Kac-Moody algebras and Lie algebras of regular vector fields on tori
arXiv:math/0109090 · doi:10.1016/S0021-8693(02)00054-6
Abstract
We consider the problem of representing the Kac-Moody algebra specified by an indecomposable generalised Cartan matrix as vector fields on the torus ${{\bb C}^*}^r$. It is shown that, if the representations are of a certain form, this is possible if and only if $\mathfrak{g}(N)\cong sl(r+1,{\bb C})$ or $\tilde{sl}(r,{\bb C})$. For $sl(r+1,{\bb C})$ and $\tilde{sl}(r,{\bb C})$, discrete families of representations are constructed. These generalise the well-known discrete families of representations of $sl(2,{\bb C})$ as regular vector fields on ${\bb C}^*$
20 pages, typo corrected in the title