On special subalgebras of derivations of Lie algebras. (English) Zbl 1376.17027
Summary: Let \(L\) be a Lie algebra, and \(\mathrm{Der}(L) \) and \(\mathrm{IDer}(L)\) be the set of all derivations and inner derivations of \(L\), respectively. Also, let \(\mathrm{Der}_{c}(L)\) denote the set of all derivations \(\alpha \in \mathrm{Der}(L)\) for which \(\alpha(x) \in \mathrm{Imad}_{x}\) for all \(x \in L\). We give necessary and sufficient conditions under which \(\mathrm{Der}_{c}(L) = \mathrm{Der}_{z}(L), \) where \(\mathrm{Der}_{z}(L) \) is the set of all derivations of \(L\) whose images lie in the center of \(L\). Moreover, it is shown that any two isoclinic Lie algebras \(L_{1}\) and \(L_{2}\) satisfy \(\mathrm{Der}_{c}(L_{1}) \cong \mathrm{Der}_{c}(L_{2})\).
MSC:
| 17B40 | Automorphisms, derivations, other operators for Lie algebras and super algebras |
| 17B30 | Solvable, nilpotent (super)algebras |
References:
| [1] | DOI: 10.1007/BF02760546 · Zbl 0654.20019 · doi:10.1007/BF02760546 |
| [2] | DOI: 10.1016/j.jalgebra.2006.08.006 · Zbl 1137.17012 · doi:10.1016/j.jalgebra.2006.08.006 |
| [3] | Jacobson N., Lie Algebras (1962) |
| [4] | DOI: 10.1215/S0012-7094-63-03067-9 · doi:10.1215/S0012-7094-63-03067-9 |
| [5] | DOI: 10.1112/jlms/s1-42.1.416 · Zbl 0166.04101 · doi:10.1112/jlms/s1-42.1.416 |
| [6] | Meng D. J., Acta. Sci. Mat. Univ. Mankai 2 pp 9– (1985) |
| [7] | Meng D. J., Acta. Math. 34 pp 191– (1991) |
| [8] | DOI: 10.1080/00927879408825142 · Zbl 0814.17003 · doi:10.1080/00927879408825142 |
| [9] | DOI: 10.1080/00927879408825141 · Zbl 0814.17004 · doi:10.1080/00927879408825141 |
| [10] | DOI: 10.1006/jabr.1995.1263 · Zbl 0836.17007 · doi:10.1006/jabr.1995.1263 |
| [11] | DOI: 10.1080/00927879608825806 · Zbl 0897.17008 · doi:10.1080/00927879608825806 |
| [12] | Moneyhun K., Algebras Groups Geom. 11 pp 9– (1994) · Zbl 0801.17005 |
| [13] | Stagg K., Int. Electron. J. Algebra 9 pp 124– (2011) |
| [14] | DOI: 10.1016/0021-8693(87)90198-0 · Zbl 0607.17005 · doi:10.1016/0021-8693(87)90198-0 |
| [15] | DOI: 10.1016/j.jalgebra.2004.09.033 · Zbl 1154.17303 · doi:10.1016/j.jalgebra.2004.09.033 |
| [16] | Tôgô S., J. Sci. Hiroshima Univ. Ser. A-I 28 pp 133– (1964) |
| [17] | DOI: 10.2307/1994323 · Zbl 0153.36001 · doi:10.2307/1994323 |
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