Loop current Lie algebra and central term

Let $\mathfrak{g}$ be a finite-dimensional real Lie algebra. Let $\kappa :\mathfrak{g}\times \mathfrak{g}\to \mathbb{R}$ be a symmetric bilinear form that is invariant, meaning $\kappa ([U,V],W)=\kappa (U,[V,W])(U,V,W\in \mathfrak{g})\mathrm{.}$ No nondegeneracy, positivity, simplicity, or compactness assumption is imposed. Put $S^1=\mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$ and $\mathcal{L}\mathfrak{g}=C^{\mathrm{\infty }}(S^1,\mathfrak{g})$, with pointwise bracket $[X,Y](\theta )=[X(\theta ),Y(\theta ){]}_{\mathfrak{g}}$. For $Y\in \mathcal{L}\mathfrak{g}$, write $Y^{\mathrm{\prime }}={\mathrm{\partial }}_{\theta }Y$ in the indicated periodic coordinate.

A Lie-algebra $2$-cocycle for the trivial scalar action is an alternating bilinear map $\omega :\mathcal{L}\mathfrak{g}\times \mathcal{L}\mathfrak{g}\to \mathbb{R}$ satisfying $\omega ([X,Y],Z)+\omega ([Y,Z],X)+\omega ([Z,X],Y)=0.$ Define the candidate current cocycle and its candidate central-term bracket by $\omega (X,Y)=\frac{1}{2\pi }{\int }_0^{2\pi }\kappa (X(\theta ),Y^{\mathrm{\prime }}(\theta ))d\theta ,\widehat{\mathcal{L}\mathfrak{g}}=\mathcal{L}\mathfrak{g}\oplus \mathbb{R},$ $[(X,a),(Y,b)]=([X,Y],\omega (X,Y))\mathrm{.}$ The real coordinates $a,b$ are central coordinates and do not enter the right-hand side. A one-dimensional central extension means a short exact sequence of Lie algebras $0\to \mathbb{R}\to \widehat{\mathcal{L}\mathfrak{g}}\to \mathcal{L}\mathfrak{g}\to 0$ whose kernel is central. The proposition on this page verifies that the displayed operation has this Lie-algebra structure. The normalization is fixed as written; no integer level, integration to a loop group, or representation-theoretic existence or unitarity follows from this definition.