## Galois theory for braided tensor categories and the modular closure (2000)

Venue: | Adv. Math |

Citations: | 31 - 7 self |

### BibTeX

@ARTICLE{Müger00galoistheory,

author = {Michael Müger},

title = {Galois theory for braided tensor categories and the modular closure},

journal = {Adv. Math},

year = {2000},

pages = {151--201}

}

### OpenURL

### Abstract

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 non-trivial 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

### Citations

430 | Representation theory of finite groups and associative algebras - Curtis, Reiner - 1962 |

253 |
Braided tensor categories
- Joyal, Street
- 1991
(Show Context)
Citation Context ...ded tensor categories without the symmetry requirement entered the scene only in the eighties. From a theoretical point of view braided tensor categories are most naturally ‘explained’ by identifying =-=[14]-=- them as 2-categories with tensor product and only one object, which in turn are just 3-categories with only one object and one 1-morphism. (All these notions are easiest to deal with in the strict ca... |

137 |
Tannakian Categories
- Deligne, Milne
- 1982
(Show Context)
Citation Context ...ka-Krein duality theory for compact groups. Further motivation for their analysis came from Grothendieck’s theory of motives and led to Saaveda Rivano’s work [28], which was corrected and extended in =-=[4]-=-. These formalisms reconstruct a group (compact topological in the first, algebraic in the second case) from the category of its representations, the latter being concrete, i.e. consisting of vector (... |

130 | Categories for the Working Mathematician, 2nd ed - Lane - 1998 |

113 |
Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics
- Doplicher, Roberts
- 1990
(Show Context)
Citation Context ...se definitions coincide. 8. Finally, we remark that there are similarities between our definition of C ⋊0 S and a construction [25] of a field algebra in algebraic quantum field theory which preceded =-=[7]-=- but where the main result of [6] was assumed. 3.2 C ⋊0 S is a Tensor Category Lemma 3.2 The operations ◦, × are bilinear and associative. 12R∗ k ☛ ✟ ρ S ∗ γ k σ Figure 3: ∗-Operation on arrows Proof... |

109 |
A new duality theory for compact groups
- Doplicher, Roberts
- 1989
(Show Context)
Citation Context ...f. [26]. This category being a category of endomorphisms of A – not of vector spaces – the existing duality theorems did not apply. This led Doplicher and Roberts to developing their characterization =-=[6]-=- of representation categories of compact groups as abstract symmetric tensor categories satisfying certain additional axioms. This result allowed the solution [7] of the longstanding problem of (re-)c... |

79 |
Quantum invariants of knots and 3-manifolds, Walter de Gruyter
- Turaev
- 1994
(Show Context)
Citation Context ...ched a certain state of maturity when it was understood that the crucial ingredient underlying these invariants of 3-manifolds is a certain class of braided tensor categories which are called modular =-=[28]-=-. A modular category is a braided tensor category which (i) has a twist [28] or balancing [14], (ii) is rational – i.e. has only finitely many equivalence classes of irreducible objects – and (iii) no... |

64 |
A theory of dimension, K-Theory 11
- Longo, Roberts
- 1997
(Show Context)
Citation Context ...ulfilled naturality of ε fails for some of the new morphisms of C ⋊0 S). Now, C ⋊0 S will be closed under direct sums, but usually not under subobjects. Thus we apply the above-mentioned procedure of =-=[19]-=- in order to obtain a category C ⋊ S which is closed under direct sums and subobjects. Then the braiding of C ⋊0 S – if it exists – extends to C ⋊ S. The result of this construction is again a tensor ... |

60 |
Braid group statistics and their superselection rules
- Rehren
- 1990
(Show Context)
Citation Context ...valent to the unit object ι. (The designation of such categories as modular is owed to the fact that they give rise to a finite dimensional representation of the modular group SL(2, Z) [28], see also =-=[23]-=-.) The role of the quantum groups then reduces just to providing several infinite families of modular categories (roughly, one for every pair (root of unity, classical Lie algebra)). Another construct... |

