Toda chains with type A(m) Lie algebra for multidimensional m-component perfect fluid cosmology

We consider a D-dimensional cosmological model describing an evolution of R icci-flat factor spaces, M-1,..., M-n (n greater than or equal to 3), in th e presence of an m-component perfect fluid source (n - 1 greater than or eq ual to m greater than or equal to 2). We find characteristic vectors, relat ed to the matter constants in the barotropic equations of state for fluid c omponents of all factor spaces. We show that, in the case where we can inte rpret these vectors as the root vectors of a Lie algebra of Cartan type A(m ) = sl(m + 1, C), the model reduces to the classical open m-body Toda chain . Using an elegant technique by Anderson for solving this system, we integr ate the Einstein equations for the model and present the metric in a Kasner -like form.