Some Results on Metric n-Lie Algebras  

作　　者：Rui Pu BAI Wan Qing WU Zhen Heng LI 

机构地区：[1]College of Mathematics and Computer Science,Hebei University [2]Department of Mathematical Sciences,University of South Carolina Aiken

出　　处：《Acta Mathematica Sinica,English Series》2012年第6期1209-1220,共12页数学学报（英文版）

基　　金：Supported by National Natural Science Foundation of China (Grant No. 10871192);Natural Science Foundation of Hebei Province, China (Grant No. A2010000194)

摘　　要：We study the structure of a metric n-Lie algebra G over the complex field C. Let G = S＋R be the Levi decomposition, where T4 is the radical of G and S is a strong semisimple subalgebra of G. Denote by re（G） the number of all minimal ideals of an indecomposable metric n-Lie algebra and R^⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R^⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to rn（G） ＋ 1. The centralizer of T4 in G is equal to the sum of all minimal ideals; it is the direct sum of R^⊥and the center of G. Finally, G has no strong semisimple ideals if and only if R^⊥ R.We study the structure of a metric n-Lie algebra G over the complex field C. Let G = S＋R be the Levi decomposition, where T4 is the radical of G and S is a strong semisimple subalgebra of G. Denote by re（G） the number of all minimal ideals of an indecomposable metric n-Lie algebra and R^⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R^⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to rn（G） ＋ 1. The centralizer of T4 in G is equal to the sum of all minimal ideals; it is the direct sum of R^⊥and the center of G. Finally, G has no strong semisimple ideals if and only if R^⊥ R.

关 键 词：Metric n-Lie algebra minimal ideal metric dimension Levi decomposition 

分 类 号：O152.5[理学—数学] TB911[理学—基础数学]