55 |
gories Tannakiennes
- Deligne, Cate
- 1991
(Show Context)
Citation Context ...tation relations. (In fact, assuming the duality theorem for abstract symmetric categories such a reconstruction result existed much earlier [25].) At the same time and independently Deligne extended =-=[5]-=- the earlier works [28, 4] by identifying a necessary and sufficient condition for an abstract symmetric tensor category to be the representation category of an algebraic group. The crucial property i... |

47 |
Projective Representations of Finite Groups
- Karpilovsky
- 1985
(Show Context)
Citation Context ...n the tensor product structure of the twisted group algebra. The claim on the center follows by specialization to an abelian group A of well-known results on the center of twisted group algebras, cf. =-=[15]-=-, or by an easy direct proof. That B is a subgroup of A is then obvious in view of (5.5) and the fact that the center is a subalgebra. Now, in restriction to B the cocycle c is symmetric, which is equ... |

42 | Higher-dimensional algebra II: 2-Hilbert spaces
- Baez
- 1997
(Show Context)
Citation Context ...he above result we did not assume irreducibility of the unit ι, viz. (ι,ι) = C idι. From now on all categories in this paper will be assumed to have this property, which has been called connectedness =-=[2]-=-. We will remark on the disconnected case in the outlook. We summarize the properties of the categories we will study. Definition 2.2 A TC ∗ is a small strict tensor ∗-category with conjugates, direct... |

41 |
Lectures on algebraic quantum field theory
- Roberts
(Show Context)
Citation Context ...eld theory it was realized around 1970 that the category of localized superselection sectors ( ∼ = physically relevant representations of the C ∗ -algebra A of observables) is symmetric monoidal, cf. =-=[26]-=-. This category being a category of endomorphisms of A – not of vector spaces – the existing duality theorems did not apply. This led Doplicher and Roberts to developing their characterization [6] of ... |

40 | Orbifold subfactors from Hecke algebras
- Evans, Kawahigashi
- 1994
(Show Context)
Citation Context ...ariant arising from the modular closure C, bypassing the construction of the latter. 5. There is an obvious connection between the crossed product C ⋊ S and the ‘orbifold constructions in subfactors’ =-=[8, 33]-=- which deserves to be worked out. 6. Generalize everything in this paper to the non-connected case where Hom(ι,ι) ̸= C idι and the compact (super)groups are replaced by compact (super)groupoids [2]. T... |

38 |
C*-Tensor categories from quantum groups
- Wenzl
- 1998
(Show Context)
Citation Context ...ollows from the C ∗ -property of M(ρ,σ). � Remark. This result is probably well known among experts, but to the best of the author’s knowledge it never appeared in print. Yet it is used implicitly in =-=[32]-=- where certain categories are proved to have a positive ∗-operation and concluded to be C ∗ -categories. In the above result we did not assume irreducibility of the unit ι, viz. Hom(ι,ι) = C idι. From... |

31 | Mapping class group actions on quantum doubles - Kerler, Roberts - 1995 |

29 | Revêtements étales et groupe fondamental (SGA1 - Grothendieck - 1971 |

28 |
Ergodic actions of compact groups on operator algebras
- Wassermann
- 1989
(Show Context)
Citation Context ...nding spectral subspace has dimension at most d2 k .) This is an instance of a well-known general result in the theory of ergodic compact group actions on von Neumann algebras, cf. [12, Prop. 2.1] or =-=[31, I]-=-. 4 Galois Correspondence and the Modular Closure Throughout the section C is a BTC ∗ , S ⊂ C is a STC ∗ and G = AutC(C ⋊ S) ∼ = Gal(S). Having defined the semidirect product C ⋊ S and established its... |

