Almost split real forms for hyperbolic Kac-Moody Lie algebras

A Borel-Tits theory was developped for almost split forms of symmetrizable Kac-Moody Lie algebras [J. of Algebra 171, 43-96 (1995)]. In this paper, we look to almost split real forms for symmetrizable hyperbolic KacMoody Lie algebras and 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. Introduction. Since their appearance in the late 1960s, as generalizations of semi-simple complex Lie algebras, the (infinite-dimensional) Kac-Moody Lie algebras have played an increasingly crucial role in various areas of mathematics as well as theoretical physics. The hyperbolic Kac-Moody Lie algebras (which constitute a subclass of Lorentzian Kac-Moody algebras [22]) and some of their (almost split) real forms have appeared, besides the affine Kac-Moody algebras, in a variety of problems in the realms of string theory ([14], [13], ...) and supergravity theories ([34], [16], ...). Almost split forms of symmetrizable Kac-Moody Lie algebras were studied in [28], [29], [30] and [2] for an arbitrary field of characteristic 0 : A Borel-Tits Theory was developed for these forms and a classification in the real case (in terms of the Satake-Tits index with the corresponding relative root system) was done for affine Lie algebras ([2]). In [27], G. Rousseau gave a realization, in terms of the loop algebras, for all the almost split real forms of affine Lie algebras. The same construction was done by V. Back for an arbitrary field of characteristic 0 instead of the real field ([2], §5). Some forms (which may be almost anisotropic or almost compact in the real case) of symmetrizable Kac-Moody algebras are defined by generators and relations ([1], [10]). Almost compact real forms of affine Kac-Moody algebras were studied in [4] and [26] and entirely classified in [7]. The conjugate classes of their Cartan subalgebras were classified in [8]. This paper is devoted to the classification (in terms of the Satake-Tits index) of almost split real forms for some symmetrizable hyperbolic Kac-Moody Lie algebras (namely, the strictly hyperbolic Kac-Moody algebras and Kac-Moody Lie algebras which are obtained as (canonical) Lorentzian extensions of affine Lie algebras) which we consider the most met in supergravity theories ([16]). The paper is organized as follows. In section 1, we recall the construction of (symmetrizable) Kac-Moody Lie algebras and groups from the so-called generalized Cartan matrices and we set the notations. We give also a description of the automorphisms group and the invariant bilinear form for any indecomposable and 2000 Mathematics Subject Classification. Primary 17B67; Secondary 83E50.