MISSING MODULES, THE GNOME LIE-ALGEBRA, AND E-10

We study the embedding of Kac-Moody algebras into Borcherds (or genera lized Kac-Moody) algebras which can be explicitly realized as Lie alge bras of physical states of some completely compactified bosonic string . The extra ''missing states'' can be decomposed into irreducible high est or lowest weight ''missing modules'' w.r.t, the relevant Kac-Moody subalgebra; the corresponding lowest weights are associated with imag inary simple roots whose multiplicities can be simply understood in te rms of certain polarization states of the associated string, We analys e in detail two examples where the momentum lattice of the string is g iven by the unique even unimodular Lorentzian lattice II1,1 or II9,1, respectively. The former leads to the Borcherds algebra g(II1,1), whic h we call ''gnome Lie algebra'', with maximal Kac-Moody subalgebra A(1 ). By the use of the denominator formula a complete set of imaginary s imple roots can be exhibited, while the DDF construction provides an e xplicit Lie algebra basis in terms of purely longitudinal states of th e compactified string in two dimensions. The second example is the Bor cherds algebra g(II9,1), whose maximal Kac-Moody subalgebra is the hyp erbolic algebra E-10. The imaginary simple roots at level 1, which giv e rise to irreducible lowest weight modules for E-10, can be completel y characterized; furthermore, our explicit analysis of two non-trivial level-2 root spaces leads us to conjecture that these are in fact the only imaginary simple roots for g(II9,1).