Allowed anyons for Chern-Simons at level $k.$ Ref.1. proves that the allowed representations of Chern-Simons $\mathrm{SU}(2)_k$ are those with dimension
$$
\dim(R)\le k+1\tag{7.53}
$$
Question: Is the generalisation of $(7.53)$ to arbitrary $N$ known? What about arbitrary (semisimple) Lie groups $G$?
Furthermore, the author also proves that the fusion rules for $\mathrm{SU}(2)_k$ are
$$
R_{j_1}\times R_{j_2}=\sum_{j_3=|j_1-j_2|}^{b(j_1,j_2)} R_{j_3}\tag{7.54}
$$
with $b(j_1,j_2)=\min(j_1+j_2,k-j_1-j_2)$.
Question: Is the generalisation of $(7.54)$ to arbitrary $N$ known? i.e., where do we truncate the Littlewood-Richardson decomposition of $R_1\times R_2\in\mathrm{Rep}(\mathrm{SU}(N))^2$? As before, what about other Lie groups $G$?
References.


*

*Pachos, J.K. - Introduction To Topological Quantum Computation.

 A: These representations are called integrable representations. In the case of a general compact semisimple Lie group, a highest weight representation descends from a highest weight:
$$\lambda = \sum_i n_i w_i, \quad i = 1, ...,r$$
Where $r$ is the rank, $w_i$ are the fundamental weights and $n_i \in \mathbb{Z}^+$.
The above representation is integrable for a level $k$ if for all $i$
$$0\le n_i \le k$$
The reasons for this condition can be understood qualitatively as follows:
The Gauss law constraint of the Chern-Simons theory on the disc in the presence of an infinitesimal Wilson loop at $x_0$ corresponding to the representation $\lambda$  is given by:
$$\frac{k}{2\pi} F^a_{12} = i T^a_{(\lambda)} \delta^2(x-x_0)$$
Witten equation 3.4. (Witten explains this subject in words in the next few paragraphs)
Where $ T^a_{(\lambda)} $ is a generator of the Lie algebra in the representation $\lambda$, which can always be diagonalized as:
$$T^a_{(\lambda)} = g H_{(\lambda)}  g^{-1}$$
Where $ H_{(\lambda)}  $ is in the Cartan subalgebra.
The holonomy of the connection solving the Gauss law has the form:
$$U = e^ {\frac{2 \pi i}{k} g H_{(\lambda)}  g^{-1} \phi}$$
Where $\phi$ is the rotation angle around the insertion point.
Since the (diagonal) matrix elements of $H_{(\lambda)}   $ are less or equal to the highest weight components, thus due to the pre-factor $\frac{2 \pi}{k}$, a change by integer multiples of $k$ does not change the holonomy.
These representations are named integrable, because the level $k$ Kac-Moody algebras that are based on them generate representations the corresponding  Kac-Moody groups, Please see Goddard and Olive.
