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QUASI-DERIVATIONS OF LOW-DIMENSIONAL LEIBNIZ ALGEBRAS
AND THEIR PROPERTIES
Musayev Sardor Xabibulla o’g’li
UNIVERSITY OF SCIENCE AND TECHNOLOGIES
Teacher of the "Exact Sciences" department
ABSTRACT
The article presents the results obtained about quasi-derivations of small-
dimension Leibniz algebras and their properties.
Key words:
Leibniz algebra, derivation, quasi-derivation, generalized
derivation, centroids and quasi-centroids.
INTRODUCTION
Currently, the class of Leibniz algebras, which is a generalization of Lie
algebras, is being intensively studied. It should be noted that algebras satisfying the
Leibniz theorem were first introduced in 1965 in the work of A. Bloch under the name
of D-algebras. However, not much attention was paid to the study of D-algebras, only
J.L. Only after the works of Lode and T. Pirashvili, Leibniz algebras began to be
intensively studied, and up to now a number of articles devoted to these algebras have
been published. Leibniz algebras were developed by the French mathematician J.L. in
the 90s of the last century. This by Lode
[𝑥, [𝑦, 𝑧]] = [[𝑥, 𝑦], 𝑧] − [[𝑥, 𝑧], 𝑦]
It was included in the science as an algebra characterized by Leibniz's equation. Since
1998, the structural theory of Leibniz algebra has been developed by Sh.A. Ayupov and
B.A. The Omirovs began to learn. The larger the size of the algebra, the more difficult
it is to describe. Ayupov Sh.A., Omirov B.A., Rakhimov I.S., Riksiboev I.M.,
Khudoyberdiyev A.Kh. with nilpotent Leibniz algebras. and others were engaged.
Since the study of nilpotent Lie algebras of large dimensions is also complicated,
nilpotent algebras are divided into several classes. For example, zero filiform, filiform,
quasi filiform and other classes.
In recent years, a number of operators considered as differentiations of non-
associative algebras and generalizations of differentiations have been widely studied.
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In particular, the concepts of quasi-differentiations were studied not only for operator
algebras but also for Lie and Leibniz algebras. This article explores the concept of
quasi-differentiations of small-scale Leibniz algebras. Quasi-differentiations of small-
scale Leibniz algebras and their properties are defined.
1.
PRELIMINARIES
This section is devoted to recalling some basic notions and concepts used
through the work.
Definition 1.
Let
(𝐿, [−, −])
be an algebra over
𝐹.
𝐿
is called Leibniz algebra,
if is satisfies for all
𝑥, 𝑦 𝑎𝑛𝑑 𝑧
:
[𝑥, [𝑦, 𝑧]] = [[𝑥, 𝑦], 𝑧] − [[𝑥, 𝑧], 𝑦].
Definition 2.
A linear map
𝐷: 𝐿 → 𝐿
is said to be a derivation of
𝐿
if it satisfies:
𝐷([𝑥, 𝑦]) = [𝐷(𝑥), 𝑦] + [𝑥, 𝐷(𝑦)],
for all
𝑥, 𝑦 ∈ 𝐿.
We denote the set of all derivations of
𝐿
by
𝐷𝑒𝑟(𝐿),
and then,
𝐷𝑒𝑟(𝐿)
provided
with the commutator is a subalgebra of
𝐸𝑛𝑑(𝐿)
and is called the derivation algebra of
𝐿.
Definition 3.
𝐷 ∈ 𝐸𝑛𝑑(𝐿)
is said to be a generalized derivation of
𝐿
, if there
exist
∃𝐷′, 𝐷
′′
∈ 𝐸𝑛𝑑(𝐿)
such that:
[𝐷(𝑥), 𝑦] + [𝑥, 𝐷
′
(𝑦)] = 𝐷
′′
([𝑥, 𝑦])
for all
𝑥, 𝑦 ∈ 𝐿.
Definition 4.
