensed matter physics to the theory of open strings. Such boundary conditions are partially characterized by the maximal vertex operator subalgebra A of the bulk chiral algebra Abulk that they respect [41, 67]. That A is respected by a boundary condition means that the correlation functions in the presence of the boundary condition satisfy the Ward