# What is a su(2) level k algebra

What is meant by su(2) level k algebra ? Is it a lie algebra of some lie group ? What is the relation with SU(2) group. I see it in the context of quantum hall edges. Googling and google-booking for a definition always takes me to explanations in the contexts of cft and wzw models. But what is the mathematical definition of this object?

The mathematical object is called the affine Kac-Moody algebra, which is sort of an infinite-dimensional generalization of the usual Lie algebra. For example, the generators $J_m^a$ of $SU(2)_k$ current algebra satisfy $[J_m^a, J_n^b]=if^{abc}J_{m+n}^c + ikm\delta^{ab}\delta_{m+n,0}$ Here $f^{abc}$ is the structure constant of SU(2) Lie algebra, and $m,n\in \mathbb{Z}$. If you restrict to $m=0$, you get exactly the Lie algebra.
• How is this related to the $SU(2)_k$ fusion rules defined on $0,\frac{1}{2},\cdots, \frac{k}{2}$ such that $j_1 \times j_2 = |j_1-j_2| + \cdots + \min \{ j_1+j_2, k - j_1-j_2\}$ ? – Ruben Verresen Jul 19 '17 at 17:04