𝐷 ∈ 𝐸𝑛𝑑(𝐿)
is said to be a quasiderivation of
𝐿
, if there exists
∃𝐷
′
∈ 𝐸𝑛𝑑(𝐿)
, such that:
𝐷(𝑥), 𝑦] + [𝑥, 𝐷(𝑦)] = 𝐷
′
([𝑥, 𝑦])
for all
𝑥, 𝑦 ∈ 𝐿.
Denote by
𝐺𝐷𝑒𝑟(𝐿)
and
𝑄𝐷𝑒𝑟(𝐿)
the sets of generalized derivations and
quasiderivations, respectively.
Definition 5.
If
𝐶(𝐿) = {𝐷 ∈ 𝐸𝑛𝑑(𝐿)} | [𝐷(𝑥), 𝑦] = [𝑥, 𝐷(𝑦)] = 𝐷([𝑥, 𝑦]}
,
for all
𝑥, 𝑦 ∈ 𝐿
, then
𝐶(𝐿)
is called the centroid of
𝐿.
Definition 6.
If
𝑄𝐶(𝐿) = {𝐷 ∈ 𝐸𝑛𝑑(𝐿)} | [𝐷(𝑥), 𝑦] = [𝑥, 𝐷(𝑦)]},
for all
𝑥, 𝑦 ∈ 𝐿
, then
𝑄𝐶(𝐿)
is called the quasicentroid of
𝐿.
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Definition
7.
If
𝑍𝐷𝑒𝑟(𝐿) = {𝐷 ∈ 𝐸𝑛𝑑(𝐿) | [𝐷(𝑥), 𝑦] = [𝑥, 𝐷(𝑦)] =
𝐷([𝑥, 𝑦]) = 0}
, for all
𝑥, 𝑦 ∈ 𝐿,
then
𝑍𝐷𝑒𝑟(𝐿)
is called the central derivation of
𝐿.
It is easy to verify that
𝑍𝐷𝑒𝑟(𝐿) ⊆ 𝐷𝑒𝑟(𝐿) ⊆ 𝑄𝐷𝑒𝑟(𝐿) ⊆ 𝐺𝐷𝑒𝑟(𝐿) ⊆ 𝐸𝑛𝑑(𝐿)
𝐶(𝐿) ⊆ 𝑄𝐶(𝐿) ⊆ 𝑄𝐷𝑒𝑟(𝐿)
Theorem 1.
If
𝐿
is a Leibniz algebra, then:
(1)
[𝐷𝑒𝑟(𝐿), 𝐶(𝐿)] ⊆ 𝐶(𝐿)
(2)
[𝑄𝐷𝑒𝑟(𝐿), 𝑄𝐶(𝐿)] ⊆ 𝑄𝐶(𝐿)
(3)
𝐷(𝐷𝑒𝑟(𝐿)) ⊆ 𝐷𝑒𝑟(𝐿), ∀𝐷 ∈ 𝐶(𝐿)
(4)
𝐶(𝐿) ⊆ 𝑄𝐷𝑒𝑟(𝐿)
(5)
[𝑄𝐶(𝐿), 𝑄𝐶(𝐿)] ⊆ 𝑄𝐷𝑒𝑟(𝐿)
(6)
𝑄𝐷𝑒𝑟(𝐿) + 𝑄𝐶(𝐿) ⊆ 𝐺𝐷𝑒𝑟(𝐿)
.
Here, the commutator
[−, −]
is defined as follows:
[𝐷
1
, 𝐷
2
] = 𝐷
1
𝐷
2
− 𝐷
2
𝐷
1
.
Proposition 1.
Let
𝐿
be a Leibniz algebra over
𝐹
, with the operation
𝐷
1
• 𝐷
2
= 𝐷
1
𝐷
2
+ 𝐷
2
𝐷
1
, ∀𝐷
1
, 𝐷
2
∈ 𝐸𝑛𝑑(𝐿)
. Then,
(𝐸𝑛𝑑(𝐿),•)
is a Jordan
algebra.
