Poisson Structures for Dispersionless Integrable Systems and Associated W-Algebras

In analogy to the KP theory, the second Poisson structure for the dispersionless KP hierarchy can be defined on the space of commutative pseudodifferential operators $L=p^n+\sum_{j=-\infty}^{n-1}u_j p^j$. The reduction of the Poisson structure to the symplectic submanifold $u_{n -1}=0$ gives rise to the w-algebras. In this paper, we discuss properties of this Poisson structure, its Miura transformation and reductions. We are particularly interested in the following two cases: a) L is pure polynomial in p with multiple roots and b) L has multiple poles at finite distance. The w-algebra corresponding to the case a) is defined as $w_ {[m_1,m_2, ... ,m_r]}$, where m_i means the multiplicity of roots and to the case b) is defined by $w(n,[m_1,m_2, ... ,m_r])$ where m_i is the multiplicity of poles. We prove that w(n,[m_1, m_2, ... , m_r])$-algebra is isomorphic via a transformation to$w_{[m_1,m_2, ... ,m_r]} \bigoplus w_{n+m} \bigoplus U(1) with $m=\sum m_i$. We also give the explicit free fields representations for these w-algebras.