On Leibniz algebras, whose subideals are ideals
We obtain a description of solvable Leibniz algebras, whose subideals are ideals. A description of certain types of Leibniz T-algebras is also obtained. In particular, it is established that the structure of Leibniz T-algebras essentially depends on the structure of its nil-radical. Отримано опис...
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Kurdachenko, L.A. Subbotin, I.Ya. Yashchuk, V.S. 2017-12-06T11:29:14Z 2017-12-06T11:29:14Z 2017 On Leibniz algebras, whose subideals are ideals / L.A. Kurdachenko, I.Ya. Subbotin, V.S. Yashchuk // Доповіді Національної академії наук України. — 2017. — № 9. — С. 15-19. — Бібліогр.: 12 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2017.09.015 https://nasplib.isofts.kiev.ua/handle/123456789/126919 512.542 We obtain a description of solvable Leibniz algebras, whose subideals are ideals. A description of certain types of Leibniz T-algebras is also obtained. In particular, it is established that the structure of Leibniz T-algebras essentially depends on the structure of its nil-radical. Отримано опис розв'язних алгебр Лейбніца, всі підідеали яких є ідеалами. Наведено теореми, що дають опис деяких типів T-алгебр Лейбніца. Зокрема, структура T-алгебр Лейбніца істотно залежить від структури її ніль-радикала. Получено описание разрешимых алгебр Лейбница, все подидеалы которых являются идеалами. Приведены теоремы, которые дают описание некоторых типов T-алгебр Лейбница. В частности, структура T-алгебр Лейбница существенно зависит от структуры ее ниль-радикала. en Видавничий дім "Академперіодика" НАН України Доповіді НАН України Математика On Leibniz algebras, whose subideals are ideals Про алгебри Лейбніца, кожен підідеал яких є ідеалом Об алгебрах Лейбница, каждый подидеал которых является идеалом Article published earlier |
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On Leibniz algebras, whose subideals are ideals |
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On Leibniz algebras, whose subideals are ideals Kurdachenko, L.A. Subbotin, I.Ya. Yashchuk, V.S. Математика |
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On Leibniz algebras, whose subideals are ideals |
| title_full |
On Leibniz algebras, whose subideals are ideals |
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On Leibniz algebras, whose subideals are ideals |
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On Leibniz algebras, whose subideals are ideals |
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on leibniz algebras, whose subideals are ideals |
| author |
Kurdachenko, L.A. Subbotin, I.Ya. Yashchuk, V.S. |
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Kurdachenko, L.A. Subbotin, I.Ya. Yashchuk, V.S. |
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Математика |
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Математика |
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2017 |
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English |
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Доповіді НАН України |
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Видавничий дім "Академперіодика" НАН України |
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Про алгебри Лейбніца, кожен підідеал яких є ідеалом Об алгебрах Лейбница, каждый подидеал которых является идеалом |
| description |
We obtain a description of solvable Leibniz algebras, whose subideals are ideals. A description of certain types of
Leibniz T-algebras is also obtained. In particular, it is established that the structure of Leibniz T-algebras essentially
depends on the structure of its nil-radical.
Отримано опис розв'язних алгебр Лейбніца, всі підідеали яких є ідеалами. Наведено теореми, що дають
опис деяких типів T-алгебр Лейбніца. Зокрема, структура T-алгебр Лейбніца істотно залежить від структури її ніль-радикала.
Получено описание разрешимых алгебр Лейбница, все подидеалы которых являются идеалами. Приведены
теоремы, которые дают описание некоторых типов T-алгебр Лейбница. В частности, структура T-алгебр
Лейбница существенно зависит от структуры ее ниль-радикала.
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| issn |
1025-6415 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/126919 |
| citation_txt |
On Leibniz algebras, whose subideals are ideals / L.A. Kurdachenko, I.Ya. Subbotin, V.S. Yashchuk // Доповіді Національної академії наук України. — 2017. — № 9. — С. 15-19. — Бібліогр.: 12 назв. — англ. |
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2025-11-26T01:39:33Z |
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| fulltext |
15ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2017. № 9
© L.A. Kurdachenko, I.Ya. Subbotin, V.S. Yashchuk , 2017
An algebra L over a field F is said to be a Leibniz algebra (more precisely, a left Leibniz algebra)
if it satisfies the Leibniz identity
[[a, b], c] = [a, [b, c]] – [b, [a, c]] for all a, b, c ∈ L. (LI)
Leibniz algebras are generalizations of Lie algebras. Indeed, a Leibniz algebra L is a Lie al geb-
ra if and only if [a, a] = 0 for every element a ∈ L. For this reason, we may consider Leibniz al-
gebras as ”non-anticommutative” analogs of Lie algebras.
