A loop algebra co-adjoint orbit construction of the generalized KdV hierarchies

Using a pencil of r-matrices on the algebra g(X)C((z)) the authors prove that the KdV hierarchies have an Adler-Kostant-Symes construction on the underlying current algebra C infinity (S 1 ,g(X)C((z))). The coadjoint orbits are reduced by Hamiltonian symmetries. The reduction process reproduces the gauge group and the bi-Hamiltonian structure. analyse the momentum maps of these reductions, obtaining the level sets and the little group. This provides a gauge fixing. Conformal symmetry of these KdV hierarchies is analysed, an expression for the energy momentum tensor is obtained. Their analysis is restricted to theories with Lax operators linear in the loop variable z.