Central extensions of generalized Kac-Moody algebras
Richard Borcherds
J. Algebra. Vol 140 No. 2, July 1991, 330-335.
The main result of Borcherds [1] states that graded Lie algebras with an
``almost positive definite'' contravariant bilinear form are essentially
the same as central extensions of generalized Kac-Moody algebras. In this
paper we calculate these central extensions. Ordinary Kac-Moody algebras
have nontrivial centers when the Cartan matrix is singular; generalized
Kac-Moody algebras turn out to have some ``extra'' center in their universal
central extensions whenever they have simple roots of multiplicity greater
than 1, and in particular the dimension of the Cartan subalgebra can be
larger than the number of rows of the Cartan matrix.