Poisson structures for dispersionless integrable systems and associated \(W\)-algebras (Q1381927)

This paper addresses some features of the second Poisson structure for the dispersionless Kadomtsev-Petviashvili (dKP) hierarchy, including its reduction and associated \(w\)-algebras. For integrable hierarchies, it is well-known that the second Poisson structures give rise to the classical realizations of conformal \(w\)-algebras. The authors proceed, by analogy with standard KP theory, to define the second Poisson structure for the dKP hierarchy on the space of commutative differential operators \[ L= P^n+ \sum_{j=-\infty}^{n-1} u_jp^j. \] The reduction of the Poisson structure to the symplectic submanifold \(u_{n-1}=0\) gives rise to \(w\)-algebras. The authors discuss properties of the Poisson structures, its Miura transformation and reductions. They concentrate on the cases where \(L\) is a pure polynomial with multiple roots, and where \(L\) has multiple poles at a finite distance.