The Socle of a Metric n-Lie Algebra |
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Authors: | Zhixue Zhang Chengming Bai |
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Institution: | 1. College of Mathematics and Computer , Hebei University , Baoding , China zzx_hb@yahoo.com.cn;3. Chern Institute of Mathematics &4. LPMC , Nankai University , Tianjin , China |
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Abstract: | We define the socle of an n-Lie algebra as the sum of all the minimal ideals. An n-Lie algebra is called metric if it is endowed with an invariant nondegenerate symmetric bilinear form. We characterize the socle of a metric n-Lie algebra, which is closely related to the radical and the center of the metric n-Lie algebra. In particular, the socle of a metric n-Lie algebra is reductive, and a metric n-Lie algebra is solvable if and only if the socle coincides with its center. We also calculate the metric dimensions of simple and reductive n-Lie algebras and give a lower bound in the nonreductive case. |
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Keywords: | Invariant scalar product Metric n-Lie algebra Socle |
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