Affine current algebra ($[J_m^a,J_n^b]=i{f}^{abc}{J}_{m+n}^c+km{\delta }_{m+n,0}{\delta }^{ab}$)

The affine current algebra is the one-dimensional central extension of the loop algebra of a finite-dimensional Lie algebra, with central element acting as the level $k$. A restricted module has $J_{n}v=0$ for all sufficiently large $n$ for each vector $v$; this makes the Sugawara construction finite on each vector after normal ordering. In a compact unitary positive-energy representation, the nontrivial integrable levels are positive integers in the standard basic normalization.