Nonlinear Lie higher derivations on triangular algebras
Nonlinear Lie higher derivations on triangular algebras
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DOI：
10.1080/03081087.2011.639373
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发表时间：
2012-07
影响因子：
1.1
通讯作者：
Zhankui Xiao;F. Wei
中科院分区：
文献类型：
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作者：
Zhankui Xiao;F. Wei
设a， B是整数代数，M是整数（a， B）-双模，它作为左a模和右B模都是可靠的。假设是由A， B和M组成的三角代数。Brešar，双加性映射的交换迹，保交换映射和李映射，译。阿米尔。数学。《社会科学》（1993），第525-546页。Yu和j - h。张，三角代数的非线性李高导，线性代数应用，432 (2010),pp. 2953-2960]，本文将研究非线性李高导。设D = {L n} n∈n是一个无加性条件的李高阶导数。在温和的假设下，证明了D = {L n} n∈n是标准形式，即每个分量L n （n≥1）可以通过一个加性高阶导数和一个非线性泛函在所有对易子上消失来表示。作为应用，对一些经典三角代数上的非线性李高阶导数进行了刻画。
Let ℛ be a commutative ring with identity, A, B be unital algebras over ℛ and M be a unital (A, B)-bimodule, which is faithful as a left A-module and also as a right B-module. Let be the triangular algebra consisting of A, B and M. Motivated by the powerful works of Brešar [M. Brešar, Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993), pp. 525–546] and Yu et al. [W.-Y. Yu and J.-H. Zhang, Nonlinear Lie derivations of triangular algebras, Linear Algebra Appl. 432 (2010), pp. 2953–2960], we will study nonlinear Lie higher derivations on 𝒯 in this article. Let D = {L n } n∈ℕ be a Lie higher derivation on 𝒯 without additive condition. Under mild assumptions, it is shown that D = {L n } n∈ℕ is of standard form, i.e. each component L n (n ≥ 1) can be expressed through an additive higher derivation and a nonlinear functional vanishing on all commutators of 𝒯. As applications, nonlinear Lie higher derivations on some classical triangular algebras are characterized.