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Extensions and Crossed Modules of \(\varvec{n}\)-Lie–Rinehart Algebras

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Abstract

We introduce a notion of n-Lie–Rinehart algebras as a generalization of Lie–Rinehart algebras to n-ary case. This notion is also an algebraic analogue of n-Lie algebroids. We develop representation theory and describe a cohomology complex of n-Lie–Rinehart algebras. Furthermore, we investigate extension theory of n-Lie–Rinehart algebras by means of 2-cocycles. Finally, we introduce crossed modules of n-Lie–Rinehart algebras to gain a better understanding of their third cohomology groups.

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Acknowledgements

Mohamed Elhamdadi was supported by Simons Foundation grant no. 712462.

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Authors and Affiliations

  1. Department of Mathematics, College of Science and Arts at Belqarn, University of Bisha, P. O. Box 60, Sabt Al-Alaya, 61985, Saudi Arabia

    A. Ben Hassine

  2. Faculty of Sciences, University of Sfax, BP 1171, 3000, Sfax, Tunisia

    A. Ben Hassine & T. Chtioui

  3. Department of Mathematics, University of South Florida, Tampa, FL, 33620, USA

    M. Elhamdadi

  4. Faculty of Sciences, University of Gafsa, BP 2100, Gafsa, Tunisia

    S. Mabrouk

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  1. A. Ben Hassine
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  2. T. Chtioui
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  3. M. Elhamdadi
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  4. S. Mabrouk
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Correspondence to M. Elhamdadi.

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Communicated by Uwe Kaehler.

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Hassine, A.B., Chtioui, T., Elhamdadi, M. et al. Extensions and Crossed Modules of \(\varvec{n}\)-Lie–Rinehart Algebras. Adv. Appl. Clifford Algebras 32, 31 (2022). https://doi.org/10.1007/s00006-022-01218-y

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