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34
CONFORMAL CORRELATION FUNCTIONS, FROBENIUS ALGEBRAS AND TRIANGULATIONS
, 2001
"... We formulate twodimensional rational conformal field theory as a natural generalization of twodimensional lattice topological field theory. To this end we lift various structures from complex vector spaces to modular tensor categories. The central ingredient is a special Frobenius algebra object A ..."
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Cited by 36 (18 self)
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We formulate twodimensional rational conformal field theory as a natural generalization of twodimensional lattice topological field theory. To this end we lift various structures from complex vector spaces to modular tensor categories. The central ingredient is a special Frobenius algebra object A in the modular category that encodes the MooreSeiberg data of the underlying chiral CFT. Just like for lattice TFTs, this algebra is itself not an observable quantity. Rather, Morita equivalent algebras give rise to equivalent theories. Morita equivalence also allows for a simple understanding of Tduality. We present a construction of correlators, based on a triangulation of the world sheet, that generalizes the one in lattice TFTs. These correlators are modular invariant and satisfy factorization rules. The construction works for arbitrary orientable world sheets, in particular for surfaces with boundary. Boundary conditions correspond to representations of the algebra A. The partition functions on the torus and on the annulus provide modular invariants and NIMreps of the fusion rules, respectively.
Bridged links and tangle presentations of cobordism categories
 Adv. Math
, 1999
"... Abstract: We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe onehandle attachements. As opposed to the usual link calculi of Kirby and others this description uses only elementary, local moves(namely modifications and isolated ..."
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Cited by 23 (5 self)
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Abstract: We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe onehandle attachements. As opposed to the usual link calculi of Kirby and others this description uses only elementary, local moves(namely modifications and isolated cancellations), and it is valid also on nonsimply connected and disconnected manifolds. In particular, it allows us to give a presentation of a 3manifold by doing surgery on any other 3manifold with the same boundary. Bridged link presentations on unions of handlebodies are used to give a Cerftheoretical derivation of presentations of 2+1dimensional cobordisms categories in terms of planar ribbon tangles and their composition rules. As an application we give a different, more natural proof of the MatveevPolyak presentations of the mapping class group, and, furthermore, find systematically surgery presentations of general mapping tori. We discuss a natural extension of the Reshetikhin Turaev invariant to the calculus of bridged links. Invariance follows now similar as for knot invariants from simple identifications of the elementary moves with elementary categorial relations for invariances or cointegrals, respectively. Hence, we avoid the lengthy computations and the unnatural FennRourke reduction of the original
KAZHDAN–LUSZTIG CORRESPONDENCE FOR THE REPRESENTATION CATEGORY OF THE TRIPLET WALGEBRA IN LOGARITHMIC CFT
, 2006
"... ... is the symmetry of the (1, p) logarithmic conformal field theory model, we propose the equivalent category Cp of finitedimensional representations of the restricted quantum group Uqsℓ(2) at q = e iπ p. We fully describe the category Cp by classifying all indecomposable representations. These ar ..."
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Cited by 20 (0 self)
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... is the symmetry of the (1, p) logarithmic conformal field theory model, we propose the equivalent category Cp of finitedimensional representations of the restricted quantum group Uqsℓ(2) at q = e iπ p. We fully describe the category Cp by classifying all indecomposable representations. These are exhausted by projective modules and three series of representations that are essentially described by indecomposable representations of the Kronecker quiver. The equivalence of the W(p) and Uqsℓ(2)representation categories is conjectured for all p �2 and proved for p = 2, the implications including the identifications of the quantumgroup center with the logarithmic conformal field theory center and of the universal Rmatrix with the braiding matrix.
Pseudo algebras and pseudo double categories
 J. Homotopy Relat. Struct
"... Abstract. As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, an ..."
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Cited by 19 (2 self)
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Abstract. As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2cells of the horizontal 2category. As a special case, strict 2algebras with one object and everything invertible are crossed modules under a group.
Frobenius monads and pseudomonoids
 2CATEGORIES COMPANION 73
, 2004
"... Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalenc ..."
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Cited by 19 (4 self)
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Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to "strongly separable " Frobenius algebras and "weak monoidal Morita equivalence". Wreath products of Frobenius algebras are discussed.
Towards an Algebraic Characterization of 3dimensional Cobordisms. ArXiv: math.GT/0106253
"... (To appear in Contemp. Math.) Abstract: The goal of this paper is to find a close to isomorphic presentation of 3manifolds in terms of Hopf algebraic expressions. To this end we define and compare three different braided tensor categories that arise naturally in the study of Hopf algebras and 3dim ..."
