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 Bruguières and Müger, 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.
