'Alex J. Feingold'
- Hermann Nicolai
title: 'Subalgebras of Hyperbolic Kac-Moody Algebras'
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[*We dedicate this work to the memory of our friend Peter Slodowy.*]{}

Introduction
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More than thirty years after their discovery, indefinite and, more specifically, hyperbolic Kac-Moody (KM) algebras remain an unsolved challenge of modern mathematics. For instance, there is not a single such algebra for which the root multiplicities are known in closed form. Even less is known about the detailed structure of the imaginary root spaces, and a manageable explicit realization analogous to the current algebra realization of affine algebras appears to be beyond reach presently. (However, there are intriguing hints that a “physical” realization might be found in an extension of Einstein’s theory of general relativity, see [@DHN] and references therein). Given this lack of knowledge, any new information about these algebras may be of potential value and importance.

In this note we point out that hyperbolic KM algebras possess a very rich structure of nonisomorphic infinite dimensional subalgebras, all of which can be generated by a set of root vectors $E_\alpha$ and $F_\alpha$, where certain roots $\alpha$ from the root system of the original algebra are taken as new simple roots. If 