QUASI-DERIVATIONS
OF
TWO-DIMENSIONAL
LEIBNIZ
ALGEBRAS:
It is known that any two-dimensional Leibniz algebra is isomorphic to one of the
following non-isomorphic Leibniz algebras:
𝐿
1
: [𝑒
1
, 𝑒
1
] = 𝑒
2
𝐿
2
: [𝑒
1
, 𝑒
2
] = −[𝑒
2
, 𝑒
1
] = 𝑒
2
𝐿
3
: [𝑒
1
, 𝑒
2
] = [𝑒
2
, 𝑒
2
] = 𝑒
1
2.
MAIN
RESULTS:
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We define derivations, quasi-derivations, centroids, quasi-centroids and
generalization derivations of these three different 2-dimensional algebras:
Theorem 2.
Derivations of algebra
𝐿
1
: [𝑒
1
, 𝑒
1
] = 𝑒
2
has the form:
𝐿
1
: [𝑒
1
, 𝑒
1
] = 𝑒
2
𝐷𝑒𝑟(𝐿
1
)
𝑄𝐷𝑒𝑟(𝐿
1
)
𝐺𝐷𝑒𝑟(𝐿
1
)
𝐶(𝐿
1
)
𝑄𝐶(𝐿
1
)
(
𝑑
11
𝑑
12
0
2𝑑
11
)
(
𝑑
11
𝑑
12
0
𝑑
22
)
(
𝑑
11
𝑑
12
0
𝑑
22
)
(
𝑑
11
𝑑
12
0
𝑑
11
)
(
𝑑
11
𝑑
12
0
𝑑
22
)
Theorem 3.
Derivations of algebra
𝐿
2
: [𝑒
1
, 𝑒
2
] = −[𝑒
2
, 𝑒
1
] = 𝑒
2
has the form:
𝐿
2
: [𝑒
1
, 𝑒
2
] = −[𝑒
2
, 𝑒
1
] = 𝑒
2
𝐷𝑒𝑟(𝐿
2
)
𝑄𝐷𝑒𝑟(𝐿
2
)
𝐺𝐷𝑒𝑟(𝐿
2
)
𝐶(𝐿
2
)
𝑄𝐶(𝐿
2
)
(
𝑑
11
𝑑
12
0
𝑑
22
)
(
𝑑
11
𝑑
12
𝑑
21
𝑑
22
)
(
𝑑
11
𝑑
12
𝑑
21
𝑑
22
)
(
𝑎
11
0
0
𝑎
11
)
(
𝑎
11
0
0
𝑎
11
)
Theorem 4.
Derivations of algebra
𝐿
3
: [𝑒
1
, 𝑒
2
] = [𝑒
2
, 𝑒
2
] = 𝑒
1
has the form:
𝐿
3
: [𝑒
1
, 𝑒
2
] = [𝑒
2
, 𝑒
2
] = 𝑒
1
𝐷𝑒𝑟(𝐿
3
)
𝑄𝐷𝑒𝑟(𝐿
3
)
𝐺𝐷𝑒𝑟(𝐿
3
)
𝐶(𝐿
3
)
𝑄𝐶(𝐿
3
)
(
𝑑
11
0
𝑑
11
0
)
(
𝑑
21
+ 𝑑
22
0
𝑑
21
𝑑
22
)
(
𝑑
11
𝑑
12
𝑑
21
𝑑
22
)
(
𝑎
11
0
0
𝑎
11
)
(
𝑎
11
0
0
𝑎
11
)
QUASI-DERIVATIONS OF THREE-DIMENSIONAL NILPOTENT
LEIBNIZ ALGEBRAS:
We are given the following three-dimensional nilpotent Leibniz algebras:
𝜆
1
: 𝑎𝑏𝑒𝑙𝑖𝑎𝑛;
𝜆
2
: [𝑒
1
, 𝑒
1
] = 𝑒
2
;
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𝜆
3
: [𝑒
2
, 𝑒
3
] = 𝑒
1
, [𝑒
3
, 𝑒
2
] = −𝑒
1
;
𝜆
4
: [𝑒
2
, 𝑒
1
] = 𝑒
3
, [𝑒
1
, 𝑒
2
] = 𝛼𝑒
3
, 𝛼 ≠ 𝛼
−1
(𝛼 ∈ 𝐶);
𝜆
5
: [𝑒
1
, 𝑒
1
] = 𝑒
3
, [𝑒
2
, 𝑒
1
] = 𝑒
3
, [𝑒
1
, 𝑒
2
] = −𝑒
3
𝜆
6
: [𝑒
1
, 𝑒
1
] = 𝑒
2
, [𝑒
2
, 𝑒
1
] = 𝑒
3
.
