Closed subsets of the Weyl alcove and TQFTs

Abstract

For an arbitrary simple Lie algebra g and an arbitrary root of unity q, we classify the closed subsets of the Weyl alcove of the quantum group Uq(g). Here a closed subset is a set such that if any two weights in the Weyl alcove are in the set, so is any weight in the Weyl alcove which corresponds to an irreducible summand of the tensor product of a pair of representations with highest weights the two original weights. The ribbon category associated to each closed subset admits a ?quotient? by a trivial subcategory as described by Bruguieres and Muger, to give a modular category and a framed three-manifold invariant or a spin modular category and a spin three-manifold invariant, as proved by the author. Most of these theories are equivalent to theories defined in Sawin, Adv. Math. 165 (2002), 1?70, but several exceptional cases represent the first nontrivial examples of theories that contain noninvertible trivial objects, making the theory much richer and more complex.