The Drinfeld-Sokolov Hierarchy Of The Twisted Affine Algebras

The Drinfeld-Sokolov hierarchies are very important in the theory of nonlinear inte-grable systems. It connects the theory of infinite dimensional Lie algebras with integrable systems and hence many other branches in mathematics. Generally speaking, for a given affine Lie algebra and a vertex fixed in its Dynkin diagram, an integrable hierarchy is associated. Drinfeld-Sokolov hierarchies are all Hamiltonian systems and most of them can also be represented as scalar Lax equations.For the non-twisted affine algebras, the associated hierarchies have a lot of good properties such as bihamiltonian structures which do not hold for the twisted affine alge-bras. So the hierarchies for non-twisted affine algebras have already been well studied, while the systems for twisted algebras are not. In this paper we studied the hierarchies associated to the twisted algebras. A crucial step is that a twisted algebra can be imbed-ded into a non-twisted algebra as a fixed point subalgebra for some suitable diagram automorphism. Then we can consider the connection between the two hierarchies.It is proved for the twisted algebras A2n(2), A2n-1(2) and Dn+1(2) that the Drinfeld-Sokolov hierarchies can be considered as the reduction from some generalised Drinfeld-Sokolov hierarchies associated to the non-twisted affine algebras. The connection between the scalar Lax operators for both the twisted and the non-twisted algebras is also considered. Hence we can consider the properties of the non-twisted hierarchies and then yield the properties of the twisted ones.It is also observed that this reduction procedure can be extended to more general situations and a typical example is given by considering the reduction from the so called constrained KP hierarchies to the Drinfeld-Sokolov hierarchies ofA2n-1(2) Then the central invariants for the bihamiltonian structure of constrained KP hierarchies are computed ex-plicitly. The result shows that this bihamiltonian structure is the topological deformation of its hydrodynamic limit.