We find a set of quasi-derivations for these algebras.
Proposition 2.
The general representation of the matrix of the space of all
derivations of
𝜆
1
: 𝑎𝑏𝑒𝑙𝑖𝑎𝑛
algebra is as follows:
(
𝛼
1
𝛼
2
𝛼
3
𝛽
1
𝛽
2
𝛽
3
𝛾
1
𝛾
2
𝛾
3
)
Theorem 5.
The general representation of the matrix of the space of all quasi-
derivations of
𝜆
2
: [𝑒
1
, 𝑒
1
] = 𝑒
2
algebra is as follows:
𝑄𝐷𝑒𝑟(𝜆
2
) = (
𝛼
1
𝛼
2
𝛼
3
0
𝛽
2
𝛽
3
0
𝛾
2
𝛾
3
)
We present the remaining derivations of the
𝜆
2
algebra without proof in the
following table:
𝜆
2
: [𝑒
1
, 𝑒
1
] = 𝑒
2
𝐷𝑒𝑟(𝜆
2
)
𝐺𝐷𝑒𝑟(𝜆
2
)
𝐶(𝜆
2
)
𝑄𝐶(𝜆
2
)
(
𝛼
1
𝛼
2
𝛼
3
0
2𝛼
1
0
0
𝛾
2
𝛾
3
)
(
𝛼
1
𝛼
2
𝛼
3
0
𝛽
2
𝛽
3
0
𝛾
2
𝛾
3
)
(
𝛼
1
𝛼
2
𝛼
3
0
𝛼
1
0
0
𝛾
2
𝛾
3
)
(
𝛼
1
𝛼
2
𝛼
3
0
𝛽
2
𝛽
3
0
𝛾
2
𝛾
3
)
Theorem 6.
The general representation of the matrix of the space of all quasi-
derivations of
𝜆
3
: [𝑒
2
, 𝑒
3
] = 𝑒
1
, [𝑒
3
, 𝑒
2
] = −𝑒
1
algebra is as follows:
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𝑄𝐷𝑒𝑟(𝜆
3
) = (
𝛼
1
0
0
𝛽
1
𝛽
2
𝛽
3
𝛾
1
𝛾
2
𝛾
3
)
𝜆
3
: [𝑒
2
, 𝑒
3
] = 𝑒
1
, [𝑒
3
, 𝑒
2
] = −𝑒
1
𝐷𝑒𝑟(𝜆
3
)
𝐺𝐷𝑒𝑟(𝜆
3
)
𝐶(𝜆
3
)
𝑄𝐶(𝜆
3
)
(
𝛽
2
+ 𝛾
3
0
0
𝛽
1
𝛽
2
𝛽
3
𝛾
1
𝛾
2
𝛾
3
)
(
𝛼
1
0
0
𝛽
1
𝛽
2
𝛽
3
𝛾
1
𝛾
2
𝛾
3
)
(
𝛼
1
0
0
𝛽
1
𝛼
1
0
𝛾
1
0
𝛼
1
)
(
𝛼
1
0
0
𝛽
1
𝛽
2
0
𝛾
1
0
𝛾
3
)
Theorem 7.
