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53
An endomorphism of the Khovanov invariant
 Adv. Math
"... Abstract. We construct an endomorphism of the Khovanov invariant to prove Hthinness and pairing phenomena of the invariants for alternating links. As a consequence, it follows that the Khovanov invariant of an oriented nonsplit alternating link is determined by its Jones polynomial, signature, and ..."
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Abstract. We construct an endomorphism of the Khovanov invariant to prove Hthinness and pairing phenomena of the invariants for alternating links. As a consequence, it follows that the Khovanov invariant of an oriented nonsplit alternating link is determined by its Jones polynomial, signature, and the linking numbers of its components. 1.
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
An invariant of link cobordisms from Khovanov’s homology theory
 Algebr. Geom. Topol
"... 1.1. Khovanov’s Homology. In [K] M.Khovanov introduced a new homology theory, which assigns to a diagram D of an oriented classical link L a bigraded family of homology groups Hi,j (D) such that the graded Euler characteristic ∑ ..."
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Cited by 51 (1 self)
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1.1. Khovanov’s Homology. In [K] M.Khovanov introduced a new homology theory, which assigns to a diagram D of an oriented classical link L a bigraded family of homology groups Hi,j (D) such that the graded Euler characteristic ∑
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.
Koszul duality for dioperads
"... Abstract. We introduce the notion of a dioperad to describe certain operations with multiple inputs and multiple outputs. The framework of Koszul duality for operads is generalized to dioperads. We show that the Lie bialgebra dioperad is Koszul. The current interests in the understanding of various ..."
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Cited by 29 (1 self)
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Abstract. We introduce the notion of a dioperad to describe certain operations with multiple inputs and multiple outputs. The framework of Koszul duality for operads is generalized to dioperads. We show that the Lie bialgebra dioperad is Koszul. The current interests in the understanding of various algebraic structures using operads is partly due to the theory of Koszul duality for operads; see eg. [K] or [L] for surveys. However, algebraic structures such as bialgebras and Lie bialgebras, which involve both multiplication and comultiplication, or bracket and cobracket, are defined using PROP’s (cf. [Ad]) rather than operads. Inspired by the theory of string topology of ChasSullivan ([ChS], [Ch], [Tr]), Victor Ginzburg suggested to the author that there should be a theory of Koszul duality for PROP’s. The present paper results from the observation that when the defining relations between the generators of a PROP are spanned over trees, then the ”treepart ” of the PROP has the structure of a dioperad. We show that one can set up a theory of Koszul duality for dioperads. In §1, we give the definition of a dioperad and other generalities. In §2, we define the notion of a quadratic dioperad, its quadratic dual, and introduce our main example of Lie bialgebra dioperad. In §3, we define the cobar dual of a dioperad. A quadratic dioperad is Koszul if its cobar dual is quasiisomorphic to its quadratic dual. The formalism in §2 and §3, in the case of operads, is due to GinzburgKapranov [GiK]. In §4, we prove a proposition to be used later in §5. This proposition is a generalization of a result of ShniderVan Osdol [SVO]. In §5, we prove that Koszulity of a quadratic dioperad is equivalent to exactness of certain Koszul complexes. In the case of operads, this is again due to GinzburgKapranov, with a different proof by ShniderVan Osdol. The Koszulity of the Lie bialgebra dioperad follows from this and an adaptation of results of Markl [M2].
A Polarized view of string topology
, 2002
"... Let M be a closed, connected manifold, and LM its loop space. In this paper we describe closed string topology operations in h∗(LM), where h ∗ is a generalized homology theory that supports an orientation of M. We will show that these operations give h∗(LM) the structure of a unital, commutative Fro ..."
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Cited by 26 (7 self)
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Let M be a closed, connected manifold, and LM its loop space. In this paper we describe closed string topology operations in h∗(LM), where h ∗ is a generalized homology theory that supports an orientation of M. We will show that these operations give h∗(LM) the structure of a unital, commutative Frobenius algebra without a counit. Equivalently they describe a positive boundary, two dimensional topological quantum field theory associated to h∗(LM). This implies that there are operations corresponding to any surface with p incoming and q outgoing boundary components, so long as q ≥ 1. The absence of a counit follows from the nonexistence of an operation associated to the disk, D 2, viewed as a cobordism from the circle to the empty set. We will study homological obstructions to constructing such an operation, and show that in order for such an operation to exist, one must take h∗(LM) to be an appropriate homological proobject associated to the loop space. Motivated by this, we introduce a prospectrum associated to LM when M has an almost complex structure. Given such a manifold its loop space has a canonical polarization of its tangent bundle, which is the fundamental feature needed to define this prospectrum. We refer to this as the “polarized Atiyah dual ” of LM. An appropriate homology theory applied to this prospectrum would be a candidate for a theory that supports string topology operations associated to any surface, including closed surfaces.
Representations of the homotopy surface category of a simply connected space
 J. Knot Theory and Ramifications
"... At the heart of the axiomatic formulation of 1+1dimensional topological field theory is the set of all surfaces with boundary assembled into a category. This category of surfaces has compact 1manifolds as objects and smooth oriented cobordisms as morphisms. Taking disjoint unions gives a monoidal ..."
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Cited by 18 (9 self)
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At the heart of the axiomatic formulation of 1+1dimensional topological field theory is the set of all surfaces with boundary assembled into a category. This category of surfaces has compact 1manifolds as objects and smooth oriented cobordisms as morphisms. Taking disjoint unions gives a monoidal
Homotopy quantum field theories and the homotopy cobordism category in dimension 1+1
"... Abstract. We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n, X) such that an HQFT is a functor ..."
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Abstract. We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n, X) such that an HQFT is a functor
The quantum Euler class and the quantum cohomology of the Grassmannians
"... The Poincaré duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with the structure of a Frobenius algebra. Any such algebra possesses a canonical “characteristic element; ” in the classical case this is the Euler class, and in the quantum case ..."
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Cited by 12 (1 self)
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The Poincaré duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with the structure of a Frobenius algebra. Any such algebra possesses a canonical “characteristic element; ” in the classical case this is the Euler class, and in the quantum case this is a deformation of the classical Euler class which we call the “quantum Euler class. ” We prove that the characteristic element of a Frobenius algebra A is a unit if and only if A is semisimple, and then apply this result to the cases of the quantum cohomology of the finite complex Grassmannians, and to the quantum cohomology of hypersurfaces. In addition we show that, in the case of the Grassmannians, the [quantum] Euler class equals, as [quantum] cohomology element and up to sign, the determinant of the Hessian of the [quantum] LandauGinzbug potential. 1