  This paper investigates the algebraic structure that exists on perturbative
BPS-states in the superstring, compactified on the product of a circle and a
Calabi-Yau fourfold. This structure was defined in a recent article by Harvey
and Moore. It shown that for a toroidal compactification this algebra is
related to a generalized Kac-Moody algebra. The BPS-algebra itself is not a
Lie-algebra. However, it turns out to be possible to construct a Lie-algebra
with the same graded dimensions, in terms of a half-twisted model. The
dimensions of these algebras are related to the elliptic genus of the
transverse part of the string algebra. Finally, the construction is applied to
an orbifold compactification of the superstring.
