On the derivations of Leibniz algebras of low dimension
Let L be an algebra over a field F. Then L is called a left Leibniz algebra if its multiplication operations [⋅, ⋅] additionally satisfy the so-called left Leibniz identity: [[a,b],c] = [a,[b,c]] – [b,[a,c]] for all elements a, b, c ∈ L. In this paper, we begin the description of the algebra of de...
Gespeichert in:
| Datum: | 2023 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Видавничий дім "Академперіодика" НАН України
2023
|
| Schriftenreihe: | Доповіді НАН України |
| Schlagworte: | |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/192998 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | On the derivations of Leibniz algebras of low dimension / L.A. Kurdachenko, M.M. Semko, V.S. Yashchuk // Доповіді Національної академії наук України. — 2023. — № 2. — С. 18-23. — Бібліогр.: 15 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-192998 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1929982023-07-30T16:20:10Z On the derivations of Leibniz algebras of low dimension Kurdachenko, L.A. Semko, M.M. Yashchuk, V.S. Математика Let L be an algebra over a field F. Then L is called a left Leibniz algebra if its multiplication operations [⋅, ⋅] additionally satisfy the so-called left Leibniz identity: [[a,b],c] = [a,[b,c]] – [b,[a,c]] for all elements a, b, c ∈ L. In this paper, we begin the description of the algebra of derivations of Leibniz algebras having dimension 3. It is clear that the description of the algebra of derivations of all Leibniz algebras, having dimension 3, is quite large. Therefore, in this article, we will focus on the description of the nilpotent Leibniz algebra, whose nilpotency class is 3, and the nilpotent Leibniz algebra, whose center has dimension 2. Нехай L — це алгебра над полем F. Тоді L називається лівою алгеброю Лейбніца, якщо її операції множення [⋅, ⋅] задовольняють так звану ліву тотожність Лейбніца: [[a, b], c] = [a, [b, c]] – [b, [a, c]] для всіх елементів a, b, c ∈ L. У статті започатковано опис алгебри похідних алгебр Лейбніца, що мають вимірність 3. Зрозуміло, що опис алгебри похідних всіх алгебр Лейбніца вимірності 3 є досить великим. Тому тут наведено опис нільпотентних алгебр Лейбніца, клас нільпотентності яких дорівнює 3, та нільпотентних алгебр Лейбніца, центр яких має розмірність 2. 2023 Article On the derivations of Leibniz algebras of low dimension / L.A. Kurdachenko, M.M. Semko, V.S. Yashchuk // Доповіді Національної академії наук України. — 2023. — № 2. — С. 18-23. — Бібліогр.: 15 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2023.02.018 http://dspace.nbuv.gov.ua/handle/123456789/192998 512.542 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Математика Математика |
| spellingShingle |
Математика Математика Kurdachenko, L.A. Semko, M.M. Yashchuk, V.S. On the derivations of Leibniz algebras of low dimension Доповіді НАН України |
| description |
Let L be an algebra over a field F. Then L is called a left Leibniz algebra if its multiplication operations [⋅, ⋅] additionally
satisfy the so-called left Leibniz identity: [[a,b],c] = [a,[b,c]] – [b,[a,c]] for all elements a, b, c ∈ L. In this paper,
we begin the description of the algebra of derivations of Leibniz algebras having dimension 3. It is clear that the
description of the algebra of derivations of all Leibniz algebras, having dimension 3, is quite large. Therefore, in this
article, we will focus on the description of the nilpotent Leibniz algebra, whose nilpotency class is 3, and the nilpotent
Leibniz algebra, whose center has dimension 2. |
| format |
Article |
| author |
Kurdachenko, L.A. Semko, M.M. Yashchuk, V.S. |
| author_facet |
Kurdachenko, L.A. Semko, M.M. Yashchuk, V.S. |
| author_sort |
Kurdachenko, L.A. |
| title |