The general representation of the matrix of the space of all quasi-
derivations of
𝜆
4
: [𝑒
2
, 𝑒
1
] = 𝑒
3
, [𝑒
1
, 𝑒
2
] = 𝛼𝑒
3
, 𝛼 ≠ 𝛼
−1
, (𝛼 ∈ 𝐶)
algebra is as
follows:
𝑄𝐷𝑒𝑟(𝜆
4
) = (
𝛼
1
0
𝛼
3
0
𝛽
2
𝛽
3
0
0
𝛾
3
)
𝜆
4
: [𝑒
2
, 𝑒
1
] = 𝑒
3
, [𝑒
1
, 𝑒
2
] = 𝛼𝑒
3
, 𝛼 ≠ 𝛼
−1
, (𝛼 ∈ 𝐶)
𝐷𝑒𝑟(𝜆
4
)
𝐺𝐷𝑒𝑟(𝜆
4
)
𝐶(𝜆
4
)
𝑄𝐶(𝜆
4
)
(
𝛼
1
0
𝛼
3
0
𝛽
2
𝛽
3
0
0
𝛼
1
+ 𝛽
2
)
(
𝛼
1
𝛼
2
𝛼
3
𝛽
1
𝛽
2
𝛽
3
0
0
𝛾
3
)
(
𝛼
1
0
𝛼
3
0
𝛼
1
𝛽
3
0
0
𝛼
1
)
(
𝛼
1
0
𝛼
3
0
𝛽
2
𝛽
3
0
0
𝛾
3
)
Theorem 8.
The general representation of the matrix of the space of all quasi-
derivations of
𝜆
5
: [𝑒
1
, 𝑒
1
] = 𝑒
3
, [𝑒
2
, 𝑒
1
] = 𝑒
3
, [𝑒
1
, 𝑒
2
] = −𝑒
3
algebra is as follows:
𝑄𝐷𝑒𝑟(𝜆
5
) = (
𝛼
1
𝛼
2
𝛼
3
0
𝛼
1
𝛽
3
0
0
𝛾
3
)
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𝜆
5
: [𝑒
1
, 𝑒
1
] = 𝑒
3
, [𝑒
2
, 𝑒
1
] = 𝑒
3
, [𝑒
1
, 𝑒
2
] = −𝑒
3
𝐷𝑒𝑟(𝜆
5
)
𝐺𝐷𝑒𝑟(𝜆
5
)
𝐶(𝜆
5
)
𝑄𝐶(𝜆
5
)
(
𝛼
1
𝛼
2
𝛼
3
0
𝛼
1
𝛽
3
0
0
2𝛼
1
)
(
𝛼
1
𝛼
2
𝛼
3
0
𝛽
2
𝛽
3
0
0
𝛾
3
)
(
𝛼
1
0
𝛼
3
0
𝛼
1
𝛽
3
0
0
𝛼
1
)
(
𝛼
1
0
𝛼
3
0
𝛼
1
𝛽
3
0
0
𝛾
3
)
Theorem 9.
The general representation of the matrix of the space of all quasi-
derivations of
𝜆
6
: [𝑒
1
, 𝑒
1
] = 𝑒
2
, [𝑒
2
, 𝑒
1
] = 𝑒
3
algebra is as follows:
𝑄𝐷𝑒𝑟(𝜆
6
) = (
𝛼
1
𝛼
2
𝛼
3
0
𝛽
2
𝛽
3
0
0
𝛾
3
)
𝜆
6
: [𝑒
1
, 𝑒
1
] = 𝑒
2
, [𝑒
2
, 𝑒
1
] = 𝑒
3
𝐷𝑒𝑟(𝜆
6
)
𝐺𝐷𝑒𝑟(𝜆
6
)
𝐶(𝜆
6
)
𝑄𝐶(𝜆
6
)
(
𝛼
1
𝛼
2
𝛼
3
0
2𝛼
1
𝛼
2
0
0
3𝛼
1
)
(
𝛼
1
𝛼
2
𝛼
3
𝛽
1
𝛽
2
𝛽
3
0
0
𝛾
3
)
(
𝛼
1
0
𝛼
3
0
𝛼
1
0
0
0
𝛼
1
)
(
𝛼
1
0
𝛼
3
0
𝛼
1
𝛽
3
0
0
𝛾
3
)
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Abdurasulov K., Kaygorodov I., Khudoyberdiyev A.: The algebraic and geometric
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