Characterizations of Additive 𝜉-Lie Derivations on Unital Algebras

• Published in: Ukrainian mathematical journal Vol. 73; no. 4; pp. 532 - 546
• Main Authors: Ashraf, M., Jabeen, A.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-09-2021; Springer
• Subjects: Algebra; Analysis; Applications of Mathematics; Geometry; Mathematics; Mathematics and Statistics; Statistics; ■
• ISSN: 0041-5995; 1573-9376
• DOI: 10.1007/s11253-021-01941-y

Let R be a commutative ring with unity and let 𝒰 be a unital algebra over ℛ (or a field 𝔽). An R-linear map
L
: 𝒰
→
𝒰 is called a Lie derivation on 𝒰 if
L
([
u, v
]) = [
L
(
u
)
, v
] + [
u,L
(
v
)] holds for all
u, v
𝜖 𝒰
.
For scalar 𝜉 𝜖 𝔽
,
an additive map
L
: 𝒰
→
𝒰 is called an additive 𝜉-Lie derivation on 𝒰 if
L
([
u, v
]
𝜉
) = [
L
(
u
)
, v
]
𝜉
+ [
u,L
(
v
)]
𝜉
,
where [
u, v
]
𝜉
=
uv −
𝜉
vu
holds for all
u, v
𝜖 𝒰
.
In the present paper, under certain assumptions imposed on 𝒰
,
it is shown that every Lie derivation (resp., additive 𝜉-Lie derivation)
L
on U is of standard form, i.e.,
L
=
δ
+
∅,
where
δ
is an additive derivation on 𝒰 and
∅
is a mapping
∅
: 𝒰
→ Z
(𝒰) vanishing at [
u, v
] with
uv
= 0 in 𝒰
.
Moreover, we also characterize the additive 𝜉-Lie derivation for 𝜉
6
= 1 by its action on zero product in a unital algebra over F
.