Almost split real forms for hyperbolic Kac–Moody Lie algebras

Abstract

A Borel–Tits theory was developed for almost split forms of symmetrizable Kac–Moody Lie algebras. In this paper, we look to almost split real forms and their restricted root systems for symmetrizable hyperbolic Kac–Moody Lie algebras. We establish a complete list of these forms, in terms of their Satake–Tits index, for the strictly hyperbolic ones and for those which are obtained as (hyperbolic) canonical Lorentzian extensions of affine Lie algebras. These forms are of particular interest in theoretical physics because of their connection to supergravity theories.