Galois Theory for Braided Tensor Categories and the Modular Closure
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 degenerate objects iff S=D. If the original category C is rational (i.e. has only finitely many equivalence classes of irreducible objects) then the same holds for the new one. The category 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.
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