$E_{7(7)}$ Duality, BPS Black-Hole Evolution and Fixed Scalars

We study the general equations determining BPS Black Holes by using a Solvable Lie Algebra representation for the homogenous scalar manifold U/H of extended supergravity. In particular we focus on the N=8 case and we perform a general group theoretical analysis of the Killing spinor equation enforcing the BPS condition. Its solutions parametrize the U-duality orbits of BPS solutions that are characterized by having 40 of the 70 scalars fixed to constant values. These scalars belong to hypermultiplets in the N=2 decomposition of the N=8 theory. Indeed it is shown that those decompositions of the Solvable Lie algebra into appropriate subalgebras which are enforced by the existence of BPS black holes are the same that single out consistent truncations of the N=8 theory to intereacting theories with lower supersymmetry. As an exemplification of the method we consider the simplified case where the only non-zero fields are in the Cartan subalgebra H of Solv(U/H) and correspond to the radii of string toroidal compactification. Here we derive and solve the mixed system of first and second order non linear differential equations obeyed by the metric and by the scalar fields. So doing we retrieve the generating solutions of heterotic black holes with two charges. Finally, we show that the general N=8 generating solution is based on the 6 dimensional solvable subalgebra Solv [(SL(2,\IR) /U(1))^3].