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Cited by 7 (0 self)
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(To appear in Contemp. Math.) Abstract: The goal of this paper is to find a close to isomorphic presentation of 3manifolds in terms of Hopf algebraic expressions. To this end we define and compare three different braided tensor categories that arise naturally in the study of Hopf algebras and 3dimensional topology. The first is the category Cob of connected surfaces with one boundary component and 3dimensional relative cobordisms, the second is a
Ovsienko, A construction of an A∞category
, 211
"... We construct an A∞category D(CB) from a given A∞category C and its full subcategory B. The construction resembles a particular case of Drinfeld’s quotient of differential graded categories [Dri02]. We use D(CB) to construct an A∞functor of Kinjective resolutions of a complex. The conventional ..."
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Cited by 6 (3 self)
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We construct an A∞category D(CB) from a given A∞category C and its full subcategory B. The construction resembles a particular case of Drinfeld’s quotient of differential graded categories [Dri02]. We use D(CB) to construct an A∞functor of Kinjective resolutions of a complex. The conventional derived category is obtained as the 0th cohomology of the quotient of differential graded category of complexes over acyclic complexes. We continue to study the 2category of A∞categories introduced in [Lyu02]. Our main subject is a quotientlike A∞category D(CB) obtained from a given A∞category C and its full subcategory B. Originally it has been defined by Drinfeld for differential graded categories [Dri02]. Bondal and Kapranov proposed to produce triangulated categories out of differential graded categories [BK90]. Drinfeld’s construction deals with their quotients, in particular, it produces derived categories. The usefulness of A∞approach is explained by our construction of an A∞functor, which assigns to a complex its Kinjective resolution, when the ground ring is a field. Plan of the article with comments. In the first section we describe conventions and notations used in the article. In particular, we recall some conventions and useful formulas from [Lyu02].
Double categories and quantum groupoids
, 2003
"... Abstract. We give the construction of a class of weak Hopf algebras (or quantum groupoids) associated to a matched pair of groupoids and certain cocycle data. This generalizes a now wellknown construction for Hopf algebras, first studied by G. I. Kac in the sixties. Our approach is based on the not ..."
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Cited by 6 (2 self)
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Abstract. We give the construction of a class of weak Hopf algebras (or quantum groupoids) associated to a matched pair of groupoids and certain cocycle data. This generalizes a now wellknown construction for Hopf algebras, first studied by G. I. Kac in the sixties. Our approach is based on the notion of double groupoids, as introduced by Ehresmann.
Bottom tangles and universal invariants
, 2006
"... A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite ..."
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Cited by 5 (2 self)
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A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite set of generators of B, which provides an especially convenient way to generate all the bottom tangles, and hence all the framed, oriented links, via closure. We also define a kind of “braided Hopf algebra action ” on the set of bottom tangles. Using the universal invariant of bottom tangles associated to each ribbon Hopf algebra H, we define a braided functor J from B to the category ModH of left H–modules. The functor J, together with the set of generators of B, provides an algebraic method to study the range of quantum invariants of links. The braided Hopf algebra action on bottom tangles is mapped by J to the standard braided Hopf algebra structure for H in ModH. Several notions in knot theory, such as genus, unknotting number, ribbon knots, boundary links, local moves, etc are given algebraic interpretations in the setting involving the category B. The functor J provides a convenient way to study the relationships between these notions and quantum invariants.
HKRtype invariants of 4thickenings of 2dimensional CWcomplexes
"... The HKR (HenningsKaufmannRadford) framework is used to construct invariants of 4thickenings of 2dimensional CWcomplexes under 1 and 2 handle slides and cancellations (2deformations). The input of the invariant is a finite dimensional unimodular ribbon Hopf algebra A and an element in a quoti ..."
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Cited by 2 (2 self)
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The HKR (HenningsKaufmannRadford) framework is used to construct invariants of 4thickenings of 2dimensional CWcomplexes under 1 and 2 handle slides and cancellations (2deformations). The input of the invariant is a finite dimensional unimodular ribbon Hopf algebra A and an element in a quotient of its center, which determines a trace function on A. We study the subset T 4 of trace elements which define invariants of 4thickenings under 2deformations. In T 4 are identified two subsets: T 3 ⊂ T 4, which produces invariants of 4thickenings normalizable to invariants of the border, and T 2 ⊂ T 4, which produces invariants of 4thickenings depending only on the 2dimensional spine and the second Whitney number of the 4thickening. The case of the quantum sl(2) is studied in details. We show that sl(2) leads to four HKRtype invariants and describe the corresponding trace elements. Moreover, the fusion algebra of the semisimple quotient of the category of representations of the quantum sl(2) is identified as a subalgebra of a quotient of its center. 1