## On Galois extensions of braided tensor categories, mapping class group representations and simple current extensions

Venue: | In preparation |

Citations: | 10 - 4 self |

### BibTeX

@INPROCEEDINGS{Müger_ongalois,

author = {Michael Müger},

title = {On Galois extensions of braided tensor categories, mapping class group representations and simple current extensions},

booktitle = {In preparation},

year = {}

}

### OpenURL

### Abstract

We show that the author’s notion of Galois extensions of braided tensor categories [22], see also [3], gives rise to braided crossed G-categories, recently introduced for the purposes of 3-manifold topology [31]. The Galois extensions C ⋊ S are studied in detail, and we determine for which g ∈ G non-trivial objects of grade g exist in C ⋊ S. 1

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Citation Context ...gorical groups classifying connected homotopy types with only π2,π3 non-trivial). On the other hand, the (III) representation categories of quasitriangular (quasi-, weak etc.) Hopf algebras, cf. e.g. =-=[15]-=-, and of (IV) quantum field theories (QFT) in low-dimensional space times [10, 11], in particular conformal field theories [10, 21], are btc. Finally, (V) the category of tangles is a btc, which is th... |

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Citation Context ...fold topology [31]. The Galois extensions C ⋊ S are studied in detail, and we determine for which g ∈ G non-trivial objects of grade g exist in C ⋊ S. 1 Introduction According to the pioneering paper =-=[14]-=-, the notion of braided tensor categories (btc for short) originated in (I) considerations in higher dimensional category theory (btc as 3categories with one object and one 1-morphism) and (II) homoto... |

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Citation Context ...ategories of quasitriangular (quasi-, weak etc.) Hopf algebras, cf. e.g. [15], and of (IV) quantum field theories (QFT) in low-dimensional space times [10, 11], in particular conformal field theories =-=[10, 21]-=-, are btc. Finally, (V) the category of tangles is a btc, which is the origin of the recent invariants of links and 3-manifolds [30, 15, 1]. It goes without saying that all five areas continue to be v... |

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The Operator Algebra of Orbifold Models
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Citation Context ...ormal field theory A carrying an action of a finite group G gives rise to a braided crossed G-category G −Loc A of ‘G-twisted representations’. This analysis has its origins in the notions of twisted =-=[7]-=- and soliton-like representations [29]. Together with [16], where the modularity of conformal representation categories is established, and Turaev’s work [30, 31] on invariants of (G-)manifolds these ... |

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Citation Context ...l). On the other hand, the (III) representation categories of quasitriangular (quasi-, weak etc.) Hopf algebras, cf. e.g. [15], and of (IV) quantum field theories (QFT) in low-dimensional space times =-=[10, 11]-=-, in particular conformal field theories [10, 21], are btc. Finally, (V) the category of tangles is a btc, which is the origin of the recent invariants of links and 3-manifolds [30, 15, 1]. It goes wi... |

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Roberts: A new duality theory for compact groups
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Citation Context ...ssions are defined, (s ◦ t) ∗ = t ∗ ◦ s ∗ and (s ⊗ t) ∗ = s ∗ ⊗ t ∗ . A ∗-operation is positive if s ∗ ◦ s = 0 implies s = 0. A category with positive ∗-operation is called ∗-category or unitary, cf. =-=[12, 8, 19, 30]-=-. The category of finite dimensional polynomial representations of a reductive proalgebraic group (in characteristic zero) is a rigid abelian symmetric tensor category with End1 = kid1. The category o... |

63 | Multi-interval subfactors and modularity of representations in conformal field theory
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Citation Context ... G gives rise to a braided crossed G-category G −Loc A of ‘G-twisted representations’. This analysis has its origins in the notions of twisted [7] and soliton-like representations [29]. Together with =-=[16]-=-, where the modularity of conformal representation categories is established, and Turaev’s work [30, 31] on invariants of (G-)manifolds these results will establish an equivariant version of the chain... |

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Citation Context ...ssions are defined, (s ◦ t) ∗ = t ∗ ◦ s ∗ and (s ⊗ t) ∗ = s ∗ ⊗ t ∗ . A ∗-operation is positive if s ∗ ◦ s = 0 implies s = 0. A category with positive ∗-operation is called ∗-category or unitary, cf. =-=[12, 8, 19, 30]-=-. The category of finite dimensional polynomial representations of a reductive proalgebraic group (in characteristic zero) is a rigid abelian symmetric tensor category with End1 = kid1. The category o... |

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Citation Context ...tinuous representations of a compact topological group has the same properties and is in addition a ∗-category. There are converses to these statements due to Doplicher and Roberts [8] and to Deligne =-=[5]-=-, respectively. For our purposes in this paper it is sufficient to consider symmetric categories with finitely many (isomorphism classes of) simple objects, corresponding to finite groups. 2.1 Definit... |

