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53
Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
"... For a copy with the handdrawn figures please email ..."
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Cited by 141 (14 self)
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For a copy with the handdrawn figures please email
On fusion categories
 Annals of Mathematics
"... Abstract. In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series ofC. For fusion categories with commutative Gr ..."
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Cited by 78 (17 self)
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Abstract. In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series ofC. For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two central series are centralizers of each other in the sense of M. Müger. Dedicated to Leonid Vainerman on the occasion of his 60th birthday 1. introduction The theory of fusion categories arises in many areas of mathematics such as representation theory, quantum groups, operator algebras and topology. The representation categories of semisimple (quasi) Hopf algebras are important examples of fusion categories. Fusion categories have been studied extensively in the literature,
Construction of Field Algebras with Quantum Symmetry from Local Observables
, 1996
"... It has been discussed earlier that ( weak quasi) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid) statistics and locality was established. This work addresses to the reco ..."
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Cited by 46 (5 self)
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It has been discussed earlier that ( weak quasi) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid) statistics and locality was established. This work addresses to the reconstruction of quantum symmetries and algebras of field operators. For every algebra A of observables satisfying certain standard assumptions, an appropriate quantum symmetry is found. Field operators are obtained which act on a positive definite Hilbert space of states and transform covariantly under the quantum symmetry. As a substitute for Bose/Fermi (anti) commutation relations, these fields are demonstrated to obey local braid relation. Contents 1 Introduction 1 2 The Notion of Quantum Symmetry 5 3 Algebraic Methods for Field Construction 9 3.1 Observables and superselection sectors in local quantum field theory . . . . 10 3.2 Localized endomorphisms and fusion structure . . . . . ....
Noncommutative Worldvolume Geometries: Branes on SU(2) and Fuzzy Spheres
, 1999
"... The geometry of Dbranes can be probed by open string scattering. If the background carries a nonvanishing Bfield, the worldvolume becomes noncommutative. Here we explore the quantization of worldvolume geometries in a curved background with nonzero NeveuSchwarz 3form field strength H = dB. ..."
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Cited by 43 (6 self)
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The geometry of Dbranes can be probed by open string scattering. If the background carries a nonvanishing Bfield, the worldvolume becomes noncommutative. Here we explore the quantization of worldvolume geometries in a curved background with nonzero NeveuSchwarz 3form field strength H = dB. Using exact and generally applicable methods from boundary conformal field theory, we study the example of open strings in the SU(2) WessZuminoWitten model, and establish a relation with fuzzy spheres or certain (nonassociative) deformations thereof. These findings could be of direct relevance for Dbranes in the presence of NeveuSchwarz 5branes; more importantly, they provide insight into a completely new class of worldvolume geometries.
Higherdimensional algebra II: 2Hilbert spaces
"... A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also ..."
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Cited by 42 (12 self)
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A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also define monoidal, braided monoidal, and symmetric monoidal versions of 2Hilbert spaces, which we call 2H*algebras, braided 2H*algebras, and symmetric 2H*algebras, and we describe the relation between these and tangles in 2, 3, and 4 dimensions, respectively. We prove a generalized DoplicherRoberts theorem stating that every symmetric 2H*algebra is equivalent to the category Rep(G) of continuous unitary finitedimensional representations of some compact supergroupoid G. The equivalence is given by a categorified version of the Gelfand transform; we also construct a categorified version of the Fourier transform when G is a compact abelian group. Finally, we characterize Rep(G) by its universal properties when G is a compact classical group. For example, Rep(U(n)) is the free connected symmetric 2H*algebra on one even object of dimension n. 1
Diagonal Crossed Products by Duals of QuasiQuantum Groups
 Rev. Math. Phys
, 1999
"... A twosided coaction δ: M → G ⊗M⊗G of a Hopf algebra (G, ∆, ǫ, S) on an associative algebra M is an algebra map of the form δ = (λ ⊗ idM) ◦ ρ = (idM ⊗ ρ) ◦ λ, where (λ, ρ) is a commuting pair of left and right Gcoactions on M, respectively. Denoting the associated commuting right and left actions ..."
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Cited by 25 (1 self)
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A twosided coaction δ: M → G ⊗M⊗G of a Hopf algebra (G, ∆, ǫ, S) on an associative algebra M is an algebra map of the form δ = (λ ⊗ idM) ◦ ρ = (idM ⊗ ρ) ◦ λ, where (λ, ρ) is a commuting pair of left and right Gcoactions on M, respectively. Denoting the associated commuting right and left actions of the dual Hopf algebra ˆ G on M by ⊳ and ⊲, respectively, we define the diagonal crossed product M ⊲ ⊳ ˆ G to be the algebra generated by M and ˆ G with relations given by ϕm = (ϕ (1) ⊲m ⊳ ˆ S −1 (ϕ (3)))ϕ (2), m ∈ M, ϕ ∈ ˆ G. We give a natural generalization of this construction to the case where G is a quasi–Hopf algebra in the sense of Drinfeld and, more generally, also in the sense of Mack and Schomerus (i.e., where the coproduct ∆ is nonunital). In these cases our diagonal crossed product will still be an associative algebra structure on M ⊗ ˆ G extending M ≡ M ⊗ ˆ1, even though the analogue of an ordinary crossed product M ⋊ ˆ G in general is not well defined as an associative algebra. Applications of our formalism include the field algebra constructions with quasiquantum
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.
Conformal Field Theory and DoplicherRoberts Reconstruction
 In
"... Abstract. After a brief review of recent rigorous results concerning the representation theory of rational chiral conformal field theories (RCQFTs) we focus on pairs (A, F) of conformal field theories, where F has a finite group G of global symmetries and A is the fixpoint theory. The comparison of ..."
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Cited by 12 (3 self)
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Abstract. After a brief review of recent rigorous results concerning the representation theory of rational chiral conformal field theories (RCQFTs) we focus on pairs (A, F) of conformal field theories, where F has a finite group G of global symmetries and A is the fixpoint theory. The comparison of the representation categories of A and F is strongly intertwined with various issues related to braided tensor categories. We
On Galois extensions of braided tensor categories, mapping class group representations and simple current extensions
 In preparation
"... We show that the author’s notion of Galois extensions of braided tensor categories [22], see also [3], gives rise to braided crossed Gcategories, recently introduced for the purposes of 3manifold topology [31]. The Galois extensions C ⋊ S are studied in detail, and we determine for which g ∈ G non ..."
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Cited by 10 (4 self)
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We show that the author’s notion of Galois extensions of braided tensor categories [22], see also [3], gives rise to braided crossed Gcategories, recently introduced for the purposes of 3manifold topology [31]. The Galois extensions C ⋊ S are studied in detail, and we determine for which g ∈ G nontrivial objects of grade g exist in C ⋊ S. 1