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Poisson Structures for Dispersionless Integrable Systems and Associated W-Algebras | | Yi Cheng
; Zhifeng Li
; | | Date: |
4 Dec 1996 | | Journal: | Lett.Math.Phys. 42 (1997) 73-83 | | Subject: | High Energy Physics - Theory; Exactly Solvable and Integrable Systems | hep-th nlin.SI solv-int | | Abstract: | 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]} igoplus w_{n+m} igoplus U(1) with $m=sum m_i$. We also give the explicit free fields representations for these w-algebras. | | Source: | arXiv, hep-th/9612044 | | Services: | Forum | Review | PDF | Favorites | |
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