Mathematics – Quantum Algebra
Scientific paper
2003-03-14
Mathematics
Quantum Algebra
18 pages, 2 figures, LaTeX using Contemporary Math proceedings style file
Scientific paper
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.
Feingold Alex J.
Nicolai Hermann
No associations
LandOfFree
Subalgebras of Hyperbolic Kac-Moody Algebras does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Subalgebras of Hyperbolic Kac-Moody Algebras, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Subalgebras of Hyperbolic Kac-Moody Algebras will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-292167