Leibniz algebra appeared first in the papers by A.M. Bloh ([1], [2], [3]), in which he call ed
them D-algebras. However, in that time these works were not in demand, and they have not been
properly developed. Real interest in Leibniz algebras arose only after two decades thanks to the
work by J.L. Loday [4], who “rediscovered” these algebras and used the term Leibniz algebras
since it was Leibniz who discovered and proved the ”Leibniz rule” for the differentiation of func-
tions. The Leibniz algebras appeared to be naturally related to several topics such as differen tial
geometry, homological algebra, classical algebraic topology, algebraic K-theory, loop spaces, non-
commutative geometry, and so on. They found some applications in physics (see, e. g. ([5—7]).
The theory of Leibniz algebras has been developing quite intensively but very uneven. Howe ver,
there are problems natural for any algebraic structure that were not previously considered for
Leibniz algebras. They are the topics concerned with the relationship of subalgebras, ideals and
subideals. It should be noted, for example, that a natural question about the structure of Leibniz
doi: https://doi.org/10.15407/dopovidi2017.09.015
УДК 512.542
L.A. Kurdachenko 1, I.Ya. Subbotin 2, V.S. Yashchuk 1
1 Oles Honchar Dnipro National University
2 National University, Los Angeles, USA
E-mail: lkurdachenko@i.ua, isubboti@nu.edu, ViktoriiaYashchuk@mail.ua
On Leibniz algebras, whose subideals are ideals
Presented by Corresponding Member of the NAS of Ukraine V.P. Motornyi
We obtain a description of solvable Leibniz algebras, whose subideals are ideals. A description of certain types of
Leibniz T-algebras is also obtained. In particular, it is established that the structure of Leibniz T-algebras essentially
depends on the structure of its nil-radical.
Keywords: Leibniz algebra, ideal, subideal, T-algebra.
16 ISSN 1025-6415. Dopov. Nac. acad. nauk Ukr. 2017. № 9
L.A. Kurdachenko, I.Ya. Subbotin, V.S. Yashchuk
algebras, whose subalgebras are ideals, was only recently considered in work [8]. If, in the case of
Lie algebras, the structure of similar algebras is very simple (they are Abelian), the picture in the
Leibniz algebras is more sophisticated and interesting. We have analogous situation for cyclic
subalgebras: if, in Lie algebras, every cyclic subalgebra has dimension 1, in Leibniz algebras, a cy-
clic subalgebra can be of very complicated structure [9].
Let L be a Leibniz algebra over a field F. If A, B are subspaces of L, then [A, B] will denote a
subspace generated by all elements [a, b], where a ∈ A, b ∈ B. As usual, a subspace A of L is called
a subalgebra of L if [x, y] ∈ A for every x, y ∈ A. It follows that [A, A] � A.
Let M be a non-empty subset of L. Then 〈M〉 denotes the subalgebra of L generated by M.
A subalgebra A is called a left (respectively, right) ideal of L if [y, x] ∈ A (respectively,
[x, y] ∈ A) for every x ∈ A, y ∈ L. In other words, if A is a left (respectively right) ideal, then
[L, A] � A (respectively [A, L] � A).
A subalgebra A of L is called an ideal of L (more precisely, two-sided ideal) if it is both a
left ideal and a right ideal, i. e., [x, y], [y, x] ∈ A for every x ∈ A, y ∈ L.
If A is an ideal of L, we can consider a factor-algebra L/A. It is not hard to see that this fac-
tor-algebra also is a Leibniz algebra.
Note that the relation “to be a subalgebra of a Leibniz algebra” is transitive. However, for
Lie algebras, the relation “to be an ideal” is not transitive.
Therefore, it is natural to ask the question about the structure of Leibniz algebras, in which
the relation “to be an ideal” is transitive.
In this context, the following important type of subalgebras naturally arises. A subalgebra A
is called a left (respectively, right) subideal of L if there is a finite series of subalgebras
A = A0 � A1 � . . . � An = L
such that Aj – 1 is a left (respectively, right) ideal of Aj, 1 � j � n.
Similarly, a subalgebra A is called a subideal of L if there is a finite series of subalgebras
A = A0 � A1 � . . . � An = L
such that Aj – 1 is an ideal of Aj, 1 � j � n.
The natural question on Leibniz algebras, in which the relation “to be an ideal” is transi-
tive, arises.
A Leibniz algebra L is called a T-algebra if a relation “to be an ideal” is transitive. In other
words, if A is an ideal of L and B is an ideal of A, then B is an ideal of L. It follows that, in a Leibniz
T-algebra, every subideal is an ideal.
Lie T-algebras have been studied by I. Stewart [10] and A.G. Gein and Yu.N. Muhin [11].
In particular, soluble T-algebras and finite dimensional T-algebras over a field of characteristic
0 were described.
