= Sugawara construction
{c}
{title2=$L_m=(2k+Q)^{-1}\sum_{a,p}:J^a_{m-p}J^a_p:$}
{wiki}
For a restricted representation of an affine current algebra with $[J_m^a,J_n^b]=if^{abc}J_{m+n}^c+km\delta^{ab}\delta_{m+n,0}$ and $f^{abc}f^{abd}=Q\delta^{cd}$, the displayed <normal ordering> construction is defined when $2k+Q\ne0$. Moving the currents past each other gives $[L_m,J_n^a]=-nJ_{m+n}^a$; the additional $Q$ in the denominator is the adjoint-Casimir reordering term. The resulting <Virasoro algebra> has <central charge> $c=k\dim\mathfrak g/(k+Q/2)$. On a compact unitary positive-energy module, $L_m^\dagger=L_{-m}$.
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