Nonlinear lie-type derivations on full matrix algebras

Xiao, Zhankui; Wei, Feng

doi:10.1007/s00605-012-0388-7

Cited by 14 publications

(9 citation statements)

References 16 publications

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“…Recall that a ring is said to be 2-torsion free if 2a = 0 implies a = 0 for any a ∈ ; otherwise, is 2-torsion. Xiao [6] generalizes the result of [7] to multiplicative Jordan derivations and obtains the same result.…”

Section: Letsupporting

confidence: 54%

“…Particularly, if δ is also assumed to be additive (linear), then δ is called additive (linear) derivation and Jordan derivation, respectively. The questions of characterizing (multiplicative) Jordan derivations and revealing the relationship between Jordan derivations and derivations have received many mathematicians' attention (for example, see [1][2][3][4][5][6][7] and the references therein).…”

Section: Letmentioning

confidence: 99%

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Multiplicative Jordan derivations on triangular n-matrix rings

Chen¹,

Qi²

2020

ScienceAsia

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Let be a triangular n-matrix ring (n 2) and δ : → a map. It is shown that δ is a multiplicative Jordan derivation if and only if one of the statements holds: (1) if is 2-torsion free, then δ is an additive derivation; (2) if is 2-torsion, under some mild assumptions, then δ(X) = d(X) + γ(X) holds for all X ∈ , where d : → is an additive derivation and γ is a map from into its center vanishing on all elements X Y + Y X for X , Y ∈. This generalizes some related known results.

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Section: Letsupporting

confidence: 54%

Section: Letmentioning

confidence: 99%

Multiplicative Jordan derivations on triangular n-matrix rings

Chen¹,

Qi²

2020

ScienceAsia

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“…Recently, some authors are interested in nonlinear Lie (or Jordan) derivation on certain associative algebras (or Lie algebras) (see [1][2][3][4][9][10][11]), the results obtained in these documents show that, although the mappings considered are assumed needless linear (or additive), however they turn out to be additive. Now, another question arises naturally: If a mapping on such an algebra is only assumed to be derivable at a single point, especially at the zero point, whether it is also additive.…”

Section: Introductionmentioning

confidence: 97%

Nonlinear mappings on upper triangular matrices derivable at zero point

Wong

Chen

2015

Linear Algebra and its Applications

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“…This result was extended to the case of generalized matrix algebras by Wang ad Wang [45]. Fosner, Wei and Xiao [17,48] studied nonlinear lie-type derivations on full matrix algebras and von Neumann algebras. As a summary, the previous works showed that nonlinear lie-type derivations of triangular algebras, generalized matrix algebras and some operator algebras are of proper forms.…”

mentioning

confidence: 94%

Nonlinear Lie-Type Derivations of finitary Incidence Algebras and Related Topics

Yang,

Wei

2021

Preprint

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This is a continuation of our earlier works [29,51,52] with respect to (non-)linear Lie-type derivations of finitary incidence algebras. Let X be a pre-ordered set, R be a 2-torsionfree and (n − 1)torsionfree commutative ring with identity, where n ≥ 2 is an integer. Let F I(X, R) be the finitary incidence algebra of X over R. In this paper, a complete clarification is obtained for the structure of nonlinear Lie-type derivations of F I(X, R). We introduce a new class of derivations on F I(X, R) named inner-like derivations, and prove that each nonlinear Lie n-derivation on F I(X, R) is the sum of an inner-like derivation, a transitive induced derivation and a quasi-additive induced Lie n-derivation. Furthermore, if X is finite, we show that a quasi-additive induced Lie n-derivation can be expressed as the sum of an additive induced Lie derivation and a central-valued map annihilating all (n − 1)-th commutators. We also provide a sufficient and necessary condition such that every nonlinear Lie n-derivation of F I(X, R) is of proper form. Some related topics for further research are proposed in the last section of this article.Let A be an associative algebra over a commutative ring R. Let [ , ] and • be the Lie product and Jordan product respectively, i.e., [x, y] = xy − yx and x • y = xy + yx for all x, y ∈ A. Then (A, [ , ]) is a Lie algebra and (A, •) is a Jordan algebra. It is a fascinating topic to study the connection between the associative, Lie and Jordan structures of A. In this field, there are two classes of important algebraic maps: algebra homomorphisms and differential operators. For example Jordan homomorphisms, Lie homomorphisms, Jordan derivations and Lie derivations. In the AMS Hour Talk of 1961, Herstein proposed many problems

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Nonlinear lie-type derivations on full matrix algebras

Cited by 14 publications

References 16 publications

Multiplicative Jordan derivations on triangular n-matrix rings

Multiplicative Jordan derivations on triangular n-matrix rings

Nonlinear mappings on upper triangular matrices derivable at zero point

Nonlinear Lie-Type Derivations of finitary Incidence Algebras and Related Topics

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