Gradient representations and affine structures in AEn

We study the indefinite Kac-Moody algebras AE(n), arising in the reduction of Einstein's theory from (n + 1) spacetime dimensions to one (time) dimension, and their distinguished maximal regular subalgebras A(n-1 equivalent to) sl(n), and A(n-2)((1)). The interplay between these two subalgebras is used, for n 3, to determine the commutation relations of the 'gradient generators' within AE(3). The low-level truncation of the geodesic or sigma-model over the coset space A E-n/K(AE(n)) is shown to map to a suitably truncated version of the SL(n)/SO(n) nonlinear or sigma-model resulting from the reduction Einstein's equations in (n + 1) dimensions to (1 + 1) dimensions. A further truncation to diagonal solutions can be exploited to define a one-to-one correspondence between such solutions, and null geodesic trajectories on the infinite-dimensional coset space h/K(h), where h is the (extended) Heisenberg group, and K(h) its maximal compact subgroup. We clarify the relation between h and the corresponding subgroup of the Geroch group.