Galois Theory for Braided Tensor Categories and the Modular Closure

Michael Müger
December 07, 1998
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 $\mathrm{Gal}(C\rtimes S/C)=\mathrm{Aut}_C(C\rtimes S)$ 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 $\mathrm{SL}(2,\mathbb{Z})$. (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.
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braided tensor *-category, crossed product