25 |
A matrix S for all simple current extensions
- Fuchs, Schellekens, et al.
- 1996
(Show Context)
Citation Context ...Lρ ⊂ Kρ ⊂ K. Remark. The result that all irreducible components of ρ appear with the same multiplicity Nρ appears as the (unproved) assumption of ‘fixpoint homogeneity’ in conformal field theory, cf. =-=[9]-=-. Corollary 5.3 The irreducible sectors (isomorphism classes of irreducible objects) of C ⋊ S are labeled by pairs (ρ,χ). Here ρ ∈ ∆/∆S is an orbit of irreducibles in ∆ under the action of the group ∆... |

24 | rmer, Compact ergodic groups of automorphisms - egh-Krohn, Landstad, et al. - 1981 |

24 |
Turaev, Quantum invariants of knots and 3–manifolds, de Gruyter
- G
- 1994
(Show Context)
Citation Context ...ched a certain state of maturity when it was understood that the crucial ingredient underlying these invariants of 3-manifolds is a certain class of braided tensor categories which are called modular =-=[29]-=-. A modular category is a braided tensor category which (i) has a twist [29] or balancing [14], (ii) is rational – i.e. has only finitely many isomorphism classes of irreducible objects – and (iii) no... |

24 |
gories Tannakiennes
- Rivano, ‘‘Cate
- 1972
(Show Context)
Citation Context ...but they are implicit in the earlier Tannaka-Krein duality theory for compact groups. Further motivation for their analysis came from Grothendieck’s theory of motives and led to Saaveda Rivano’s work =-=[27]-=-, which was corrected and extended in [4]. These formalisms reconstruct a group (compact topological in the first, algebraic in the second case) from the category of its representations, the latter be... |

18 | Lane: Categories for the Working Mathematician, 2nd edition - Mac - 1998 |

17 | A theory of dimension
- Longo, Roberts
- 1997
(Show Context)
Citation Context ...ulfilled naturality of ε fails for some of the new morphisms of C ⋊0 S). Now, C ⋊0 S will be closed under direct sums, but usually not under subobjects. Thus we apply the above-mentioned procedure of =-=[19]-=- in order to obtain a category C ⋊ S which is closed under direct sums and subobjects. Then the braiding of C ⋊0 S – if it exists – extends to C ⋊ S. The result of this construction is again a tensor ... |

16 | Semisimple and modular categories from link invariants - Turaev, Wenzl - 1997 |

13 |
Introduction to the algebraic theory of superselection sectors (space-time dimension=2strictly localized morphisms
- Kastler, Mebkhout, et al.
- 1990
(Show Context)
Citation Context ...from link invariants, cf. [28, Chap. XII], [29]. Finally, braided tensor categories appear naturally also in the superselection theory of quantum field theories in low dimensional spacetimes, cf. eg. =-=[17]-=-. In many cases, as for the WZW and orbifold models, these categories actually turn out to be modular. Let A be a quantum field theory in 1 + 1 dimensions and let C be the braided category of supersel... |

13 | On charged fields with group symmetry and degeneracies of Verlinde’s matrix
- Müger
- 1999
(Show Context)
Citation Context ...[23] Rehren conjectured that the representation category of F is non-degenerate. Under the assumption that there are only finitely many irreducible degenerate sectors this was proved by the author in =-=[21]-=-. The aim of the present paper is to give a purely categorical analogue of this construction (without the finiteness restriction). More precisely, given a braided tensor category C which is enriched o... |

11 |
Orbifold construction in subfactors
- Xu
- 1994
(Show Context)
Citation Context ...ariant arising from the modular closure C, bypassing the construction of the latter. 5. There is an obvious connection between the crossed product C ⋊ S and the ‘orbifold constructions in subfactors’ =-=[8, 33]-=- which deserves to be worked out. 6. Generalize everything in this paper to the non-connected case where Hom(ι,ι) ̸= C idι and the compact (super)groups are replaced by compact (super)groupoids [2]. T... |

