Subalgebras of Hyperbolic Kac-Moody Algebras

Document Type

Book Chapter

Publication Date



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.


The work is dedicated in memory of Peter Slodowy.

Publisher Attribution

Final version published in Contemporary Mathematics, 343 (January 2003), published by the American Mathematical Society.