= Coset construction
{c}
{title2=$L_n^{\mathfrak g/\mathfrak h}=L_n^{\mathfrak g}-L_n^{\mathfrak h}$}
{wiki}
For a current subalgebra $\mathfrak h\subset\mathfrak g$, its currents have conformal weight one under both <Sugawara construction> stress tensors. Their difference therefore commutes with all $\mathfrak h$ currents and with the $\mathfrak h$ stress tensor. It obeys the <Virasoro algebra> with <central charge> $c_{\mathfrak g}-c_{\mathfrak h}$. For a positive-level compact unitary vacuum representation, $\|L_{-2}^{\mathfrak g/\mathfrak h}|0\rangle\|^2=(c_{\mathfrak g}-c_{\mathfrak h})/2\ge0$. A <Cartan subalgebra> has $c_{\mathfrak h}=\operatorname{rank}\mathfrak g$, giving the lower bound on $c_{\mathfrak g}$ under these hypotheses.
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