Orbits of exceptional groups, duality and BPS states in string theory

We give an invariant classification of orbits of the fundamental representations of the exceptional groups E-7(7) and E-6(6) which classify BPS states in string and M theories toroidally compactified to d = 4 and 5. The exceptional Jordan algebra and the exceptional Freudenthal triple system and their cubic and quartic invariants play a major role in this classification. The cubic and quartic invariants correspond to the black hole entropy in d = 5 and d = 4, respectively. The classification of BPS states preserving different numbers of supersymmetries is in close parallel to the classification of the little groups and the orbits of timelike, lightlike and spacelike vectors in Minkowski space. The orbits of BPS black holes in N = 2 Maxwell-Einstein supergravity theories in d = 4 and d = 5 with symmetric space geometries are also classified including the exceptional N = 2 theory, which has E7(-25) and E6(-26) as its symmetry in the respective dimensions.