Subalgebras of Hyperbolic Kac-Moody Algebras
Quantum Algebra, High Energy Physics - Theory
The hyperbolic (and more generally, Lorentzian) Kac-Moody (KM) Lie algebras $\cA$ 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 $\cA$ such that the difference of any two is not a root of $\cA$. 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, $\cF$, we find that $\cF$ 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 $\cA$ also contains Borcherds algebras, obtained by taking all of the root spaces of $\cA$ 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.
Final version published in Contemporary Mathematics, 343 (January 2003), published by the American Mathematical Society.
Feingold, A. J., & Nicolai, H. (2004). Subalgebras of hyperbolic Kac-Moody algebras. Contemporary Mathematics, 343, 97-114. doi: doi: 10.1090/conm/343/06184