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Lie Subalgebras of Differential Operators in One Variable

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Abstract

Let \(\mathrm{Witt}\) be the Lie algebra generated by the set \(\{L_i\,\vert \, i \in {{\mathbb {Z}}}\}\) and \(\mathrm{Vir}\) its universal central extension. Let \(\mathrm{Diff}(V)\) be the Lie algebra of differential operators on \(V={{\mathbb {C}}}(\!(z)\!)\), or \(V={{\mathbb {C}}}(z)\). We explicitly describe all Lie algebra homomorphisms from \(\mathfrak {sl}(2)\), \(\mathrm{Witt}\), and \(\mathrm{Vir}\) to \(\mathrm{Diff}(V)\), such that \(L_0\) acts on V as a first-order differential operator.

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Acknowledgements

The authors would like to thank the anonymous referee for the careful reading of the paper and pointing out many typos and some inaccuracies that have helped to greatly improve this work.

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  1. Departamento de Matemáticas and IUFFyM, Universidad de Salamanca, Plaza de la Merced 1-4, 37008, Salamanca, Spain

    F. J. Plaza Martín & C. Tejero Prieto

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  1. F. J. Plaza Martín
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  2. C. Tejero Prieto
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Correspondence to F. J. Plaza Martín.

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This work is supported by the research contracts MTM2017-86042-P and MTM2015-66760-P of MINECO (Spain) and FS/29-2017 Fundación Solórzano.

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Plaza Martín, F.J., Tejero Prieto, C. Lie Subalgebras of Differential Operators in One Variable. Mediterr. J. Math. 16, 147 (2019). https://doi.org/10.1007/s00009-019-1416-9

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