ON E(10) AND THE DDF CONSTRUCTION

An attempt is made to understand the root spaces of Kac Moody algebras of hyperbolic type, and in particular E(10), in terms of a DDF constr uction appropriate to a subcritical compactified bosonic string. While the level-one root spaces can be completely characterized in terms of transversal DDF states (the level-zero elements just span the affine subalgebra), longitudinal DDF states are shown to appear beyond level one, In contrast to previous treatments of such algebras, we find it n ecessary to make use of a rational extension of the self-dual root lat tice as an auxiliary device, and to admit non-summable operators (in t he sense of the vertex algebra formalism). We demonstrate the utility of the method by completely analyzing a non-trivial level two root spa ce, obtaining an explicit and comparatively simple representation for it. We also emphasize the occurrence of several Virasoro algebras, who se interrelation is expected to be crucial for a better understanding of the complete structure of the Kac Moody algebra.