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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 hyperbolic state sum invariants of 3– manifolds
"... Any triple (W, L, ρ), where W is a compact closed oriented 3manifold, L is a link in W and ρ is a flat principal Bbundle over W (B is the Borel subgroup of upper triangular matrices of SL(2, C)), can be encoded by suitable distinguished and decorated triangulations T = (T, H, D). For each T, for e ..."
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Cited by 5 (0 self)
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Any triple (W, L, ρ), where W is a compact closed oriented 3manifold, L is a link in W and ρ is a flat principal Bbundle over W (B is the Borel subgroup of upper triangular matrices of SL(2, C)), can be encoded by suitable distinguished and decorated triangulations T = (T, H, D). For each T, for each odd integer N ≥ 3, one defines a state sum KN(T), based on the FaddeevKashaev quantum dilogarithm at ω = exp(2πi/N), such that KN(W,L, ρ) = KN(T) is a welldefined complex valued invariant. The purely topological, conjectural invariants KN(W,L) proposed earlier by Kashaev correspond to the special case of the trivial flat bundle. Moreover, we extend the definition of these invariants to the case of flat bundles on W \ L with not necessarily trivial holonomy along the meridians of the link’s components, and also to 3manifolds endowed with a Bflat bundle and with arbitrary nonspherical parametrized boundary components. As a matter of fact the distinguished and decorated triangulations are strongly reminiscent of the way one represents the classical refined scissors congruence class ̂ β(F), belonging to the extended Bloch group, of any given finite volume hyperbolic 3manifold F by using any hyperbolic ideal triangulation of F. We point out some remarkable specializations of the invariants; among these, the so called Seiferttype invariants, when W = S 3: these seem to be good candidates in order to fully reconstruct the Jones polynomials in the main stream of quantum hyperbolic invariants. Finally, we try to set our results against the heuristic backgroud of the Euclidean analytic continuation of (2+1) quantum gravity with negative cosmological constant, regarded as a gauge theory with the noncompact group SO(3, 1) as gauge group.
Congruence subgroups and generalized FrobeniusSchur indicators
"... Abstract. We define generalized FrobeniusSchur indicators for objects in a linear pivotal category C. An equivariant indicator of an object is defined as a functional on the Grothendieck algebra of the quantum double Z(C) of C using the values of the generalized FrobeniusSchur indicators. In a sph ..."
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Abstract. We define generalized FrobeniusSchur indicators for objects in a linear pivotal category C. An equivariant indicator of an object is defined as a functional on the Grothendieck algebra of the quantum double Z(C) of C using the values of the generalized FrobeniusSchur indicators. In a spherical fusion category C with FrobeniusSchur exponent N, we prove that the set of all equivariant indicators admits a natural action of the modular group, and the kernel of the canonical modular representation of Z(C) is a congruence subgroup of level N. Moreover, if C is modular, then the kernel of the projective modular representation of C is also a congruence subgroup of level N, and every modular representation of C has a finite image.
HIGHER FROBENIUSSCHUR INDICATORS FOR PIVOTAL CATEGORIES
, 2005
"... Abstract. We define higher FrobeniusSchur indicators for objects in linear pivotal monoidal categories. We prove that they are category invariants, and take values in the cyclotomic integers. We also define a family of natural endomorphisms of the identity endofunctor on a klinear semisimple rigid ..."
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Cited by 4 (2 self)
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Abstract. We define higher FrobeniusSchur indicators for objects in linear pivotal monoidal categories. We prove that they are category invariants, and take values in the cyclotomic integers. We also define a family of natural endomorphisms of the identity endofunctor on a klinear semisimple rigid monoidal category, which we call the FrobeniusSchur endomorphisms. For a klinear semisimple pivotal monoidal category — where both notions are defined —, the FrobeniusSchur indicators can be computed as traces of the FrobeniusSchur endomorphisms.
Tensor categories: A selective guided tour ∗
, 2008
"... These are the – only lightly edited – lecture notes for a short course on tensor categories. The coverage in these notes is relatively nontechnical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something int ..."
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These are the – only lightly edited – lecture notes for a short course on tensor categories. The coverage in these notes is relatively nontechnical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something interesting in them. Once the basic definitions are given, the focus is mainly on klinear categories with finite dimensional homspaces. Connections with quantum groups and low dimensional topology are pointed out, but these notes have no pretension to cover the latter subjects at any depth. Essentially, these notes should be considered as annotations to the extensive bibliography. We also recommend the recent review [33], which covers less ground in a deeper way. 1 Tensor categories 1.1 Strict tensor categories