# Article

Full entry | PDF   (0.2 MB)
Keywords:
simple Lie algebras; parabolic subalgebras; triple automorphisms of Lie algebras
Summary:
An invertible linear map $\varphi$ on a Lie algebra $L$ is called a triple automorphism of it if $\varphi ([x,[y,z]])=[\varphi (x),[ \varphi (y),\varphi (z)]]$ for $\forall x, y, z\in L$. Let $\frak {g}$ be a finite-dimensional simple Lie algebra of rank $l$ defined over an algebraically closed field $F$ of characteristic zero, $\mathfrak {p}$ an arbitrary parabolic subalgebra of $\mathfrak {g}$. It is shown in this paper that an invertible linear map $\varphi$ on $\mathfrak {p}$ is a triple automorphism if and only if either $\varphi$ itself is an automorphism of $\mathfrak {p}$ or it is the composition of an automorphism of $\mathfrak {p}$ and an extremal map of order $2$.
References:
[1] Humphreys, J. E.: Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics 9, New York-Heidelberg-Berlin, Springer-Verlag (1972). MR 0499562 | Zbl 0254.17004
[2] Ji, P., Wang, L.: Lie triple derivations of TUHF algebras. Linear Algebra Appl. 403 (2005), 399-408. DOI 10.1016/j.laa.2005.02.004 | MR 2140293 | Zbl 1114.46048
[3] Lu, F.-Y.: Lie triple derivations on nest algebras. Math. Nach. 280 (2007), 882-887. DOI 10.1002/mana.200410520 | MR 2326061 | Zbl 1124.47054
[4] Miers, C. R.: Lie triple derivations of von Neumann algebras. Proc. Am. Math. Soc. 71 (1978), 57-61. DOI 10.1090/S0002-9939-1978-0487480-9 | MR 0487480 | Zbl 0384.46047
[5] Li, Q.-G., Wang, H.-T.: Lie triple derivation of the Lie algebra of strictly upper triangular matrix over a commutative ring. Linear Algebra Appl. 430 (2009), 66-77. MR 2460499 | Zbl 1163.17014
[6] Cao, H.-X., Wu, B.-W., Zhang, J.-H.: Lie triple derivations of nest algebras. Linear Algebra Appl. 416 (2006), 559-567. DOI 10.1016/j.laa.2005.12.003 | MR 2242444 | Zbl 1102.47060

Partner of