Representation theory of su(2)_k

Question

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? 

Answer 1

You can think of the representations of $SU(2)_k$ as decomposing the Hilbert space of the $SU(2)_k$ WZW CFT. The "highest weight vectors" are the conformal primaries, where the weight is minus the scaling dimension.

You can think of the current algebra objects $J^a_n$ as defining raising and lowering operators by $J^\pm_n :=J^1_n \pm i J^2_n$ just like in usual $SU(2)$ except now the charge can be carried by any mode of the string. We find $[J^3_m,J^\pm_n] = \pm J^{\pm}_{n+m}$ so $J^3_0$ can play the role of the weight.