Double derivations of n-Lie algebras

Document Type : Research Paper


College of Mathematics and Information Science‎, ‎Hebei University‎, ‎Baoding 071002‎, ‎P.R‎. ‎China.


After introducing double derivations of $n$-Lie algebra $L$ we‎ ‎describe the relationship between the algebra $\mathcal D(L)$ of double derivations and the usual‎ ‎derivation Lie algebra $\mathcal Der(L)$‎. ‎In particular‎, ‎we prove that the inner derivation algebra $ad(L)$‎ ‎is an ideal of the double derivation algebra $\mathcal D(L)$; we also show that if $L$ is a perfect $n$-Lie algebra‎ ‎with certain constraints on the base field then the centralizer of $ad(L)$ in $\mathcal D(L)$ is‎ ‎trivial and $\mathcal D(L)$ is centerless‎. ‎In addition‎, ‎we obtain that for every perfect $n$-Lie‎ ‎algebra $L$ with zero center‎, ‎the triple derivations of the derivation algebra $\mathcal Der(L)$ are exactly‎ ‎the derivations of $\mathcal Der(L)$‎, ‎and the triple derivations of the inner derivation algebra $ad(L)$ are‎ ‎precisely the derivations of $ad(L)$‎.


Main Subjects

R.P. Bai and P.P. Jia, The real compact n-Lie algebra and invariant bilinear form, Acta Math. Sci. Ser. A Chin. Ed. 27 (2007), no. 6, 1074--1081.
R.P. Bai, J. Wang and Z. Li, Derivations of the 3-Lie algebra realizated by gl(n;C), J. Nonlinear Math. Phys. 18 (2011), no. 1, 151--160.
R.P. Bai, W.Wu, Y. Li and Z. Li, Module extensions of 3-Lie algebras, Linear Multilinear Algebra 60 (2012), no. 4, 433--447.
R.P. Bai, Z. Zhang, H. Li and H. Shi, The inner derivation algebras of (n+1)-dimensional n-Lie algebras, Comm. Algebra 28 (2000), no. 6, 2927--2934.
W. Cheung, Lie derivations of triangular algebras, Linear Multilinear Algebra 51 (2003), no. 3, 299--310.
A. Dzhumadil'daev, Representations of n-Lie algebras, Comm. Algebra 32 (2004), no. 9, 3315--3326.
V. Filippov, n-Lie algebras, Sibirsk. Mat. Zh. 26 (1985), no. 6, 126--140, 191.
F. Lu, Lie triple derivations on nest algebras, Math. Nachr. 280 (2007), no. 8, 882--887.
Y. Su and L. Zhu, Derivation algebras of centerless perfect Lie algebras are complete, J. Algebra 285 (2005), no. 2, 508--515.
D. Wang and Q. Yu, Derivations of the parabolic subalgebras of the general linear Lie algebra over a commutative ring, Linear Algebra Appl. 418 (2006), no. 2-3, 763--774.
W. Yu and J. Zhang, Nonlinear Lie derivations of triangular algebras, Linear Algebra Appl. 432 (2010), no. 11, 2953--2960.
J. Zhou, Triple derivations of perfect Lie algebras, Comm. Algebra 41 (2013), no. 5, 1647--1654.