10 | Galois extensions of braided tensor categories and braided crossed G-categories
- Müger
(Show Context)
Citation Context ...paper.) We refrain from giving details since that would use too much space and will not be used in this paper. The claimed fact will be contained as a special case in a more general result, proved in =-=[22]-=-. � 3 Crossed Product of Braided Tensor ∗-Categories by Symmetric Subcategories 3.1 Definition of the Crossed Product We assume that C has direct sums and subobjects, which can be interpreted by sayin... |

8 |
The fundamental theorem of Galois theory
- Janelidze
- 1989
(Show Context)
Citation Context ...pact (super)groupoids [2]. The resulting Galois theory should resemble the Galois theory for commutative rings instead of the one for fields. 7. Since Janelidze’s general Galois theory for categories =-=[13]-=- was modeled on the Galois theory for commutative rings as expounded by Magid, it should be possible to show that with the proper identifications our Galois correspondence fits into Janelidze’s formal... |

7 |
Markov traces as characters for local algebras
- Rehren
- 1990
(Show Context)
Citation Context ...gument we see that naturality of the braiding ˜ε(ρ,σ) in C ⋊0 S fails for ˜ S = S ⊗ ψk ∈ HomC⋊0S(σ,η). � Remark. It is instructive to relate this result to what happens in the quantum field framework =-=[24, 21]-=-. There the observables A are extended by fields implementing the sectors in a symmetric semigroup ∆ of DHR endomorphisms and the localized sectors of A are extended to the fields F. If ∆ contains non... |

6 |
On the equality of q-dimension and intrinsic dimension
- Roberts, Tuset
(Show Context)
Citation Context ...ex of an inclusion of factors, cf. [19]. Note that the braiding does not play a role here, yet the categorical dimension coincides with the q-dimension for representation categories of quantum groups =-=[27]-=-.) The more specific notion of C ∗ -tensor categories will not be needed explicitly in this paper. But since we wish to make use of results of [10, 6, 19] we will prove that many tensor ∗-categories a... |

4 |
egh-Krohn, Ergodic actions by compact groups
- Albeverio, Ho
- 1980
(Show Context)
Citation Context ...), and HomC⋊S(ρ,ρ) is the twisted group algebra C c Kρ. (This result could also have been derived from the general theory of ergodic actions of compact abelian groups on von Neumann algebras, cf. eg. =-=[1]-=-.) Due to Te ∈ Hom(ρ,ρ) ∈ C idρ we can choose Te = idρ, which will always be assumed in the sequel. Now we need some group theoretical results. Lemma 5.1 Let A be a finite abelian group and c ∈ Z 2 (A... |

4 |
Rivano: Catégories Tannakiennes
- Saaveda
- 1972
(Show Context)
Citation Context ...but they are implicit in the earlier Tannaka-Krein duality theory for compact groups. Further motivation for their analysis came from Grothendieck’s theory of motives and led to Saaveda Rivano’s work =-=[28]-=-, which was corrected and extended in [4]. These formalisms reconstruct a group (compact topological in the first, algebraic in the second case) from the category of its representations, the latter be... |

3 | Revêtement Etales et Groupe Fondamental (SGA1), Lecture Note - Grothendieck - 1971 |

3 |
The reconstruction problem, unpublished manuscript
- Roberts
- 1971
(Show Context)
Citation Context ...tations of A and have nice properties like Bose-Fermi commutation relations. (In fact, assuming the duality theorem for abstract symmetric categories such a reconstruction result existed much earlier =-=[25]-=-.) At the same time and independently Deligne extended [5] the earlier works [27, 4] by identifying a necessary and sufficient condition for an abstract symmetric tensor category to be the representat... |

2 |
Baez: Higher-dimensional algebra. II. 2-Hilbert spaces
- C
- 1997
(Show Context)
Citation Context ...above result we did not assume irreducibility of the unit ι, viz. Hom(ι,ι) = C idι. From now on all categories in this paper will be assumed to have this property, which has been called connectedness =-=[2]-=-. We will remark on the disconnected case in the outlook. 5We summarize the properties of the categories we will study. Definition 2.2 A TC ∗ is a small strict tensor ∗-category with conjugates, dire... |

2 |
Tensor categories for operator algebraists, unpublished notes
- Yamagami
- 1998
(Show Context)
Citation Context ...so full and faithful, C and C are equivalent as categories, cf. [20, Sect. IV.4]. That this is in fact an equivalence of tensor categories requires an additional argument for which we refer, e.g., to =-=[34]-=-. Definition 3.12 C ⋊ S = C ⋊0 S. C is identified with a subcategory of C ⋊ S via the embedding ρ ↦→ (ρ,idρ), Hom(ρ,σ) ∋ S ↦→ S ⊗ Ω ∈ Hom((ρ,idρ),(σ,idσ)). Theorem 3.13 C⋊S is a C ∗ -tensor category w... |

1 |
Mebkhout & K.-H. Rehren: Introduction to the algebraic theory of superselection sectors (space-time dimension=2 - strictly localized morphisms
- Kastler, M
(Show Context)
Citation Context ...from link invariants, cf. [29, Chap. XII], [30]. Finally, braided tensor categories appear naturally also in the superselection theory of quantum field theories in low dimensional spacetimes, cf. eg. =-=[17]-=-. In many cases, as for the WZW and orbifold models, these categories actually turn out to be modular. Let A be a quantum field theory in 1 + 1 dimensions and let C be the braided category of supersel... |

1 |
Rehren: Braid group statistics and their superselection rules. In: [16
- K-H
(Show Context)
Citation Context ...valent to the unit object ι. (The designation of such categories as modular is owed to the fact that they give rise to a finite dimensional representation of the modular group SL(2, Z) [29], see also =-=[23]-=-.) The role of the quantum groups then reduces just to providing several infinite families of modular categories (roughly, one for every pair (root of unity, classical Lie algebra)). Another construct... |

1 |
Roberts: The reconstruction problem. Unpublished manuscript
- E
- 1971
(Show Context)
Citation Context ...tations of A and have nice properties like Bose-Fermi commutation relations. (In fact, assuming the duality theorem for abstract symmetric categories such a reconstruction result existed much earlier =-=[25]-=-.) At the same time and independently Deligne extended [5] the earlier works [28, 4] by identifying a necessary and sufficient condition for an abstract symmetric tensor category to be the representat... |

1 |
Roberts: Lectures on algebraic quantum field theory. In: [16
- E
(Show Context)
Citation Context ...eld theory it was realized around 1970 that the category of localized superselection sectors ( ∼ = physically relevant representations of the C ∗ -algebra A of observables) is symmetric monoidal, cf. =-=[26]-=-. This category being a category of endomorphisms of A – not of vector spaces – the existing duality theorems did not apply. This led Doplicher and Roberts to developing their characterization [6] of ... |

1 |
On mapping class group representations and simple current extensions
- Müger
(Show Context)
Citation Context ...paper.) We refrain from giving details since that would use too much space and will not be used in this paper. The claimed fact will be contained as a special case in a more general result, proved in =-=[22]-=-. � 3 Crossed Product of Braided Tensor ∗-Categories by Symmetric Subcategories 93.1 Definition of the Crossed Product We assume that C has direct sums and subobjects, which can be interpreted by say... |

1 |
Tensor categories for operator algebraists
- Yamagami
(Show Context)
Citation Context ... is also full and faithful, C and C are equivalent as categories, cf. [20, Sect. IV.4]. That this is in fact an equivalence of tensor categories requires another argument for which we refer, e.g., to =-=[33]-=-. Definition 3.12 C ⋊ S = C ⋊0 S. C is identified with a subcategory of C ⋊ S via the embedding ρ ↦→ (ρ,idρ), (ρ,σ) ∋ S ↦→ S ⊗ Ω ∈ ((ρ,idρ),(σ,idσ)). Theorem 3.13 C⋊S is a C ∗ -tensor category with co... |