Nilpotent Lie algebras of Kac–Moody affine type

The nilpotent Lie algebras are grouped into Kac–Moody types: simple, affine, hyperbolic. We study the affine type and show that it can be reduced to the simple type concerning the discrete series whereas the continuous families require in addition the study of grassmanians. The explicit list for the rank 2 illustrates the whole article.