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LIE $n$-HIGHER DERIVATIONS AND LIE $n$-HIGHER DERIVABLE MAPPINGS

Part of: Rings and algebras with additional structure Special classes of linear operators

Published online by Cambridge University Press:  09 June 2017

YANA DING Open the ORCID record for YANA DING [Opens in a new window]  and
JIANKUI LI Open the ORCID record for JIANKUI LI [Opens in a new window]
YANA DING
Affiliation:
Department of Mathematics, East China University of Science and Technology, Shanghai, China email [email protected]
JIANKUI LI*
Affiliation:
Department of Mathematics, East China University of Science and Technology, Shanghai, China email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

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Let ${\mathcal{A}}$ be a unital torsion-free algebra over a unital commutative ring ${\mathcal{R}}$. To characterise Lie $n$-higher derivations on ${\mathcal{A}}$, we give an identity which enables us to transfer problems related to Lie $n$-higher derivations into the same problems concerning Lie $n$-derivations. We prove that: (1) if every Lie $n$-derivation on ${\mathcal{A}}$ is standard, then so is every Lie $n$-higher derivation on ${\mathcal{A}}$; (2) if every linear mapping Lie $n$-derivable at several points is a Lie $n$-derivation, then so is every sequence $\{d_{m}\}$ of linear mappings Lie $n$-higher derivable at these points; (3) if every linear mapping Lie $n$-derivable at several points is a sum of a derivation and a linear mapping vanishing on all $(n-1)$th commutators of these points, then every sequence $\{d_{m}\}$ of linear mappings Lie $n$-higher derivable at these points is a sum of a higher derivation and a sequence of linear mappings vanishing on all $(n-1)$th commutators of these points. We also give several applications of these results.

Keywords

Lie higher derivationLie n-higher derivable mappingLie n-higher derivationLie triple higher derivation

MSC classification

Primary: 16W25: Derivations, actions of Lie algebras
Secondary: 47B47: Commutators, derivations, elementary operators, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 2 , October 2017 , pp. 223 - 232
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

This work is partially supported by the National Natural Science Foundation of China, Grant No. 11371136.

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