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Pseudodifferential Symbols on Riemann Surfaces and Krichever–Novikov Algebras

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Abstract

We define the Krichever-Novikov-type Lie algebras of differential operators and pseudodifferential symbols on Riemann surfaces, along with their outer derivations and central extensions. We show that the corresponding algebras of meromorphic operators and symbols have many invariant traces and central extensions, given by the logarithms of meromorphic vector fields. Very few of these extensions survive after passing to the algebras of operators and symbols holomorphic away from several fixed points. We also describe the associated Manin triples and KdV-type hierarchies, emphasizing the similarities and differences with the case of smooth symbols on the circle.

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Authors and Affiliations

  1. Dept. of Mathematics, University of Toronto, Toronto, Ont, M5S 2E4, Canada

    Dmitry Donin & Boris Khesin

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  1. Dmitry Donin
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  2. Boris Khesin
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Correspondence to Boris Khesin.

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Communicated by L. Takhtajan

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Donin, D., Khesin, B. Pseudodifferential Symbols on Riemann Surfaces and Krichever–Novikov Algebras. Commun. Math. Phys. 272, 507–527 (2007). https://doi.org/10.1007/s00220-007-0234-2

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  • DOI: https://doi.org/10.1007/s00220-007-0234-2

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