On Equality of Certain Derivations of Lie Algebras

Azita Amiri 1 , Farshid Saeedi 1  and Mohammad Reza Alemi 1
  • 1 Department of Mathematics, Mashhad Branch, Mashhad


Let L be a Lie algebra. A derivation α of L is a commuting derivation (central derivation), if α (x) ∈ CL (x) (α (x) ∈ Z (L)) for each xL. We denote the set of all commuting derivations (central derivations) by 𝒟 (L) (Derz (L)). In this paper, first we show 𝒟 (L) is subalgebra from derivation algebra L, also we investigate the conditions on the Lie algebra L where commuting derivation is trivial and finally we introduce the family of nilpotent Lie algebras in which Derz (L) = 𝒟 (L).

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  • [1] H.E. Bell and W.S. Martindale, Centralizing mappings of semiprime rings, Canad. Math. Bull 30 (1987) 92–101. doi:10.1016/j.laa.2011.06.037

  • [2] S. Cicalo, W.A. de Graaf and C. Schneider, Six-dimensional nilpotent Lie algebras, Linear Algebra Appl. 436 (2012) 163–189. doi:10.1007/BF02638378

  • [3] M. Deaconescu, G. Silberberg and G.L. Walls, On commuting automorphisms of groups, Arch. Math. (Basel) 79 (2002) 423–429. doi:10.1006/jabr.2000.8655

  • [4] M. Deaconescu and G.L. Walls, Right 2-Engel elements and commuting automorphisms of groups, J. Algebra 238 (2001) 479–484. doi:10.1006/jabr.2000.8655

  • [5] N. Divinsky, On commuting automorphisms of rings, Trans. Roy. Soc. Canada. Sect. III 49 (1955) 19–22. doi:10.12691/jmsa-2-3-1

  • [6] S. Fouladi and R. Orfi, Commuting automorphisms of some finite groups, Glas. Mat. Ser. III 48 (2013) 91–96. doi:10.3336/gm.48.1.08

  • [7] W.A. de Graaf, Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2, J. Algebra 309 (2007) 640–653. doi:10.1016/j.jalgebra.2006.08.006

  • [8] I.N. Herstein, Problem proposal, Amer. Math. Monthly 91 (1984), 203.

  • [9] I.N. Herstein, T.J. La ey and J. Thomas, Problems and solutions: solutions of elementary problems E3039, Amer. Math. Monthly 93 (1986) 816–817. doi:10.2307/2322945

  • [10] J. Luh, A note on commuting automorphisms of rings, Amer. Math. Monthly 77 (1970) 61–62. doi:10.1080/00029890.1970.11992420

  • [11] E.I. Marshall, The Frattini subalgebra of a Lie algebra, J. London Math. Soc. 42 (1967) 41–422. doi:10.1112/jlms/s1-42.1.416

  • [12] M. Pettet, Personal communication.

  • [13] F. Saeedi and S. Sheikh-Mohseni, A characterization of stem algebras in terms of central derivations, Algebr. Represent. Theory 20 (2017) 1143–1150. doi:10.1007/s10468-017-9680-5

  • [14] S. Sheikh-Mohseni, F. Saeedi and M. Badrkhani Asl, On special subalgebras of derivations of Lie algebras, Asian-Eur. J. Math. 8 (2015), 1550032. doi:10.1142/S1793557115500321

  • [15] S. Tôgô, Derivations of Lie algebras, J. Sci. Hiroshima Univ. Ser. A, Series A-I 28 (1964) 133–158. doi:10.32917/hmj/1206139393

  • [16] F. Vosooghpour, Z. Kargarian and M. Akhavan-Malayeri, Commuting automorphism of p-groups with cyclic maximal subgroups, Commun. Korean Math. Soc. 28 (2013) 643–647. doi:10.1142/S0219498819502086


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