$D$-dimensional $p$-brane cosmological models associated with a Lie algebra of the type $A_m$

We study a $D$-dimensional cosmological model on the manifold $\mathbf M= \mathbb R\times M_0\times\cdots\times M_n$ describing an evolution of $n+1$ Einstein factor spaces $M_i$ in a theory with several dilatonic scalar fields and differential forms admitting an interpretation in terms of intersecting $p$-branes. The equations of motion of the model are reduced to the Euler–Lagrange equations for the so-called pseudo-Euclidean Toda-like system. Assuming that the characteristic vectors related to the configuration of $p$-branes and their couplings to the dilatonic scalar fields can be interpreted as the root vectors of a Lie algebra of the type $A_m\equiv sl(m+1,\mathbb C)$, 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 $M_0$ e Einsteinian conformal gaugeraction of the internal factor spaces $M_1,\dots,M_n$.