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Citation Context ...l). On the other hand, the (III) representation categories of quasitriangular (quasi-, weak etc.) Hopf algebras, cf. e.g. [15], and of (IV) quantum field theories (QFT) in low-dimensional space times =-=[10, 11]-=-, in particular conformal field theories [10, 21], are btc. Finally, (V) the category of tangles is a btc, which is the origin of the recent invariants of links and 3-manifolds [30, 15, 1]. It goes wi... |

52 | From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories
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Citation Context ...Γ) is a monoid, (Γ,∆ : Γ → Γ 2 ,ε : Γ → 1) is a comonoid and the condition idΓ ⊗ m ◦ ∆ ⊗ idΓ = ∆ ◦ m = ∆ ⊗ idΓ ◦ idΓ ⊗ m holds. A Frobenius algebra in a k-linear category is called strongly separable =-=[24]-=- if m ◦ ∆ = α idΓ, ε ◦ η = β id1, α,β ∈ k ∗ . 2.6 Proposition [24] Let G be a finite group and k an algebraically closed field whose characteristic does not divide |G|. There exists a strongly separab... |

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Citation Context ...g-de Vries Institute, Amsterdam, Netherlands email: mmueger@science.uva.nl February 24, 2008 Abstract We show that the author’s notion of Galois extensions of braided tensor categories [22], see also =-=[3]-=-, gives rise to braided crossed G-categories, recently introduced for the purposes of 3-manifold topology [31]. The Galois extensions C ⋊ S are studied in detail, and we determine for which g ∈ G non-... |

29 | Galois theory for braided tensor categories and the modular closure
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Citation Context ...nds and Korteweg-de Vries Institute, Amsterdam, Netherlands email: mmueger@science.uva.nl February 24, 2008 Abstract We show that the author’s notion of Galois extensions of braided tensor categories =-=[22]-=-, see also [3], gives rise to braided crossed G-categories, recently introduced for the purposes of 3-manifold topology [31]. The Galois extensions C ⋊ S are studied in detail, and we determine for wh... |

27 |
Local quantum physics (2nd ed
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Citation Context ...⋊ S is a braided crossed G-category, where we also clarify for which g ∈ G there exist Xg ∈ C ⋊ S with ∂X = g. In a companion paper [26] we will show, in the context of algebraic quantum field theory =-=[13]-=-, that a chiral conformal field theory A carrying an action of a finite group G gives rise to a braided crossed G-category G −Loc A of ‘G-twisted representations’. This analysis has its origins in the... |

24 | Spherical categories
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Citation Context ... to consider symmetric categories with finitely many (isomorphism classes of) simple objects, corresponding to finite groups. 2.1 Definition 1. A TC is a semisimple k-linear spherical tensor category =-=[2]-=- with finite dimensional Hom-spaces and End1 = kid1, where k is an algebraically closed field. It is called finite if the set of isomorphism classes of simple objects is finite. The dimension of a fin... |

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Citation Context ...al space times [10, 11], in particular conformal field theories [10, 21], are btc. Finally, (V) the category of tangles is a btc, which is the origin of the recent invariants of links and 3-manifolds =-=[30, 15, 1]-=-. It goes without saying that all five areas continue to be very active fields of research and the connections continue to be explored. In this paper we are concerned with a recent extension of the no... |

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Lane: Categories for the Working Mathematician, 2nd edition
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Citation Context ...at will be needed later. Some of those are well known, others relatively recent. We assume known the notions of abelian, monoidal (or tensor) braided, symmetric, rigid and ribbon categories, cf. e.g. =-=[20, 14, 15, 1]-=-. All categories considered in this paper will be k-linear semisimple (thus in particular abelian) over an algebraically closed field k with finite dimensional Hom-spaces and monoidal with End1 = kid1... |

17 |
On the structure of modular categories
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Citation Context ... c w.r.t. the second variable holds iff S ⊂ Z2(C), which is the case iff Γ ∈ Z2(C). Here Z2(C) ⊂ C is the full subcategory of objects X satisfying cX,Y ◦ cY,X = idY X for all Y ∈ C, called central in =-=[25]-=- and transparent in [3]. In order to understand the general case S ̸⊂ Z2(C) we need some preliminary considerations. 4.2 Lemma Let X,Y ∈ C, Z ∈ C ∩ S ′ and s ∈ Hom Ĉ (X,Y ) = HomC(ΓX,Y ). Then Proof. ... |

14 |
Catégories tensorielles
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Citation Context ...egral but non-positive dimensions, respectively, it still is the representation category of a supergroup, i.e. a pair (G,k) where G is a group and k ∈ Z(G) is involutive, cf. [8, Section 7], see also =-=[6]-=-. This generalization will not be used in this paper. ✷ 42.5 Definition Let C be a strict tensor category. A Frobenius algebra in C is a quintuple (Γ,m,η,∆,ε) such that (Γ ∈ C,m : Γ 2 → Γ,η : 1 → Γ) ... |

