Galois Theory for Braided Tensor Categories and the Modular Closure

• Published in: Advances in mathematics (New York. 1965) 2000-03, Vol.150 (2), p.151-201
• Main Author: Müger, Michael
• Format: Article
• Language: English
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1006/aima.1999.1860

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 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.