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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This Research supported by NSFC (11471139), NSF of Jilin Province (20170101050JC) and Nan Hu Scholar Development Program of XYNU.
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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
Keywords
- 3-Lie algebra
- Representation
- Generalized representation
- Cohomology
- Abelian extension
- Maurer-Cartan element