exclusively based on a decomposition into irreducible representations of their affine subalgebras. (Decompositions with respect to finite subalgebras seem to have received scant attention so far.) Although the theory of representations of affine algebras is reasonably well understood, this approach has not yielded much information about the behavior of the algebra “deep within the light-cone”. A complete analysis was carried out up to affine level two in [@FF] for the rank 3 hyperbolic algebra, $\cF$ (also referred to as $HA_1^{(1)}$), containing the affine algebra $A_1^{(1)}$. This work was generalized in [@KMW] to the rank 10 hyperbolic KM algebra $E_{10}$, where the first two levels with respect to the affine subalgebra $E_9 = E_8^{(1)}$ were analyzed. This analysis has been extended as far as level 5 for $\cF$ and also carried out for some levels for some other algebras [@BKM1; @BKM2; @BKM3; @Ka1; @Ka2; @Ka3; @Ka4; @Ka5; @KaM1; @KaM2], but appears rather hopeless as a method for giving all levels. Finally, DDF operators have been used in [@GN] to probe some root spaces in detail.

We conclude this introduction with a brief description of the main results in each section. In section 2 we give the necessary background and details about 