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Lie triple derivation of the Lie algebra of strictly upper triangular matrix over a commutative ring. (English) Zbl 1163.17014

Summary: Let \(\mathcal N(n, R)\) be the nilpotent Lie algebra consisting of all strictly upper triangular \(n\times n\) matrices over a 2-torsionfree commutative ring \(R\) with identity 1. In this paper, we prove that any Lie triple derivation of \(\mathcal N(n, R)\) can be uniquely decomposed as a sum of an inner triple derivation, diagonal triple derivation, central triple derivation and extremal triple derivation for \(n \geq 6\). In the cases \(1\leq n\leq 5\), the results are trivial.

MSC:

17B30 Solvable, nilpotent (super)algebras
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
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[1] Cao, Y.-A., Automorphisms of certain Lie algebras of upper triangular matrices over a commutative ring, J. Algebra, 189, 506-513 (1997) · Zbl 0878.17016
[2] Cao, Y.-A.; Wang, J., A note on algebra automorphisms of strictly upper triangular matrices over commutative rings, Linear Algebra Appl., 311, 187-193 (2000) · Zbl 0954.15011
[3] Doković, D.Ž., Automorphisms of the Lie algebra of upper triangular matrices over a connected commutative ring, J. Algebra, 170, 101-110 (1994) · Zbl 0822.17017
[4] Ji, P.-S.; Wang, L., Lie triple derivations of TUHF algebras, Linear Algebra Appl., 403, 399-408 (2005) · Zbl 1114.46048
[5] Ou, S.-K.; Wang, D.-Y.; Yao, R.-P., Derivations of the Lie algebra of strictly upper triangular matrices over a commutative ring, Linear Algebra Appl., 424, 378-383 (2007) · Zbl 1135.17010
[6] Wang, D.-Y.; Yu, Q., Derivations of the parabolic subalgebras of the general linear Lie algebra over a commutative ring, Linear Algebra Appl., 418, 763-774 (2006) · Zbl 1161.17313
[7] Zhang, J.-H.; Wu, B.-W.; Cao, H.-X., Lie triple derivations of nest algebras, Linear Algebra Appl., 416, 559-567 (2006) · Zbl 1102.47060
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