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 se t 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 relate d to semisimple Lie algebras is suggested. This conjecture is proved for th e Lie algebras: A(m), Cm+1, m greater than or equal to 1. For simple Lie al gebras the powers of polynomials coincide with the components of the dual W eyl vector in the basis of simple roots. The coefficients of polynomials de pend upon the extremality parameter mu > 0. In the extremal case mu = 0 suc h polynomials were considered previously by H Lu, J Maharana, S Mukherji an d C N Pope. Explicit formulae for the A(2)-solution are obtained. Two examp les of A(2)-dyon solutions, i.e., dyon in D = 11 supergravity with M2 and M 5 branes intersecting at a point and the Kaluza-Klein dyon, are considered.