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From the \(w_\infty\)-algebra to its central extension: a \(\tau\)-function approach. (English) Zbl 0925.58031
Summary: The KP hierarchy, deformations of pseudodifferential operators \(L\) of order one, admits a \(w_\infty\)-algebra of symmetries \(\vec Y_{z^\alpha}(\partial/\partial z)^\beta\), which are vector fields transversal to and commuting with the KP hierarchy. Expressed in terms of \(L\) and another pseudodifferential operator \(M\) (introduced by Orlov and coworkers) satisfying \([ L,M ]=1\), these vector fields act on the wave function \(\Psi\) (a properly normalized eigenfunction of \(L\)) as \[ \vec Y_{z^\alpha}(\partial/\partial z)^\beta\Psi\equiv-(M^\beta L^\alpha)_-\Psi. \] Introducing a generating function \(\vec Y_{N}\Psi{}= N_{-}\Psi \), with \[ N\equiv (\mu-\lambda)\exp [(\mu-\lambda)M]\delta(\lambda,L), \] for the algebra of symmetries \(w_{\infty}\) on \(\Psi\) and taking into account the well-known representation of \[ \Psi(t,z)=[e^{-\eta}\tau(\overline t)/\tau(\overline t)]\exp (\sum_{1}^{\infty}\overline t_{i}z^{i}), \] in terms of the \(\tau\)-function, where \(\eta=\sum_{i=1}^{\infty}(z^{-i}/i)(\partial/\partial t_{i})\). We show a precise relationship between \(\vec Y_{N}\) and the Date-Jimbo-Kashiwara-Miwa vertex operator \[ \vec X(t,\lambda,\mu) \equiv\exp [\sum_{i=1}^{\infty}(\mu^{i}-\lambda^{i})t_{i}] \exp [\sum_{i=1}^{\infty}(\lambda^{-i}-\mu^{-i})(1/ i)(\partial/\partial t_{i})], \] a generating function of the \(W_{\infty}\)-algebra of symmetries (with central extension) on \(\tau\), to wit \(\vec Y_{N}\log \Psi=(e^{-\eta}-1)\vec X \log \tau\), where \(\vec Y_{N}\log\) and \(\vec X \log\) act on \(\Psi\) and \(\tau\) as logarithmic derivatives, with respect to the vector fields \(\vec Y_{N}\) and \(\vec X\) .

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q58 Other completely integrable PDE (MSC2000)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
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