D-dimensional p-brane cosmological models associated with a Lie algebra ofthe type A(m)

We study a D-dimensional cosmological model on the manifold M = R x M-o x . .. x M-n, describing an evolution of n+1 Einstein factor spaces M-i in a th eory with several dilatonic scalar fields and differential forms admitting an interpretation in terms of intersecting p-branes. The equations of motio n of the model are reduced to the Euler-Lagrange equations for the so-calle d pseudo-Euclidean Toda-like system. Assuming that the characteristic vecto rs related to the configuration of p-branes and their couplings to the dila tonic scalar fields can be interpreted as the root vectors of a Lie algebra of the type A(m) = sl(m+1, C), we reduce the model to an open Toda chain, which is integrable by the customary methods. The resulting metric has the form of the Kasner solution. We single out the particular model describing the Friedman-like evolution of the three-dimensional external factor space M-o (in the Einsteinian conformal gauge) and the contraction of the interna l factor spaces M-1,..., M-n.