Using the tensor category theory developed by Lepowsky, Zhang and the second author, we construct a braided tensor category structure with a twist on a semisimple category of modules for an affine Lie algebra at an admissible level. We conjecture that this braided tensor category is rigid and thus is a ribbon category. We also give conjectures on the modularity of this category and on the equivalence with a suitable quantum group tensor category. In the special case that the affine Lie algebra is sl 2 , we prove the rigidity and modularity conjectures.Proposition 7.2 Let C be a ribbon category, s.t. for any simple object X the ribbon twist θ X = θ X Id X with | θ X | = 1. Let X, Y be simple objects of C, then θ X⊠Y is a scalar if and only if |S X,Y | = dim(X) dim(Y ).
Proof. Recall the balancing axiom of a ribbon categoryAssuming that θ X⊠Y is a scalar and then taking the absolute value of the trace of both sides, one gets the identityBut dim(X ⊠ Y ) = dim(X) dim(Y ). Conversely, assume that |S X,Y | = dim(X) dim(Y ). The balancing axiom implies that |S X,Y | = |Tr X⊠Y (θ X⊠Y ) |. So that by Cauchy-Schwarz inequality this is only possible if the twist acts by the same scalar on each simple summand of X ⊠ Y .