# On dimension of a special subalgebra of derivations of nilpotent Lie algebras

Document Type : Research Paper

Authors

Abstract

‎Let $L$ be a Lie algebra‎, ‎$\mathrm{Der}(L)$ be the set of all derivations of $L$ and $\mathrm{Der}_c(L)$ denote the set of all derivations $\alpha\in\mathrm{Der}(L)$ for which $\alpha(x)\in [x,L]:=\{[x,y]\vert y\in L\}$ for all $x\in L$‎. ‎We obtain an upper bound for dimension of $\mathrm{Der}_c(L)$ of the finite dimensional nilpotent Lie algebra $L$ over algebraically closed fields‎. ‎Also‎, ‎we classify all finite dimensional nilpotent Lie algebras $L$ over algebraically closed fields for which dim$\mathrm{Der}_c(L)$ attains its maximum value.

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#### References

S. Cicalo, W. A. de Graaf and C. Schneider, Six-dimensional nilpotent Lie algebras, Linear Algebra Appl. 436 (2012), no. 1, 163--189.
W. A. de Graaf, Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2, J. Algebra 309 (2007), no. 2, 640--653.
E. I. Marshall, The Frattini subalgebra of a Lie algebra, J. Lond. Math. Soc. 42 (1967) 416--422.
S. Sheikh-Mohseni and F. Saeedi, On Camina Lie algebras, submitted.
K. Stagg, Analogues of the Frattini subalgebra, Int. Electron. J. Algebra 9 (2011) 124--132.
M. Yadav, Class preserving automorphisms of finite p-groups, J. Lond. Math. Soc. (2) 75 (2007), no. 3, 755--772.

### History

• Receive Date: 25 November 2014
• Revise Date: 13 October 2015
• Accept Date: 14 October 2015
• First Publish Date: 22 February 2017