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Twisted Forms of Differential Lie Algebras over \({\mathbb {C}}(t)\) Associated with Complex Simple Lie Algebras

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Abstract

Discussed here is descent theory in the differential context where everything is equipped with a differential operator. To answer a question personally posed by A. Pianzola, we determine all twisted forms of the differential Lie algebras over \({\mathbb {C}}(t)\) associated with complex simple Lie algebras. Hopf–Galois Theory, a ring-theoretic counterpart of theory of torsors for group schemes, plays a role when we grasp the above-mentioned twisted forms from torsors.

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Acknowledgements

The authors thank Professor Arturo Pianzola who kindly brought to them an interesting problem, informing of Steinberg’s Theorem, and gave helpful comments during their revision of the manuscript. They also thank the referees for helpful suggestions and valuable comments.

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Authors and Affiliations

  1. Institute of Mathematics, University of Tsukuba, Ibaraki, 305-8571, Japan

    Akira Masuoka

  2. Graduate School of Pure and Applied Sciences, University of Tsukuba, Ibaraki, 305-8571, Japan

    Yuta Shimada

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  1. Akira Masuoka
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  2. Yuta Shimada
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Correspondence to Akira Masuoka.

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Dedicated to Professor Jun Morita on the occasion of his retirement.

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Supported by JSPS Grant-in-Aid for Scientific Research (C) 17K05189.

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Masuoka, A., Shimada, Y. Twisted Forms of Differential Lie Algebras over \({\mathbb {C}}(t)\) Associated with Complex Simple Lie Algebras. Arnold Math J. 7, 107–134 (2021). https://doi.org/10.1007/s40598-020-00155-7

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  • DOI: https://doi.org/10.1007/s40598-020-00155-7

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