zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Nonlinear Lie derivations of triangular algebras. (English) Zbl 1193.16030
Let $\cal A$ be an algebra over a commutative ring $\cal R$. A map $\delta\colon\cal A\to\cal A$ is called an additive derivation if it is additive and satisfies $\delta(xy)=\delta(x)y+x\delta(y)$ for all $x,y\in\cal A$. If there exists an element $a\in\cal A$ such that $\delta(x)=[x,a]$ for all $x\in\cal A$, where $[x,a]=xa-ax$ is the Lie product or the commutator of the elements $x,a\in\cal A$, then $\delta$ is said to be an inner derivation. Let $\varphi\colon\cal A\to\cal A$ be a map (without the additivity assumption). We say that $\varphi$ is a nonlinear Lie derivation if $\varphi([x,y])=[\varphi(x),y]+[x,\varphi(y)]$ for all $x,y\in\cal A$. The structure of additive or linear Lie derivations on matrix algebras has been studied by many authors. {\it W.-S. Cheung} [in Linear Multilinear Algebra 51, No. 3, 299-310 (2003; Zbl 1060.16033)] initiated the study of Lie derivations of triangular algebras and showed that every Lie derivation of a triangular algebra is of standard form. Recently, {\it L. Chen, J.-H. Zhang}, [Linear Multilinear Algebra 56, No. 6, 725-730 (2008; Zbl 1166.16016)], described nonlinear Lie derivations of upper triangular matrix algebras. Motivated by the previous works, the authors prove that under mild conditions any nonlinear Lie derivation of triangular algebras is the sum of an additive derivation and a map into its center sending commutators to zero. This result is applied to some special triangular algebras, for example to block upper triangular matrix algebras and to nest algebras.

16W25Derivations, actions of Lie algebras (associative rings and algebras)
16S50Endomorphism rings: matrix rings
16W10Associative rings with involution, etc.
47B47Commutators, derivations, elementary operators, etc.
47L35Nest algebras, CSL algebras
Full Text: DOI
[1] Brešar, M.: Commuting traces of biadditive mappings, commutativity preserving mappings and Lie mappings, Trans. amer. Math. soc. 335, 525-546 (1993) · Zbl 0791.16028 · doi:10.2307/2154392
[2] Benkovič, D.; Eremita, D.: Commuting traces and commutativity preserving maps on triangular algebras, J. algebra 280, 797-824 (2004) · Zbl 1076.16032 · doi:10.1016/j.jalgebra.2004.06.019
[3] D. Benkovič, Biderivations triangular algebras, Linear Algebra Appl., in press. · Zbl 1185.16045
[4] Cheung, W. S.: Commuting maps of triangular algebras, J. London math. Soc. 63, 117-127 (2001) · Zbl 1014.16035 · doi:10.1112/S0024610700001642
[5] Cheung, W. S.: Lie derivation of triangular algebras, Linear and multilinear algebra 51, 299-310 (2003) · Zbl 1060.16033 · doi:10.1080/0308108031000096993
[6] Chen, L.; Zhang, J. H.: Nonlinear Lie derivation on upper triangular matrix algebras, Linear and multilinear algebra 56, No. 6, 725-730 (2008) · Zbl 1166.16016 · doi:10.1080/03081080701688119
[7] Coelho, S. P.; Milies, C. P.: Derivations of upper triangular matrix rings, Linear algebra appl. 187, 263-267 (1993) · Zbl 0781.16020 · doi:10.1016/0024-3795(93)90141-A
[8] Davidson, K. R.: Nest algebras, Pitan research notes in mathematics series (1988)
[9] Han, D. G.: Additive derivations of nest algebras, Proc. amer. Math. soc. 119, 1165-1169 (1993) · Zbl 0810.47040 · doi:10.2307/2159979
[10] Han, D. G.: Continuity and linearity of additive derivations of nest algebras on Banach spaces, Chinese ann. Math. ser. B 17, 227-236 (1996) · Zbl 0856.47028
[11] Johnson, B. E.: Symmetric amenability and the nonexistence of Lie and Jordan derivations, Math. proc. Cambridge philos. Soc. 120, 455-473 (1996) · Zbl 0888.46024 · doi:10.1017/S0305004100075010
[12] Jøndrup, S.: Automorphisms and derivations of upper triangular matrix rings, Linear algebra appl. 221, 205-218 (1995) · Zbl 0826.16034 · doi:10.1016/0024-3795(93)00255-X
[13] Killam, E.: Lie derivations on skew elements in prime rings with involution, Canad. math. Bull. 30, 344-350 (1987) · Zbl 0607.16024 · doi:10.4153/CMB-1987-049-5
[14] Mathieu, M.; Villena, A. R.: The structure of Lie derivations on C*-algebras, J. funct. Anal. 202, 504-525 (2003) · Zbl 1032.46086 · doi:10.1016/S0022-1236(03)00077-6
[15] Iii, W. S. Martindale: Lie derivations of primitive rings, Michigan math. J. 11, 183-187 (1964) · Zbl 0123.03201 · doi:10.1307/mmj/1028999091
[16] Miers, C. R.: Lie derivations of von Neumann algebras, Duke math. J. 40, 403-409 (1973) · Zbl 0264.46064 · doi:10.1215/S0012-7094-73-04032-5
[17] Qi, X. F.; Hou, J. C.: Additive Lie ($\xi $-Lie) derivations and generalized Lie ($\xi $-Lie) derivations on nest algebras, Linear algebra appl. 431, 843-854 (2009) · Zbl 1207.47081 · doi:10.1016/j.laa.2009.03.037
[18] Swain, G. A.: Lie derivations of the skew elements of prime rings with involution, J. algebras 184, 679-704 (1996) · Zbl 0856.16037 · doi:10.1006/jabr.1996.0281
[19] Wong, T. -L.: Jordan isomorphisms of triangular algebras, Linear algebra appl. 418, 225-233 (2006)
[20] Zhang, J. -H.; Yu, W. -Y.: Jordan derivations of triangular algebras, Linear algebra appl. 419, 251-255 (2006) · Zbl 1103.47026 · doi:10.1016/j.laa.2006.04.015
[21] Zhang, J. -H.: Lie derivations on nest subalgebras of von Neumann algebras, Acta math. Sinica 46, 657-664 (2003) · Zbl 1054.47061