On the derivations of Leibniz algebras of low dimension |
| title_short |
On the derivations of Leibniz algebras of low dimension |
| title_full |
On the derivations of Leibniz algebras of low dimension |
| title_fullStr |
On the derivations of Leibniz algebras of low dimension |
| title_full_unstemmed |
On the derivations of Leibniz algebras of low dimension |
| title_sort |
on the derivations of leibniz algebras of low dimension |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| publishDate |
2023 |
| topic_facet |
Математика |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/192998 |
| citation_txt |
On the derivations of Leibniz algebras of low dimension / L.A. Kurdachenko, M.M. Semko, V.S. Yashchuk // Доповіді Національної академії наук України. — 2023. — № 2. — С. 18-23. — Бібліогр.: 15 назв. — англ. |
| series |
Доповіді НАН України |
| work_keys_str_mv |
AT kurdachenkola onthederivationsofleibnizalgebrasoflowdimension AT semkomm onthederivationsofleibnizalgebrasoflowdimension AT yashchukvs onthederivationsofleibnizalgebrasoflowdimension |
| first_indexed |
2025-07-16T18:54:24Z |
| last_indexed |
2025-07-16T18:54:24Z |
| _version_ |
1837830837166407680 |
| fulltext |
18 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2023. No 2: 18—23
C i t a t i o n: Kurdachenko L.A., Semko M.M., Yashchuk V.S. On the derivations of Leibniz algebras of low
dimension. Dopov. Nac. akad. nauk Ukr. 2023. No 2. P. 18—23. https://doi.org/10.15407/dopovidi2023.02.018
© Видавець ВД «Академперіодика» НАН України, 2023. Стаття опублікована за умовами відкритого до-
ступу за ліцензією CC BY-NC-ND (https://creativecommons.org/licenses/by-nc-nd/4.0/)
МАТЕМАТИКА
MATHEMATICS
https://doi.org/10.15407/dopovidi2023.02.018
UDC 512.542
L.A. Kurdachenko1
, https://orcid.org/0000-0002-6368-7319
M.M. Semko2
, https://orcid.org/0000-0003-0123-4872
V.S. Yashchuk1
, https://orcid.org/0000-0001-6211-6410
1 Oles Honchar Dnipro National University, Dnipro
2 State Tax University, Irpin
E-mail: lkurdachenko@i.ua, dr.mykola.semko@gmail.com, viktoriia.s.yashchuk@gmail.com
On the derivations of Leibniz algebras of low dimension
Presented by Corresponding Member of the NAS of Ukraine V.P. Motornyi
Let L be an algebra over a field F. Then L is called a left Leibniz algebra if its multiplication operations [⋅, ⋅] addition-
ally satisfy the so-called left Leibniz identity: [[a,b],c] = [a,[b,c]] – [b,[a,c]] for all elements a, b, c ∈ L. In this paper,
we begin the description of the algebra of derivations of Leibniz algebras having dimension 3. It is clear that the
description of the algebra of derivations of all Leibniz algebras, having dimension 3, is quite large. Therefore, in this
article, we will focus on the description of the nilpotent Leibniz algebra, whose nilpotency class is 3, and the nilpotent
Leibniz algebra, whose center has dimension 2.
Keywords: dimension, derivation, hypercenter, Leibniz algebra, nilpotent Leibniz algebra
Let L be an algebra over a field F with the binary operations + and [⋅, ⋅]. Then L is called a left
Leibniz algebra if it satisfies the left Leibniz identity
[[a, b], c] = [a, [b, c]] – [b, [a, c]]
for all a, b, c ∈ L. We will also use another form of this identity:
[a, [b, c]] = [[a, b], c] + [b, [a, c]].
Leibniz algebras appeared first in the paper of A. Blokh [1], but the term “Leibniz algebra”
appears in the book of J.-L. Loday [2], and the article of J.-L. Loday [3]. In [4] J.-L. Loday and
T. Pirashvili began the real study of the properties of Leibniz algebras. The theory of Leibniz alge-
bras has developed very intensively in many different directions. Some of the results of this theory
have been presented in the book [5].
Same as in Lie algebras, the structure of Leibniz algebras is greatly influenced by their alge-
bras of derivations.
19ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2023. № 2
On the derivations of Leibniz algebras of low dimension
Denote by EndF(L) the set of all linear transformations of L, then L is an associative algebra
by the operation + and ∘. As usual, EndF(L) is a Lie algebra by the operations + and [ , ], where
[f, g] = f ∘ g – g ∘ f for all f, g ∈ EndF(L).
A linear transformation f of a Leibniz algebra L is called a derivation, if
f ([a, b]) = [f (a), b] + [a, f (b)] for all a, b ∈ L.
Let Der(L) be the subset of all derivations of L. It is possible to prove that Der(L) is a subal-
gebra of a Lie algebra EndF(L). Der(L) is called the algebra of derivations of a Leibniz algebra L.
The influence on the structure of the Leibniz algebra of their algebras of derivations can be
seen from the following result: if A is an ideal of a Leibniz algebra, then the factor-algebra of L by
the annihilator of A is isomorphic to some subalgebra of Der(A) [6, Proposition 3.2].
It is natural to start studying the algebra of derivations of Leibniz algebras, the structure of
which has been studied quite extensively. A description of the structure of algebras of derivations
of finite-dimensional cyclic Leibniz algebras was obtained in papers [7—9]. The question natural-
ly arises about an algebra of derivations of Leibniz algebras, having a small dimension. In contrast
to Lie algebras, the situation with Leibniz algebras of dimension 3 is very diverse. Leibniz algebras
of dimension 3 are mostly described, and the description of Leibniz algebras of dimensions 4, and
5 are carried out quite intensively. Here we only note that the study of right Leibniz algebras of
dimension 3 is the subject of section 3.1 of a book [5] and works [10—14].
In this paper, we begin the description of the algebra of derivations of Leibniz algebras, having
dimension 3. It is clear that the description of the algebra of derivations of all Leibniz algebras,
having dimension 3, is quite large. Therefore, in this article, we will focus on the description of
nilpotent Leibniz algebra, whose nilpotency class is 3, and of nilpotent Leibniz algebra, whose
center has dimension 2.
1. Some preliminary results. Let’s start with some general properties of the algebra of deri-
vations of Leibniz algebras. We will show in this section some basic elementary properties of deri-
vations, which have been proved in a paper [7]. First of all, let’s recall some definitions.
Every Leibniz algebra L has one specific ideal. Denote by Leib(L) the subspace, generated by
the elements [a, a], a ∈ L. It is possible to prove that Leib(L) is an ideal of L. The ideal Leib(L) is
called the Leibniz kernel of algebra L. By its definition, a factor-algebra L/Leib(L) is a Lie algebra.
And conversely, if K is an ideal of L such that L/K is a Lie algebra, then K includes a Leibniz kernel.
Let L be a Leibniz algebra. Define the lower central series of L as
1 2 1( ) ( ) ( ) ( ) ( ) ( ). . . . . .L L L L L L Lα α+ δ ∞= γ γ γ γγ γ =
by the following rule: γ1(L) = L, γ2(L) = [L, L], and recursively γα+1(L) = [L, γα(L)] for all ordinals
α and γλ(L) = ∩μ < λγμ(L) for the limit ordinals λ. The last term γδ(L) = γ∞(L) is called the lower
hypocenter of L. We have γδ(L) = [L, γδ(L)].
If α = k is a positive integer, then γk(L) = [L, [L, [L, …, L] … L] is the left normed commutator
of k copies of L.
As usual, we say that a Leibniz algebra L is called nilpotent if there exists a positive integer k
such that γk(L) = ❬0❭. More precisely, L is said to be nilpotent of nilpotency class c if γc+1(L) = ❬0❭,
but γc(L) ≠ ❬0❭.
20 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2023. No 2
L.A. Kurdachenko, M.M. Semko, V.S. Yashchuk
The left (respectively right) center ζleft(L) (respectively ζright(L)) of a Leibniz algebra L is
defined by the rule:
ζ = ∈left ( ) { | [ , ]L x L x y = 0 for each element y ∈ L}
(respectively,
left ( ) { | [ ] ,L x L y xζ = ∈ = 0 for each element y ∈ L}).
