Toda p-brane black holes and polynomials related to Lie algebras

fBlack hole generalized p-brane solutions for a wide class of intersection rules are obtained. The solutions are defined on a manifold that contains a product of n - 1 Ricci-flat 'internal' spaces. They are defined up to a set of functions H-s obeying non-linear differential equations equivalent to Toda-type equations with certain boundary conditions imposed. A conjecture on polynomial structure of governing functions H-s for intersections related to semisimple Lie algebras is suggested. This conjecture is proved for the Lie algebras: A(m), Cm+1, m greater than or equal to 1. For simple Lie algebras the powers of polynomials coincide with the components of the dual Weyl vector in the basis of simple roots. The coefficients of polynomials depend upon the extremality parameter mu > 0. In the extremal case mu = 0 such polynomials were considered previously by H Lu, J Maharana, S Mukherji and C N Pope. Explicit formulae for the A(2)-solution are obtained. Two examples of A(2)-dyon solutions, i.e., dyon in D = 11 supergravity with M2 and M5 branes intersecting at a point and the Kaluza-Klein dyon, are considered.