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31
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
Representations of compact quantum groups and subfactors
 J. Reine Angew. Math
, 1999
"... Abstract: We associate Popa systems ( = standard invariants of subfactors, cf. [P3],[P4]) to the finite dimensional representations of compact quantum groups. We characterise the systems arising in this way: these are the ones which can be “represented ” on finite dimensional Hilbert spaces. This is ..."
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Cited by 32 (18 self)
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Abstract: We associate Popa systems ( = standard invariants of subfactors, cf. [P3],[P4]) to the finite dimensional representations of compact quantum groups. We characterise the systems arising in this way: these are the ones which can be “represented ” on finite dimensional Hilbert spaces. This is proved by an universal construction. We explicitely compute (in terms of some free products) the operation of going from representations of compact quantum groups to Popa systems and then back via the universal construction. This is related with our previous work [B2]. We prove a Kesten type result for the coamenability of compact quantum groups, which allows us to compare it with the amenability of subfactors.
Galois theory for braided tensor categories and the modular closure
 Adv. Math
, 2000
"... Given a braided tensor ∗category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C ⋊ S. This construction yields a tensor ∗category with conjugates and an irreducible unit. (A ∗category is a category enriched over VectC ..."
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Cited by 29 (6 self)
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Given a braided tensor ∗category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C ⋊ S. This construction yields a tensor ∗category with conjugates and an irreducible unit. (A ∗category is a category enriched over VectC with positive ∗operation.) A Galois correspondence is established between intermediate categories sitting between C and C ⋊S and closed subgroups of the Galois group Gal(C⋊S/C) = AutC(C⋊S) of C, the latter being isomorphic to the compact group associated to S by the duality theorem of Doplicher and Roberts. Denoting by D ⊂ C the full subcategory of degenerate objects, i.e. objects which have trivial monodromy with all objects of C, the braiding of C extends to a braiding of C⋊S iff S ⊂ D. Under this condition C⋊S has no nontrivial degenerate objects iff S = D. If the original category C is rational (i.e. has only finitely many isomorphism classes of irreducible objects) then the same holds for the new one. The category C ≡ C ⋊ D is called the modular closure of C since in the rational case it is modular, i.e. gives rise to a unitary representation of the modular group SL(2, Z). (In passing we prove that every braided tensor ∗category with conjugates automatically is a ribbon category, i.e. has a twist.) If all simple objects of S have dimension one the structure of the category C ⋊ S can be clarified quite explicitly in terms of group cohomology. 1
On exotic modular tensor categories
 Commun. Contemp. Math
"... Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular Smatrix S and −S has both topological and physical significance, so in our convention there are a tot ..."
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Cited by 13 (7 self)
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Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular Smatrix S and −S has both topological and physical significance, so in our convention there are a total of 70 UMTCs of rank ≤ 4. In particular, there are two trivial UMTCs with S = (±1). Each such UMTC can be obtained from 10 nontrivial prime UMTCs by direct product, and some symmetry operations. Explicit data of the 10 nontrivial prime UMTCs are given in Section 5. Relevance of UMTCs to topological quantum computation and various conjectures are given in Section 6. 1.
3manifold invariants from cosets
"... Abstract. We construct unitary modular categories for a general class of coset conformal field theories based on our previous study of these theories in the algebraic quantum field theory framework using subfactor theory. We also consider the calculations of the corresponding 3manifold invariants. ..."
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Cited by 8 (4 self)
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Abstract. We construct unitary modular categories for a general class of coset conformal field theories based on our previous study of these theories in the algebraic quantum field theory framework using subfactor theory. We also consider the calculations of the corresponding 3manifold invariants. It is shown that under certain index conditions the link invaraints colored by the representations of coset factorize into the products of the the link invaraints colored by the representations of the two groups in the coset. But the 3manifold invariants do not behave so simply in general due to the nontrivial branching and selection rules of the coset. Examples in the parafermion cosets and diagonal cosets show that 3manifold invariants of the coset may be finer than the products of the 3manifold invariants associated with the two groups in the coset, and these two invariants do not seem to be simply related in some cases, for an example, in the cases when there are issues of “fixed point resolutions”. In the later case our framework provides a mathematical understanding of the underlying unitary modular categories which has not been obtained by other methods.
From Quantum Groups to Unitary Modular Tensor Categories
 CONTEMPORARY MATHEMATICS
"... Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently propos ..."
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Cited by 8 (6 self)
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Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently proposed applications to quantum computing have provided an impetus to understand and describe these examples as explicitly as possible, especially those that are “physically feasible.” We survey the current status of the problem of producing unitary modular tensor categories from quantum groups, emphasizing explicit computations.
On a family of nonunitarizable ribbon categories
 Math Z
, 2005
"... Abstract. We consider several families of categories. The first are quotients of H. Andersen’s tilting module categories for quantum groups of Lie type B at odd roots of unity. The second consists of categories of type BC constructed from idempotents in BMWalgebras. Our main result is to show that ..."
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Cited by 7 (5 self)
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Abstract. We consider several families of categories. The first are quotients of H. Andersen’s tilting module categories for quantum groups of Lie type B at odd roots of unity. The second consists of categories of type BC constructed from idempotents in BMWalgebras. Our main result is to show that these families coincide as braided tensor categories using a recent theorem of Tuba and Wenzl. By appealing to similar results of Blanchet and Beliakova we obtain another interesting equivalence with these two families of categories and the quantum group categories of Lie type C at odd roots of unity. The morphism spaces in these categories can be equipped with a Hermitian form, and we are able to show that these categories are never unitary, and no braided tensor category sharing the Grothendieck semiring common to these families is unitarizable. 1.
Lowdimensional unitary representations of B3
 Proc. Amer. Math. Soc
"... Abstract. We characterize all simple unitarizable representations of the braid group B3 on complex vector spaces of dimension d ≤ 5. In particular, we prove that if σ1 and σ2 denote the two generating twists of B3, then a simple representation ρ: B3 → GL(V) (for dim V ≤ 5) is unitarizable if and onl ..."
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Cited by 7 (2 self)
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Abstract. We characterize all simple unitarizable representations of the braid group B3 on complex vector spaces of dimension d ≤ 5. In particular, we prove that if σ1 and σ2 denote the two generating twists of B3, then a simple representation ρ: B3 → GL(V) (for dim V ≤ 5) is unitarizable if and only if the eigenvalues λ1, λ2,..., λd of ρ(σ1) are distinct, satisfy λi  = 1 and µ (d) 1i> 0 for 2 ≤ i ≤ d, where the µ (d) 1i are functions of the eigenvalues, explicitly described in this paper. 1.
A FINITENESS PROPERTY FOR BRAIDED FUSION CATEGORIES
"... Abstract. We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has property F if the associated braid group representations ..."
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Cited by 5 (3 self)
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Abstract. We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has property F if the associated braid group representations factor over a finite group, and suggest that categories of integral FrobeniusPerron dimension are precisely those with property F. 1.
Unitarizability of premodular categories
 J. Pure Appl. Algebra
"... Abstract. We study the unitarizability of premodular categories constructed from representations of quantum group at roots of unity. We introduce Grothendieck unitarizability as a natural generalization of unitarizability to any class of premodular categories with a common Grothendieck semiring. We ..."
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Cited by 5 (4 self)
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Abstract. We study the unitarizability of premodular categories constructed from representations of quantum group at roots of unity. We introduce Grothendieck unitarizability as a natural generalization of unitarizability to any class of premodular categories with a common Grothendieck semiring. We obtain new results for quantum groups of Lie types F4 and G2, and improve the known results for Lie types B and C.