It is not hard to prove that the left center of L is an ideal, but it is not true for the right center.
Moreover, left )( ) (Leib L z L , so that L/ζleft(L) is a Lie algebra. The right center is a subalgebra
of L, and, in general, the left and right centers are different; they even may have different dimen-
sions (see [6]).
The center ζ(L) of L is defined by the rule:
ζ = ∈ = ={ |( ) ,[ ] 0 [ , ]L x L x y y x for each element y ∈ L}.
The center is an ideal of L.
We define now the upper central series
0 1 2 1( ) ( ) ( ) ( ) ( ) ( ) ( )0 . . . . . .L L L L z L L Lα α+ γ ∞= ζ ζ ζ ζ ζ = ζ á ñ
of a Leibniz algebra L by the following rule: ζ1(L) = ζ(L) is the center of L, and recursively,
ζα + 1(L)/ζα(L) = ζ(L/ζα(L)) for all ordinals α, and ζλ(L) = ∪μ < λζμ(L) for the limit ordinals λ. By
definition, each term of this series is an ideal of L. The last term ζ∞(L) of this series is called the
upper hypercenter of L. If L = ζ∞(L) then L is called a hypercentral Leibniz algebra.
Lemma 1. Let L bea Leibniz algebra over a field F and f be a derivation of L. Then
left left( ( )) ( )f L Lζ ζ , right right( ( )) ( )f L Lζ ζ and ( ) ( )( )f L Lζ ζ .
Corollary. Let L bea Leibniz algebra over a field F and f be a derivation of L. Then
)( ( )) (f L Lα αζ ζ for every ordinal α.
Lemma 2. Let L bea Leibniz algebra over a field F and f be a derivation of L. Then
)( ( )) (f L Lα αγ γ for all ordinals α, in particular, )( ( )) (f L L∞ ∞γ γ .
It is natural to first give a description of the algebra of derivations of the Leibniz algebras of
dimension 2. The description of Leibniz algebra, having dimension 2, is given in several papers,
one of the first of which was [15]. The Leibniz algebras, having dimension 2, which are not Lie
algebras, are limited to the algebras of the following two types
Lei1(2, F) = Fa1 ⊕ Fa2 where [a1, a1] = a2, [a1, a2] = [a2, a1] = [a2, a2] = 0;
Lei2(2, F) = Fa1 ⊕ Fa2 where [a1, a1] = a2, [a1, a2] = a2, [a2, a1] = [a2, a2] = 0.
Let L be a Lie algebra. We say that L is a semidirect sum of an ideal A and a subalgebra B if
L = A + B and A ∩ B = ❬0❭.
Proposition 1. Let D be the algebra of derivations of the Leibniz algebra Lei1(2, F). Then D is
a semidirect sum of an ideal ofdimension 1 and a subalgebra of dimension 1. More precisely, D is
21ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2023. № 2
On the derivations of Leibniz algebras of low dimension
isomorphic to a subalgebra of matrices, having the following form
1
2 1
0
2
α⎛ ⎞
⎜ ⎟α α⎝ ⎠
, where α1, α2 ∈ F.
Proposition 2. Let D be the algebra of derivations of the Leibniz algebra Lei2(2, F). Then D is
abelian and has dimension 1, D = Ff where f (a1) = a2, f (a2) = a2.
2. Algebra of derivations of some Leibniz algebras, having dimension 3. Now, let’s move
on to the main part of our work, namely the consideration of the algebra of derivations of a Leib-
niz algebra with dimension 3. Naturally, we will only consider Leibniz algebras that are not Lie
algebras, which means their Leibniz kernel is not zero. The first type of Leibniz algebras we will
consider is the nilpotent Leibniz algebras, and specifically, the nilpotent Leibniz algebras of nilpo-
tency class 3. There is only one type of such algebra, which is the following Lei1(3, F):
Lei1(3, F) = Fa1 ⊕ Fa2 ⊕ Fa3 where [a1, a1] = a2, [a1, a2] = a3, [a1, a3] = 0,
[a2, a1] = [a3, a1] = [a2, a2] = [a2, a3] = [a3, a3] = 0.
