The aim of this paper was to provide a characterization of nonlinear generalized Lie $ n $-higher derivations for a certain class of triangular algebras. It was shown that, under some mild conditions, each component $ G_r $ of a nonlinear generalized Lie $ n $-higher derivation $ \{G_r\}_{r\in N} $ of the triangular algebra $ \mathcal{U} $ could be expressed as the sum of an additive generalized higher derivation and a nonlinear mapping vanishing on all ($ n-1 $)-th commutators on $ \mathcal{U} $.
Citation: He Yuan, Qian Zhang, Zhendi Gu. Characterizations of generalized Lie $ n $-higher derivations on certain triangular algebras[J]. AIMS Mathematics, 2024, 9(11): 29916-29941. doi: 10.3934/math.20241446
The aim of this paper was to provide a characterization of nonlinear generalized Lie $ n $-higher derivations for a certain class of triangular algebras. It was shown that, under some mild conditions, each component $ G_r $ of a nonlinear generalized Lie $ n $-higher derivation $ \{G_r\}_{r\in N} $ of the triangular algebra $ \mathcal{U} $ could be expressed as the sum of an additive generalized higher derivation and a nonlinear mapping vanishing on all ($ n-1 $)-th commutators on $ \mathcal{U} $.
| [1] |
W. S. Cheung, Lie derivations of triangular algebras, Linear Multilinear Algebra, 51 (2003), 299–310. https://doi.org/10.1080/0308108031000096993 doi: 10.1080/0308108031000096993
|
| [2] |
W. Y. Yu, J. H. Zhang, Nonlinear Lie derivations of triangular algebras, Linear Algebra Appl., 432 (2010), 2953–2960. https://doi:10.1016/j.laa.2009.12.042 doi: 10.1016/j.laa.2009.12.042
|
| [3] |
Z. K. Xiao, F. Wei, Nonlinear Lie higher derivations on triangular algebras, Linear Multilinear Algebra, 60 (2012), 979–994. https://doi.org/10.1080/03081087.2011.639373 doi: 10.1080/03081087.2011.639373
|
| [4] |
X. F. Qi, Characterization of Lie higher derivations on triangular algebras, Acta Math. Sin. (Engl. Ser.), 29 (2013), 1007–1018. https://doi:10.1007/s10114-012-1548-3 doi: 10.1007/s10114-012-1548-3
|
| [5] |
M. Ashraf, M. A. Ansari, M. S. Akhtar, Characterization of Lie-type higher derivations of triangular rings, Georgian Math. J., 30 (2023), 33–46. https://doi.org/10.1515/gmj-2022-2195 doi: 10.1515/gmj-2022-2195
|
| [6] |
D. Benkovič, Generalized Lie derivations on triangular algebras, Linear Algebra Appl., 434 (2011), 1532–1544. https://doi.org/10.1016/j.laa.2010.11.039 doi: 10.1016/j.laa.2010.11.039
|
| [7] |
W. H. Lin, Nonlinear generalized Lie $n$-derivations on triangular algebras, Commun. Algebra, 46 (2018), 2368–2383. https://doi.org/10.1080/00927872.2017.1383999 doi: 10.1080/00927872.2017.1383999
|
| [8] |
D. Benkovič, Generalized Lie $n$-derivations of triangular algebras, Commun. Algebra, 47 (2019), 5294–5302. https://doi.org/10.1080/00927872.2019.1617875 doi: 10.1080/00927872.2019.1617875
|
| [9] |
M. Ashraf, A. Jabeen, Nonlinear generalized Lie triple higher derivation on triangular algebras, Bull. Iran. Math. Soc., 44 (2018), 513–530. https://doi:10.1007/s41980-018-0035-8 doi: 10.1007/s41980-018-0035-8
|
| [10] |
W. S. Cheung, Commuting maps of triangular algebras, J. London Math. Soc., 63 (2001), 117–127. https://doi.org/10.1112/S0024610700001642 doi: 10.1112/S0024610700001642
|
| [11] |
D. Benkovič, D. Eremita, Multiplicative Lie $n$-derivations of triangular rings, Linear Algebra Appl., 436 (2012), 4223–4240. https://doi:10.1016/j.laa.2012.01.022 doi: 10.1016/j.laa.2012.01.022
|
He Yuan, Qian Zhang, Zhendi Gu. Characterizations of generalized Lie $ n $-higher derivations on certain triangular algebras[J]. AIMS Mathematics, 2024, 9(11): 29916-29941. doi: 10.3934/math.20241446
DownLoad: