Galois Theory for Braided Tensor Categories and the Modular Closure
✍ Scribed by Michael Müger
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 533 KB
- Volume
- 150
- Category
- Article
- ISSN
- 0001-8708
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✦ Synopsis
Given a braided tensor V-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 V -category with conjugates and an irreducible unit. (A V -category is a category enriched over Vect C with positive V -operation.) A Galois correspondence is established between intermediate categories sitting between C and C < S and closed subgroups of the Galois group Gal(C < SÂC)=Aut C (C < S) of C, the latter being isomorphic to the compact group associated with 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). 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.