It is cyclic Leibniz algebra,
Leib(Lei1(3, F)) = ζleft(Lei1(3, F)) = [Lei1(3, F), Lei1(3, F)] = Fa2 ⊕ Fa2,
ζright(Lei1(3, F)) = ζ(Lei1(3, F)) = γ3(Lei1(3, F)) = Fa3.
Theorem 1. Let D be the algebra of derivations of the Leibniz algebra Lei1(3, F). Then D is a
semidirect sum of an ideal N of dimension 1 and a subalgebra of dimension 1, generated by deriva-
tion f1 such that f1(a1) = a1, f1(a2) = 2a2, f1(a3) = 3a3. Furthermore, N is abelian, N = Ff2 ⊕ Ff3,
where f2(a1) = a2, f2(a2) = a3, f2(a3) = 0, f3(a1) = a3, f3(a2) = 0, f3(a3) = 0. An algebra D is iso-
morphic to a Lie subalgebra of matrices, having the following form
1
2 1
3 2 1
0 0
2 0
3
α⎛ ⎞
⎜ ⎟α α⎜ ⎟
⎜ ⎟α α α⎝ ⎠
, where α1, α2, α3 ∈ F.
Theorem 2. Let D be the algebra of derivations of the Leibniz algebra Lei2(3, F). Then D has a se-
ries of ideals 0 N C A D á ñ such that N is abelian, N = Ff3 ⊕ Ff4, C = N ⊕ Ff2, A = C ⊕ Ff1,
D = A ⊕ Ff0, where f0, f1, f2, f3, f4 are the derivation, defined by the rules:
f0(a1) = a1, f0(a2) = 2a2, f0(a3) = 0;
f1(a1) = 0, f1(a2) = 0, f1(a3) = a3;
f2(a1) = a3, f2(a2) = 0, f2(a3) = 0;
f3(a1) = a2, f3(a2) = 0, f3(a3) = 0;
f4(a1) = 0, f4(a2) = 0, f4(a3) = a2.
22 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2023. No 2
L.A. Kurdachenko, M.M. Semko, V.S. Yashchuk
Moreover,
f3 ∘ f4 = f4 ∘ f3, f3 ∘ f2 = f2 ∘ f3,
[f4, f2] = f3, [f1, f2] = f2,
f3 ∘ f1 = f1 ∘ f3, [f4, f1] = f4,
f0 ∘ f1 = f1 ∘ f0,
[f2, f0] = f2, [f0, f3] = f3, [f0, f4] = 2f4.
REFERENCES
1. Blokh, A. M. (1965). A generalization of the concept of a Lie algebra. Dokl. Akad. Nauk SSSR, 165, No. 3,
pp. 471-473 (in Russian).
2. Loday, J.-L. (1998). Cyclic homology. Grundlehren der mathematischen Wissenschaften (Vol. 301). Berlin,
Heidelberg: Springer. https://doi.org/10.1007/978-3-662-11389-9
3. Loday, J.-L. (1993). Une version non commutative des algèbres de Lie: les algèbres de Leibniz. Enseign.
Math., 39, pp. 269-293.
4. Loday, J.-L. & Pirashvili, T. (1993). Universal enveloping algebras of Leibniz algebras and (co)homology.
Math. Ann., 296, No. 1, pp. 139-158. https://doi.org/10.1007/BF01445099
5. Ayupov, Sh., Omirov, B. & Rakhimov, I. (2020). Leibniz algebras: Structure and classification. Boca Raton,
London, New York: CRC Press, Taylor & Francis Group.
