Purely imaginary roots of kac-moody algebras

In this paper, we introduce a new class of imaginary roots, called purely imaginary roots and give a necessary and sufficient condition for an imaginary root to be purely imaginary,thereby giving a complete classification of those Kac-Moody algebras with the purely imaginary property, namely, all their imaginary roots are purely imaginary. We also define a new class of non-hyperbolic indefinite Kac-Moody algebras, called the extended hyperbolic kac-Moody algebras possessing the purely imaginary property.