I'm trying to understand the representation theory of the affine Lie algebra su(2) level k (which describe anyonic theories), which can be thought of as a deformation of the standard su(2) Lie algebra.

In particular, is there an analogous way of constructing the Hilbert space described by su(2)_k to that of su(2)? For the latter, one can start with a highest weight state, and repeatedly apply lowering operators to span the entire Hilbert space.

Hence, my question is: is there such a raising/lowering operator and what charge, if any, does it lower?