6. Kurdachenko, L. A., Otal, J. & Pypka, A. A. (2016). Relationships between factors of canonical central series
of Leibniz algebras. Eur. J. Math., 2, No. 2, pp. 565-577. https://doi.org/10.1007/s40879-016-0093-5
7. Kurdachenko, L. A., Subbotin, I. Ya. & Yashchuk, V. S. (2022). On the endomorphisms and derivations of
some Leibniz algebras. J. Algebra Its Appl. https://doi.org/10.1142/S0219498824500026
8. Semko, M. M., Skaskiv, L. V. & Yarovaya, O. A. (2022). On the derivations of cyclic Leibniz algebras. Car-
pathian Math. Publ., 14, No. 2, pp. 345-353. https://doi.org/10.15330/cmp.14.2.345-353
9. Kurdachenko, L. A., Semko, N. N. & Yashchuk, V. S. (2021). On the structure of the algebra of derivations of
cyclic Leibniz algebras. Algebra Discret. Math., 32, No. 2, pp. 241-252. https://doi.org/10.12958/adm1898
10. Casas, J. M., Insua, M. A., Ladra, M. & Ladra, S. (2012). An algorithm for the classification of 3-di men sio nal
complex Leibniz algebras. Linear Algebra Appl., 436, No. 9, pp. 3747-3756.
https://doi.org/10.1016/j.laa.2011.11.039
11. Demir, I., Misra, K. C. & Stitzinger, E. (2014). On some structures of Leibniz algebras. In Recent advances in
representation theory, quantum groups, algebraic geometry, and related topics. Contemporary Mathematics
(Vol. 623) (pp. 41-54). Providence: American Mathematical Society. https://doi.org/10.1090/conm/623/12456
12. Khudoyberdiyev, A. Kh., Kurbanbaev, T. K. & Omirov, B. A. (2010). Classification of three-dimensional solv-
able p-adic Leibniz algebras. p-Adic Num. Ultrametr. Anal. Appl., 2, No. 3, pp. 207-221.
https://doi.org/10.1134/S2070046610030039
13. Rakhimov, I. S., Rikhsiboev, I. M. & Mohammed, M. A. (2018). An algorithm for a classification of three-di-
mensional Leibniz algebras over arbitrary fields. JP J. Algebra, Number Theory Appl., 40, No. 2, pp. 181-198.
https://doi.org/10.17654/NT040020181
14. Yashchuk, V. S. (2019). On some Leibniz algebras, having small dimension. Algebra Discret. Math., 27, No. 2,
pp. 292-308.
15. Cuvier, C. (1994). Algèbres de Leibnitz: définitions, propriétés. Ann. Scient. Éc. Norm. Sup., 4e série, 27,
pp. 1-45. https://doi.org/10.24033/asens.1687
Received 10.02.2023
23ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2023. № 2
On the derivations of Leibniz algebras of low dimension
Л.А. Курдаченко1, https://orcid.org/0000-0002-6368-7319
М.М. Семко2, https://orcid.org/0000-0003-0123-4872
В.С. Ящук1, https://orcid.org/0000-0001-6211-6410
1 Дніпровський національний університет ім. Олеся Гончара
2 Державний податковий університет, Ірпінь
E-mail: lkurdachenko@i.ua, dr.mykola.semko@gmail.com, viktoriia.s.yashchuk@gmail.com
ПРО ПОХІДНІ АЛГЕБР ЛЕЙБНІЦА МАЛОЇ ВИМІРНОСТІ
Нехай L — це алгебра над полем F. Тоді L називається лівою алгеброю Лейбніца, якщо її операції множення
[⋅, ⋅] задовольняють так звану ліву тотожність Лейбніца: [[a, b], c] = [a, [b, c]] – [b, [a, c]] для всіх елементів
a, b, c ∈ L. У статті започатковано опис алгебри похідних алгебр Лейбніца, що мають вимірність 3. Зрозу-
міло, що опис алгебри похідних всіх алгебр Лейбніца вимірності 3 є досить великим. Тому тут наведено
опис нільпотентних алгебр Лейбніца, клас нільпотентності яких дорівнює 3, та нільпотентних алгебр
Лейбніца, центр яких має розмірність 2.
Ключові слова: вимірність, похідна, гіперцентр, алгебра Лейбніца, нільпотентна алгебра Лейбніца.
|