×

Multiplicative Lie \(n\)-derivations of triangular rings. (English) Zbl 1247.16040

Let \(R\) be an associative ring and let \([x,y]=xy-yx\) denote the Lie product of \(x,y\in R\). An additive map \(\delta\) is called a Lie derivation on \(R\) if it is a derivation with respect to the Lie product. In this paper the concept of Lie derivation is generalized as follows. Let \(p_1(x)=x\) and, for \(n\geq 2\), let \(p_n(x_1,\dots,x_n)=[p_{n-1}(x_1,\dots,x_{n-1}),x_n]\), where \(x,x_1,\dots,x_n\in R\). A mapping \(\varphi\colon R\to R\) is a multiplicative Lie \(n\)-derivation if \[ \varphi(p_n(x_1,\dots,x_n))=\sum_{i=1}^np_n(x_1,\dots,x_{i-1},\varphi(x_i),x_{i+1},\dots,x_n) \] holds for all \(x_1,\dots,x_n\in R\). For instance, let \(\delta\) be a Lie derivation on \(R\) and \(\gamma\colon R\to Z(R)\) such that \(\gamma(p_n(R,\dots,R))\) is trivial. Then \(\delta+\gamma\) is a multiplicative \(n\)-Lie derivation on \(R\) and multiplicative \(n\)-Lie derivations of this type are said to be of the standard form.
One of the results of the paper gives a necessary and sufficient condition for a multiplicative \(n\)-Lie derivation on an \((n-1)\)-torsion free triangular ring to be of the standard form. The main theorem gives sufficient conditions on an \((n-1)\)-torsion free triangular ring for every multiplicative \(n\)-Lie derivation on it being of the standard form.

MSC:

16W25 Derivations, actions of Lie algebras
16S50 Endomorphism rings; matrix rings
47B47 Commutators, derivations, elementary operators, etc.
16R60 Functional identities (associative rings and algebras)
15A78 Other algebras built from modules
47L35 Nest algebras, CSL algebras
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Abdullaev, I.Z., n-Lie derivations on von Neumann algebras, Uzbek. mat. zh ., 5-6, 3-9, (1992) · Zbl 0902.47037
[2] Benkovič, D., Biderivations of triangular algebras, Linear algebra appl., 431, 1587-1602, (2009) · Zbl 1185.16045
[3] Benkovič, D., Generalized Lie derivations on triangular algebras, Linear algebra appl., 434, 1532-1544, (2011) · Zbl 1216.16032
[4] Benkovič, D.; Eremita, D., Commuting traces and commutativity preserving maps on triangular algebras, J. algebra, 280, 797-824, (2004) · Zbl 1076.16032
[5] Brešar, M., Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. amer. math. soc., 335, 525-546, (1993) · Zbl 0791.16028
[6] Brešar, M.; Šemrl, P., Commuting traces of biadditive maps revisited, Comm. algebra, 31, 381-388, (2003) · Zbl 1046.16019
[7] Cheung, W.-S., Commuting maps of triangular algebras, J. London math. soc., 63, 117-127, (2001) · Zbl 1014.16035
[8] Cheung, W.-S., Lie derivations of triangular algebras, Linear and multilinear algebra, 51, 299-310, (2003) · Zbl 1060.16033
[9] Davidson, K.R., Nest algebras, Pitman research notes in mathematics series 191, (1988), Longman Harlow
[10] Forrest, B.E.; Marcoux, L.W., Derivations of triangular Banach algebras, Indiana univ. math. J., 45, 441-462, (1996) · Zbl 0890.46035
[11] Han, D.; Wei, F., Jordan \((\alpha, \beta)\)-derivations on triangular algebras and related mappings, Linear algebra appl., 434, 259-284, (2011) · Zbl 1208.16033
[12] Lu, F., Lie triple derivations on nest algebras, Math. nachr., 280, 882-887, (2007) · Zbl 1124.47054
[13] Martindale, W.S., Lie derivations of primitive rings, Michigan math. J., 11, 183-187, (1964) · Zbl 0123.03201
[14] Miers, R., Lie triple derivations of von Neumann algebras, Proc. amer. math. soc., 71, 57-61, (1978) · Zbl 0384.46047
[15] Ji, P.; Wang, L., Lie triple derivations of TUHF algebras, Linear algebra appl., 403, 399-408, (2005) · Zbl 1114.46048
[16] Posner, E.C., Derivations in prime rings, Proc. amer. math. soc., 8, 1093-1100, (1957) · Zbl 0082.03003
[17] Villena, A.R., Lie derivations on Banach algebras, J. algebra, 226, 390-409, (2000) · Zbl 0957.46032
[18] Wei, F.; Xiao, Z., Higher derivations of triangular algebras and its generalizations, Linear algebra appl., 435, 1034-1054, (2011) · Zbl 1226.16028
[19] Wong, T.-L., Jordan isomorphisms of triangular rings, Proc. amer. math. soc., 133, 3381-3388, (2005) · Zbl 1077.47056
[20] Xiao, Z.; Wei, F., Jordan higher derivations on triangular algebras, Linear algebra appl., 432, 2615-2622, (2010) · Zbl 1185.47034
[21] Yu, W.-Y.; Zhang, J.-H., Nonlinear Lie derivations of triangular algebras, Linear algebra appl., 432, 2953-2960, (2010) · Zbl 1193.16030
[22] Zhang, J.-H., Lie derivations on nest subalgebras of von Neumann algebras, Acta math. sinica, 46, 657-664, (2003) · Zbl 1054.47061
[23] Zhang, J.-H.; Yu, W.-Y., Jordan derivations of triangular algebras, Linear algebra appl., 419, 251-255, (2006) · Zbl 1103.47026
[24] 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.