A note on f-derivations of triangular algebras

The notion of f -derivations was introduced by Beidar and Fong to unify several kinds of linear maps including derivations, Lie derivations and Jordan derivations. In this paper we introduce the notion of f -biderivations as a natural "biderivation" counterpart of the notion of "f -derivations". We first show, under some conditions, that any f -biderivation is a Jordan biderivation. Then, we turn to study f -biderivations of a unital algebra with an idempotent. Our second main result shows, under some conditions, that every Jordan biderivation can be written as a sum of a biderivation, an antibiderivation and an extremal biderivation. As a consequence we show that every Jordan biderivation on a triangular algebra is a biderivation.

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Zero-Sum Coefficient Derivations in Three Variables of Triangular Algebras

Kim

Lee²

2016

JMR

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Under mild assumptions Benkovi\v{c} showed that an $f$-derivation of a triangular algebra is a derivation when the sum of the coefficients of the multilinear polynomial $f$ is nonzero. We investigate the structure of $f$-derivations of triangular algebras when $f$ is of degree 3 and the coefficient sum is zero. The zero-sum coeffient derivations include Lie derivations (degree 2) and Lie triple derivations (degree 3), which have been previously shown to be not necessarily derivations but in standard form, i.e., the sum of a derivation and a central map. In this paper, we present sufficient conditions on the coefficients of $f$ to ensure that any $f$-derivations are derivations or are in standard form.<br /><br />

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A note on f-derivations of triangular algebras

Cited by 3 publications

References 16 publications

Lie (Jordan) derivations of arbitrary triangular algebras

Lie (Jordan) derivations of arbitrary triangular algebras

f-Biderivations and Jordan biderivations of unital algebras with idempotents

Zero-Sum Coefficient Derivations in Three Variables of Triangular Algebras

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