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Moyal deformation of KdV and Virasoro action. (English) Zbl 1065.81070

Summary: The Lie algebra of pseudodifferential symbols \(\Psi D(S^1)\) on \(S^1\) has a natural Lie-Poisson structure. The Moyal deformation of dispersionless Korteweg-de Vries (KdV) equation is constructed from the action of Vect(\(S^1\)) on \(\Psi D(S^1\)). We also study the Moyal deformation of supersymmetric two-boson KdV equation. We offer their Lax representations. Using Souriau-Kravchenko-Khesin cocycle we are able to construct KdV equation.

MSC:

81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
81S10 Geometry and quantization, symplectic methods
53D55 Deformation quantization, star products
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