Hamiltonian Structures of KdV-Type Hierarchies and Associated W-Algebras

The $(n,m)^{\th}$ KdV hierarchy is a restriction of the KP hierarchy to a submanifold of pseudo-differential operators in a radio form. Explicit formula of the restricted Hamiltonian structure of KP is given which provides a new, more constructive proof of the isomorphism between the associated $W(n,m)$-algebra to $W_{n+m}\oplus W_m\oplus U(1)$ algebra, and the Hamiltonian property of the $(n,m)^{\th}$ KdV hierarchy as well as its Lax-Manakov triad representation. Similarly the Hamiltonian property for a version of modified $n^{\th}$ KdV and the isomorphism between $W_n$-algebra to $W_l\oplus W_m\oplus U(1)$ algebra are shown, where $l+m=n$. The role of U(1) current in both cases is also explained.