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Extensions and contractions of the Lie algebra of q-pseudodifferential symbols

Boris Khesin·Volodymyr Lyubashenko·Claude Roger
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Abstract

We construct cocycles on the Lie algebra of pseudo- and q-pseudodifferential symbols of one variable and on their close relatives: the sine-algebra and the Poisson algebra on two-torus. A ``quantum'' Godbillon-Vey cocycle on (pseudo)-differential operators appears in this construction as a natural generalization of the Gelfand-Fuchs 3-cocycle on periodic vector fields. We describe a nontrivial embedding of the Virasoro algebra into (a completion of) q-pseudodifferential symbols, and propose q-analogs of the KP and KdV-hierarchies admitting an infinite number of conserved charges.

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Boris Khesin, Volodymyr Lyubashenko, Claude Roger. 2008-02-20. Extensions and contractions of the Lie algebra of q-pseudodifferential symbols. https://arxiv.org/abs/hep-th/9403189

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