Loop algebra

Summary

The loop algebra of a Lie algebra $\mathfrak{g}$ is the Lie algebra $\mathfrak{g} \otimes k[t, t^{-1}]$ of Laurent polynomial maps into $\mathfrak{g}$, with the pointwise bracket. Over $\C$, its elements can be seen as maps from the circle $\abs{t} = 1$ to $\mathfrak{g}$ with finite Fourier series. It is the infinitesimal version of a loop group. Adding a one-dimensional center and a derivation to it gives the untwisted affine Lie algebras.