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Abstract

The hyperbolic (and more generally, Lorentzian) Kac-Moody (KM) Lie algebras A of rank r+2 > 2 are shown to have a rich structure of indefinite KM subalgebras which can be described by specifying a subset of positive real roots of A such that the difference of any two is not a root of A. Taking these as the simple roots of the subalgebra gives a Cartan matrix, generators and relations for the subalgebra. Applying this to the canonical example of a rank 3 hyperbolic KM algebra, F, we find that F contains all of the simply laced rank 2 hyperbolics, as well as an infinite series of indefinite KM subalgebras of rank 3. It is shown that A also contains Borcherds algebras, obtained by taking all of the root spaces of A whose roots are in a hyperplane (or any proper subspace). This applies as well to the case of rank 2 hyperbolics, where the Borcherds algebras have all their roots on a line, giving the simplest possible examples.