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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 $$\vec X$$ .

MSC:
 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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