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\rtimes S. This construction yields a tensor *category with conjugates and an irreducible unit. (A *category is a category enriched over Vect_C with positive *operation.) A Galois correspondence is established between intermediate categories sitting between C and C\rtimes S and closed subgroups of the Galois group Gal(C\rtimes S/C)=Aut_C(C\rtimes 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\subset 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\rtimes S iff S\subset D. Under this condition C\rtimes 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\rtimes 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\rtimes S can be clarified quite explicitly in terms of group cohomology.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 December 1998
 DOI:
 10.48550/arXiv.math/9812040
 arXiv:
 arXiv:math/9812040
 Bibcode:
 1998math.....12040M
 Keywords:

 Mathematics  Category Theory;
 High Energy Physics  Theory
 EPrint:
 latex2e, 39 pages. Style file included via filecontents command. Final version, to appear in Adv. Math. Purely notational improvements