# zbMATH — the first resource for mathematics

##### Examples
 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.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 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)
Characterizations of Lie derivations of triangular algebras. (English) Zbl 1226.16026

A triangular algebra $T=T\left(A,X,B\right)$ has the form of an upper triangular matrix ring with elements having diagonal entries in $A$ and $B$ and upper right entries in $X$; $A$ and $B$ are unital algebras over a commutative ring $R$ with 1, and $X$ is an $A$-$B$-bimodule that is faithful on each side. The center of $T$ is $Z\left(T\right)=\left\{\text{diag}\left(a,b\right)\in T\mid am=mb$ for all $m\in X\right\}$. The projection of $Z\left(T\right)$ to either $A$ or $B$ is central.

The main results in the paper assume that these projections are all of $Z\left(A\right)$ and $Z\left(B\right)$. The first theorem considers an $R$-linear map $\delta$ of $T$ satisfying $\delta \left(\left[x,y\right]\right)=\left[\delta \left(x\right),y\right]+\left[x,\delta \left(y\right)\right]$ for all $x,y\in T$ so that $xy=0$ and proves that on $T$, $\delta \left(x\right)=d\left(x\right)+\tau \left(x\right)$ for $d$ a derivation of $T$ and $\tau$ an $R$-linear map from $T$ to $Z\left(T\right)$ with $\tau \left(\left[x,y\right]\right)=0$ when $xy=0$. The second main result is the first with the assumptions that $xy=0$ replaced with $xy=\text{diag}\left({1}_{A},{0}_{B}\right)$, but also requires the additional assumption that for any $a\in A$ there is an integer $j$ so that $j{1}_{A}-a$ is invertible.

##### MSC:
 16W25 Derivations, actions of Lie algebras (associative rings and algebras) 16S50 Endomorphism rings: matrix rings 47B47 Commutators, derivations, elementary operators, etc.
##### Keywords:
Lie derivations; triangular algebras; linear maps; commutators
##### References:
 [1] Alaminos, J.; Extremera, J.; Villena, A. R.; Bresar, M.: Characterizing homomorphisms and derivations on C$*$-algebras, Proc. roy. Soc. Edinburgh sect. A 137, 1-7 (2007) · Zbl 1144.47030 · doi:10.1017/S0308210505000090 [2] Alaminos, J.; Mathieu, M.; Villena, A. R.: Symmetric amenability and Lie derivations, Math. proc. Cambridge philos. Soc. 137, 433-439 (2004) · Zbl 1063.46033 · doi:10.1017/S0305004104007637 [3] An, R.; Hou, J. C.: Characterizations of derivations on triangular rings: additive maps derivable at idempotents, Linear algebra appl. 431, 1070-1080 (2009) · Zbl 1173.47023 · doi:10.1016/j.laa.2009.04.005 [4] Bresar, M.: Characterizing homomorphisms, derivations and multipliers in rings with idempotents, Proc. roy. Soc. Edinburgh sect. A 137, 9-21 (2007) · Zbl 1130.16018 · doi:10.1017/S0308210504001088 [5] Chebotar, M. A.; Ke, W. -F.; Lee, P. -H.: Maps characterized by action on zero products, Pacific J. Math. 216, 217-228 (2004) · Zbl 1078.16034 · doi:10.2140/pjm.2004.216.217 [6] Cheng, W. S.: Commuting maps of triangular algebras, J. London math. Soc. 63, 117-127 (2001) · Zbl 1014.16035 · doi:10.1112/S0024610700001642 [7] Cheng, W. S.: Lie derivations of triangular algebras, Linear and multilinear algebra 51, 299-310 (2003) [8] Christensen, E.: Derivations of nest algebras, Math. ann. 229, 155-161 (1977) · Zbl 0356.46057 · doi:10.1007/BF01351601 [9] K.R. Davision, Nest Algebras, Pitman Research Notes in Mathematics Series, vol. 191, Longman Scientific and Technical, Burnt Mill Harlow, Essex, UK, 1988. [10] Hou, J. C.; Qi, X. F.: Additive maps derivable at some points on J-subspace lattice algebras, Linear algebra appl. 429, 1851-1863 (2008) · Zbl 1153.47062 · doi:10.1016/j.laa.2008.05.013 [11] Jiao, M.; Hou, J. C.: Additive maps derivable at zero point on nest algebras, Linear algebra appl. 432, 2984-2994 (2010) [12] Jing, W.: On Jordan all-derivable points of $B\left(H\right)$, Linear algebra appl. 430, 941-946 (2009) · Zbl 1163.47030 · doi:10.1016/j.laa.2008.09.006 [13] Jing, W.; Lu, S.; Li, P.: Characterisations of derivations on some operator algebras, Bull. austral. Math. soc. 66, 227-232 (2002) · Zbl 1035.47019 · doi:10.1017/S0004972700040077 [14] Lu, F. Y.; Jing, W.: Characterizations of Lie derivations of $B\left(X\right)$, Linear algebra appl. 432, 89-99 (2010) · Zbl 1188.47029 · doi:10.1016/j.laa.2009.07.026 [15] Qi, X. F.; Hou, J. C.: Characterizations of derivations of Banach space nest algebras: all-derivable points, Linear algebra appl. 432, 3183-3200 (2010) · Zbl 1192.47071 · doi:10.1016/j.laa.2010.01.020 [16] Zhu, J.: All-derivable points of operator algebras, Linear algebra appl. 427, 1-5 (2007) · Zbl 1128.47062 · doi:10.1016/j.laa.2007.05.049 [17] Zhu, J.; Xiong, C.: Derivable mappings at unit operator on nest algebras, Linear algebra appl. 422, 721-735 (2007) · Zbl 1140.47059 · doi:10.1016/j.laa.2006.12.002 [18] Zhu, J.; Xiong, C. P.: All-derivable points in continuous nest algebras, J. math. Anal. appl. 340, 845-853 (2008) · Zbl 1134.47054 · doi:10.1016/j.jmaa.2007.08.055