We study the embedding of Kac-Moody algebras into Borcherds (or generalized
Kac-Moody) algebras which can be explicitly realized as Lie algebras of
physical states of some completely compactified bosonic string. The extra
``missing states'' can be decomposed into irreducible highest or lowest weight
``missing modules'' w.r.t. the relevant Kac-Moody subalgebra; the corresponding
lowest weights are associated with imaginary simple roots whose multiplicities
can be simply understood in terms of certain polarization states of the
associated string. We analyse in detail two examples where the momentum lattice
of the string is given by the unique even unimodular Lorentzian lattice
II1,1 or II9,1, respectively. The former leads to the Borcherds
algebra g1,1, which we call ``gnome Lie algebra", with maximal Kac-Moody
subalgebra A1. By the use of the denominator formula a complete set of
imaginary simple roots can be exhibited, whereas the DDF construction provides
an explicit Lie algebra basis in terms of purely longitudinal states of the
compactified string in two dimensions. The second example is the Borcherds
algebra g9,1, whose maximal Kac-Moody subalgebra is the hyperbolic algebra
E10. The imaginary simple roots at level 1, which give rise to irreducible
lowest weight modules for E10, can be completely 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
g9,1.Comment: 31 pages, LaTeX2e, AMS packages, PSTRICK