Generalized Kac-Moody algebras, 
Richard Borcherds
J.Alg. Vol 115, No. 2, June 1988, p. 501-512. 

We study a class of Lie algebras which have a contravariant bilinear form 
which is almost positive definite. These algebras generalize Kac-Moody 
algebras, and can be thought of as Kac-Moody algebras with imaginary simple 
roots. Most facts about Kac-Moody algebras generalize to these new algebras; 
for example, we prove a version of the Kac-Weyl character formula, which is 
like the usual one except that it has an extra correction term for the 
imaginary simple roots. 

There are several ways in which these new algebras turn up. The fixed point 
algebra of any Kac-Moody algebra under a diagram automorphism is not usually 
a Kac-Moody algebra, but is one of these more general algebras. There is also 
a generalized Kac-Moody algebra associated to any even Lorentzian lattice of 
dimension at most 26 or any Lorentzian lattice of dimension at most 10, and 
we give a simple formula for the multiplicities of the roots of these 
algebras (but unfortunately I do not know what the Cartan matrices are)! The 
numbers 10 and 26 come from the ``no ghost''theorem.