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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?

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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.

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    $\begingroup$ How is this related to the $SU(2)_k$ fusion rules defined on $0,\frac{1}{2},\cdots, \frac{k}{2}$ such that $j_1 \times j_2 = |j_1-j_2| + \cdots + \min \{ j_1+j_2, k - j_1-j_2\}$ ? $\endgroup$ – Ruben Verresen Jul 19 '17 at 17:04

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