Topics on n-ary algebras
Topics on n-ary algebras
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DOI：
10.1088/1742-6596/284/1/012019
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发表时间：
2011-02
期刊：
影响因子：
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通讯作者：
J. A. Azcárraga;J. M. Izquierdo
中科院分区：
文献类型：
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作者：
J. A. Azcárraga;J. M. Izquierdo
我们描述了两个n元代数，广义李代数（GLAs），特别是Filippov（≡n-Lie）代数（FAs）的基本性质，并评论了它们的n元泊松对立物，广义泊松（GP）和nambu -泊松（N-P）结构。我们描述了与FAs的中心扩展和无穷小变形相关的Filippov代数上同调。可见，半简单方程不允许中心扩展，而且它们是刚性的。这将熟悉的Whitehead引理扩展到所有n≥2个fa， n = 2是标准李代数的情况。当fa的n括号不再要求完全偏对称时，就会导致n-莱布尼茨（或Loday）代数结构。利用fa是n-莱布尼兹代数的一种特殊情况，即具有反交换n-括号的代数，我们研究了一类简单fa的n-莱布尼兹变形，它们在n-莱布尼兹括号的前n−1个整体上保持偏对称性。
We describe the basic properties of two n-ary algebras, the Generalized Lie Algebras (GLAs) and, particularly, the Filippov (≡ n-Lie) algebras (FAs), and comment on their n-ary Poisson counterparts, the Generalized Poisson (GP) and Nambu-Poisson (N-P) structures. We describe the Filippov algebra cohomology relevant for the central extensions and infinitesimal deformations of FAs. It is seen that semisimple FAs do not admit central extensions and, moreover, that they are rigid. This extends the familiar Whitehead's lemma to all n ≥ 2 FAs, n = 2 being the standard Lie algebra case. When the n-bracket of the FAs is no longer required to be fully skewsymmetric one is lead to the n-Leibniz (or Loday's) algebra structure. Using that FAs are a particular case of n-Leibniz algebras, those with an anticommutative n-bracket, we study the class of n-Leibniz deformations of simple FAs that retain the skewsymmetry for the first n − 1 entires of the n-Leibniz bracket.