In this thesis, we consider several aspects of over-extended and
very-extended Kac-Moody algebras in relation with theories of gravity coupled
to matter. In the first part, we focus on the occurrence of KM algebras in the
cosmological billiards. We analyse the billiards in the simplified situation of
spatially homogeneous cosmologies. The most generic cases lead to the same
algebras as those met in the general inhomogeneous case, but also sub-algebras
of the "generic" ones appear. Next, we consider particular gravitational
theories which, upon toroidal compactification to D=3 space-time dimensions,
reduce to a theory of gravity coupled to a symmetric space non-linear
sigma-model. We show that the billiard analysis gives direct information on
possible dimensional oxidations (or on their obstructions) and field content of
the oxidation endpoint. We also consider all hyperbolic Kac-Moody algebras and
completely answer the question of whether or not a specific theory exists
admitting a billiard characterised by the given hyperbolic algebra. In the
second part, we turn to the set up of such gravity-matter theories through the
building of an action explicitly invariant under a Kac-Moody group. As a first
step to include fermions, we check the compatibility of the presence of a Dirac
fermion with the (hidden duality) symmetries appearing in the toroidal
compactification down to 3 space-time dimensions. Next, we investigate how the
fermions (with spin 1/2 or 3/2) fit in the conjecture for hidden over-extended
symmetry G++. Finally, in the context of G+++ invariant actions, we derive all
the possible signatures for all the GB++ theories that can be obtained from the
conventional one (1,D-1) by "dualities" generated by Weyl reflections. This
generalizes the results obtained for E8++.Comment: Ph.D. Thesis, Universite Libre de Bruxelles, June 2006 (232 pages