14 |
On braiding and dyslexia
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Citation Context ... idY . The full subcategory Γ −Mod 0 C ⊂ Γ −ModC consisting of the objects (X,µ) satisfying µ ◦ cX,Γ ◦ cΓ,X = µ is monoidal and braided. 2.11 Remark The above definition and facts are due to Pareigis =-=[28]-=- and were rediscovered in [17]. The special case where Γ ∈ Z2(C), implying Γ−Mod 0 C = Γ−ModC, was considered in [3]. ✷ Recall that the dimension of a finite TC is the sum over the squared dimensions ... |

12 | Conformal Field Theory and Doplicher-Roberts Reconstruction. In: [29 - Müger |

8 |
Categorical G-crossed modules and 2-fold extensions
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Citation Context ...ed to non-strict tensor categories, cf. [31]. The G-action can be generalized by relaxing the γg to be self-equivalences satisfying natural isomorphisms γgγh ∼ = γgh with suitable coherence, cf. e.g. =-=[4]-=-. For our purposes, in particular the application to conformal field theory, the above strict version is sufficient. ✷ In view of Definition 4.7, Propositions 4.3, 4.6 essentially amount to the follow... |

4 |
On fusion categories. math.QA/0203060
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Citation Context ...f it is absolutely simple (End X = kidX). We will therefore just speak of simple objects. By dropping the assumption of sphericity one arrives at the notion of fusion categories which were studied in =-=[9]-=-. There are remarkably strong results like the automatic positivity of dim C when k = C. (Yamagami has shown that a ∗-structure gives rise to an essentially unique spherical structure, and one might s... |

4 | Spin-statistics and CPT for solitons
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(Show Context)
Citation Context ...on of a finite group G gives rise to a braided crossed G-category G −Loc A of ‘G-twisted representations’. This analysis has its origins in the notions of twisted [7] and soliton-like representations =-=[29]-=-. Together with [16], where the modularity of conformal representation categories is established, and Turaev’s work [30, 31] on invariants of (G-)manifolds these results will establish an equivariant ... |

4 |
Turaev: Homotopy field theory in dimension 3 and crossed group-categories. math.GT/0005291
- G
(Show Context)
Citation Context ... that the author’s notion of Galois extensions of braided tensor categories [22], see also [3], gives rise to braided crossed G-categories, recently introduced for the purposes of 3-manifold topology =-=[31]-=-. The Galois extensions C ⋊ S are studied in detail, and we determine for which g ∈ G non-trivial objects of grade g exist in C ⋊ S. 1 Introduction According to the pioneering paper [14], the notion o... |

3 |
Ostrik: On q-analog of McKay correspondence and ADE classification of sl ( 2) conformal field theories. math.AQ/0101219
- unknown authors
(Show Context)
Citation Context ...−Mod 0 C ⊂ Γ −ModC consisting of the objects (X,µ) satisfying µ ◦ cX,Γ ◦ cΓ,X = µ is monoidal and braided. 2.11 Remark The above definition and facts are due to Pareigis [28] and were rediscovered in =-=[17]-=-. The special case where Γ ∈ Z2(C), implying Γ−Mod 0 C = Γ−ModC, was considered in [3]. ✷ Recall that the dimension of a finite TC is the sum over the squared dimensions of its simple objects, cf. e.g... |

2 |
Tuset: Regular representations of algebraic quantum groups and embedding theorems
- Müger, L
(Show Context)
Citation Context ...orical Frobenius algebra in H −Mod is strongly separable iff H is semisimple and cosemisimple, cf. [24]. 3. Some of the structure survives for infinite compact groups and discrete quantum groups, cf. =-=[27]-=-. ✷ 2.8 Remark Given the monoid part of the above Frobenius algebra, one can obtain a fiber functor E : C → Vectk as follows: E(X) = HomC(1,Γ ⊗ X), E(s)φ = s ⊗ idX ◦ φ, s : X → Y, φ ∈ E(X). The natura... |

1 |
Turaev: Quantum groups and ribbon G-categories. math.QA/0103017
- V
(Show Context)
Citation Context ... ⊗ t = X t ⊗ s ◦ cX,Y ∀s : X → X ′ , t : Y → Y ′ . Of the various generalizations permitted by this definition we will need only the admission of inhomogeneous objects, cf. Section 4. As to (III): In =-=[18]-=- it was shown that some crossed G-categories can be obtained from quantum groups. With a view towards applications to algebraic topology (II), in [4] a notion of categorical G-crossed module was defin... |

1 |
Conformal orbifold theories, crossed G-categories and quasiabelian cohomology
- Müger
(Show Context)
Citation Context ... given). Dropping this latter condition we show in Section 4 that C ⋊ S is a braided crossed G-category, where we also clarify for which g ∈ G there exist Xg ∈ C ⋊ S with ∂X = g. In a companion paper =-=[26]-=- we will show, in the context of algebraic quantum field theory [13], that a chiral conformal field theory A carrying an action of a finite group G gives rise to a braided crossed G-category G −Loc A ... |