Abstract

It is well known that local  bihamiltonian structure (compatible  Poisson structures) of hydrodynamic  type on a loop space is  the leading term of a local bihamiltonian structure admitting a dispersionless limit and under certain conditions it defines Dubrovin-Frobenius manifold. We consider Drinfeld-Sokolov bihamiltonian structure associated with a distinguished nilpotent element of semisimple type in a simple Lie algebra which does not always  admit a dispersionless limit.  We show that its  leading term defines a finite dimensional polynomial completely integrable system. Moreover, its reduction to the space of common equilibrium points of this integrable system leads to a local algebraic bihamiltonian structure admitting a dispersionless limit.  In addition, the leading term of the new local bihamiltonian structure leads to an algebraic Dubrovin-Frobenius manifold which  supports a conjecture due to Dubrovin about their classification.