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A New Approach to Representations of 3-Lie Algebras and Abelian Extensions

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Abstract

In this paper, we introduce the notion of generalized representation of a 3-Lie algebra, by which we obtain a generalized semidirect product 3-Lie algebra. Moreover, we develop the corresponding cohomology theory. Various examples of generalized representations of 3-Lie algebras and computation of 2-cocycles of the new cohomology are provided. Also, we show that a split abelian extension of a 3-Lie algebra is isomorphic to a generalized semidirect product 3-Lie algebra. Furthermore, we describe general abelian extensions of 3-Lie algebras using Maurer-Cartan elements.

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Acknowledgements

This Research supported by NSFC (11471139), NSF of Jilin Province (20170101050JC) and Nan Hu Scholar Development Program of XYNU.

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Authors and Affiliations

  1. Department of Mathematics, Xinyang Normal University, Xinyang, 464000, Henan, China

    Jiefeng Liu & Yunhe Sheng

  2. Laboratoire de Mathématiques, Informatique et Applications, University of Haute Alsace, Mulhouse, France

    Abdenacer Makhlouf

  3. Department of Mathematics, Jilin University, Changchun, 130012, China

    Yunhe Sheng

Authors
  1. Jiefeng Liu
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  2. Abdenacer Makhlouf
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  3. Yunhe Sheng
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Correspondence to Abdenacer Makhlouf.

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Presented by Henning Krause.

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Liu, J., Makhlouf, A. & Sheng, Y. A New Approach to Representations of 3-Lie Algebras and Abelian Extensions. Algebr Represent Theor 20, 1415–1431 (2017). https://doi.org/10.1007/s10468-017-9693-0

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  • DOI: https://doi.org/10.1007/s10468-017-9693-0

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