Kac–Moody algebras and Lie algebras of regular vector fields on tori

We consider the problem of representing the Kac–Moody algebra specified by an r×r indecomposable generalized Cartan matrix N as vector fields on the torus . It is shown that, if the representations are of a certain form, this is possible if and only if is isomorphic to either or . For and , discrete families of representations are constructed. These generalize the well-known discrete families of representations of as regular vector fields on .