| Title:
|
Triple automorphisms of simple Lie algebras (English) |
| Author:
|
Wang, Dengyin |
| Author:
|
Yu, Xiaoxiang |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
61 |
| Issue:
|
4 |
| Year:
|
2011 |
| Pages:
|
1007-1016 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $2$. (English) |
| Keyword:
|
simple Lie algebras |
| Keyword:
|
parabolic subalgebras |
| Keyword:
|
triple automorphisms of Lie algebras |
| MSC:
|
17B20 |
| MSC:
|
17B30 |
| MSC:
|
17B40 |
| idZBL:
|
Zbl 1249.17026 |
| idMR:
|
MR2886252 |
| DOI:
|
10.1007/s10587-011-0043-9 |
| . |
| Date available:
|
2011-12-16T15:42:46Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141802 |
| . |
| Reference:
|
[1] Humphreys, J. E.: Introduction to Lie algebras and representation theory.Graduate Texts in Mathematics 9, New York-Heidelberg-Berlin, Springer-Verlag (1972). Zbl 0254.17004, MR 0499562 |
| Reference:
|
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| Reference:
|
[3] Lu, F.-Y.: Lie triple derivations on nest algebras.Math. Nach. 280 (2007), 882-887. Zbl 1124.47054, MR 2326061, 10.1002/mana.200410520 |
| Reference:
|
[4] Miers, C. R.: Lie triple derivations of von Neumann algebras.Proc. Am. Math. Soc. 71 (1978), 57-61. Zbl 0384.46047, MR 0487480, 10.1090/S0002-9939-1978-0487480-9 |
| Reference:
|
[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. Zbl 1163.17014, MR 2460499 |
| Reference:
|
[6] Cao, H.-X., Wu, B.-W., Zhang, J.-H.: Lie triple derivations of nest algebras.Linear Algebra Appl. 416 (2006), 559-567. Zbl 1102.47060, MR 2242444, 10.1016/j.laa.2005.12.003 |
| . |
