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| Constructions of Metric $(n+1)$-Lie Algebras |
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Citation： |
Ruipu BAI,Shuangshuang CHEN.Constructions of Metric $(n+1)$-Lie Algebras[J].Chinese Annals of Mathematics B,2016,37(5):729~742 |
| Page view： 1154
Net amount： 1331 |
Authors： |
Ruipu BAI; Shuangshuang CHEN |
Foundation： |
This work was supported by the National Natural Science
Foundation of China (No.11371245) and the Natural Science
Foundation of Hebei Province (No.A2014201006). |
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| Abstract： |
Metric $n$-Lie algebras have wide applications in mathematics and
mathematical physics. In this paper, the authors introduce two
methods to construct metric $(n+1)$-Lie algebras from metric $n$-Lie
algebras for $n\geq 2$. For a given $m$-dimensional metric $n$-Lie
algebra $(\frak g, [ , \cdots, ], B_g)$, via one and two dimensional
extensions $\frak L=\frak g\dot+\mathbb Fc$ and $\frak g_0$ $=\frak
g\dot+ $ $\mathbb F x^{-1}\dot+ \mathbb F x^0$ of the vector space
$\frak g$ and a certain linear function $f$ on $\frak g$, we
construct $(m+1)$- and $(m+2)$-dimensional $(n+1)$-Lie algebras
$(\frak L, [ , \cdots, ]_{cf})$ and $(\frak g_0, [ , \cdots, ]_1)$,
respectively. Furthermore, if the center $Z(\frak g)$ is
non-isotropic, then we obtain metric $(n+1)$-Lie algebras $(\frak
L, [ , \cdots, ]_{cf}, B)$ and $(\frak g_0, [ , \cdots, ]_1, B)$
which satisfy $B|_{\frak g\times \frak g}=B_g$. Following this
approach the extensions of all $(n+2)$-dimensional metric $n$-Lie
algebras are discussed. |
Keywords： |
$n$-Lie
algebra, Metric $n$-Lie algebra, Extension, Isotropic center |
Classification： |
17B05, 17B30 |
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