• DOI:10.1006/AIMA.1999.1860
• Corpus ID: 1858268

Galois Theory for Braided Tensor Categories and the Modular Closure

@article{Mueger1998GaloisTF,
  title={Galois Theory for Braided Tensor Categories and the Modular Closure},
  author={Michael Mueger},
  journal={Advances in Mathematics},
  year={1998},
  volume={150},
  pages={151-201}
}

• Michael Mueger
• Published 7 December 1998
• Mathematics
• Advances in Mathematics

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 Vect C 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 )=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.