In this paper, we will consider some generalized soluble Leibniz T-algebras. Let L be a Leib-
niz algebra. We define the lower central series of L
L = γ1(L) � γ2(L) � . . . � γα(L) � γα + 1(L) � . . . �γδ(L)
by the following rule: γ1(L) = L, γ2(L) = [L, L], and, recursively, γα + 1(L) = [L, γα(L)] for all or di-
nals α, γλ(L) = ∩ μ < λ γμ(L) for the limit ordinals λ. It is possible to show that every term of
17ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2017. № 9
On Leibniz algebras, whose subideals are ideals
this series is an ideal of L. The last term γδ(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] … ] is a left normed product of k co-
pies of L.
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 a nilpotent of the nilpotency class c if γc + 1(L) = 〈0〉, but γc(L) ≠ 〈0〉.
We denote, by ncl (L), the nilpotency class of L.
As usual, a Leibniz Algebra L is called Abelian if [x, y] = 0 for all elements x, y ∈ L. In an
Abelian Leibniz algebra, every subspace is a subalgebra and an ideal.
The center ζ(L) of a Leibniz algebra L is defined in the following way:
ζ(L) = {x ∈ L | [x, y] = 0 = [y, x] for each element y ∈ L}.
Clearly, ζ(L) is an ideal of L.
Let L be a Leibniz algebra. The subalgebra Nil(L) generated by all nilpotent ideals of L is
called the nil radical of L. Clearly Nil(L) is an ideal of L. If L = Nil(L), then L is called a Leib-
niz nil-algebra. Every nilpotent Leibniz algebra is a nil-algebra, but converse is not true even for a
Lie algebra. Note also that if L is a finite-dimensional nil-algebra, then L is nilpotent.
The subalgebra Ba(L) generated by all nilpotent subideal of L is called the Baer radical of L.
It is possible to show that Ba(L) is an ideal of L and Nil(L) � Ba(L). If L = Ba(L), then L is call ed
a Leibniz Baer algebra. Every nil-algebra is a Baer algebra, but converse is not true even for
a Lie algebra (see, e. g., [12]). Note also that if L is a finite-dimensional Baer algebra, then L is
nilpotent.
As in the cases mentioned above, the situation for the Leibniz algebra is much more complex
and diverse, than it was for Lie algebras. Here are few simple examples illustrating this point. Let
F be an arbitrary field, and L be a vector space over F with a basis {a, c}. Define the operation [ , ]
on L by the following rule: [a, a] = c, [c, a] = [a, c] = [c, c] = 0. Then L is a cyclic Leibniz algebra,
and Fc is its unique non-zero subalgebra. Moreover, Fc is the center of L, in particular, Fc is an
ideal of L. Thus, every subalgebra of L is an ideal.
Let now F = F2 and let L be a Leibniz algebra constructed above. Put A = L ⊕ Fv and let
[v, v] = [v, c] = [c, v] = 0, [v, a] = [a, v] = a. It is not hard to check that A is a Leibniz algebra and
L is an ideal of A. Moreover, if B is a non-zero ideal of A and L does not include B, then B = A.
As we have seen above, Fc is a unique non-zero ideal of L. But Fc = ζ(L), thus, Fc is an ideal of A.
Thus, A is a Leibniz T-algebra.
Let again F = F2 and D = L ⊕ Fu. Put now [u, u] = [u, c] = [c, u] = 0, [u, a] = [a, u] = a + c. It is
not hard to check that D is a Leibniz algebra and L is an ideal of A. As above, we can check that
D is a Leibniz T-algebra.
As we will see further, these examples are typical in some sense.
A Leibniz algebra L is called an extraspecial if it satisfies the following condition:
• ζ(L) is non-trivial and has dimension 1;
• L/ζ(L) is Abelian.
Theorem A. Let L be a Leibniz T-algebra over a field F. If L is a Baer algebra, then either L is
Abelian or L = E ⊕ Z, where Z � ζ(L) and E is an extraspecial subalgebra such that [a, a] ≠ 0 for
every element a ∉ ζ(E).
18 ISSN 1025-6415. Dopov. Nac. acad. nauk Ukr. 2017. № 9
L.A. Kurdachenko, I.Ya. Subbotin, V.S. Yashchuk
A Leibniz algebra L is called hyper-Abelian if it has an ascending series
〈0〉 = L0 ≤ L1 � . . . � Lα � Lα + 1 � . . . �Lγ = L
of ideals, whose factors Lα + 1/Lα are Abelian for all α < γ. If this series is finite, then L is called a
soluble Leibniz algebra.
The structure of Leibniz T-algebras essentially depends of the structure of its nil-radical.
Theorem B. Let L be a hyper-Abelian Leibniz T-algebra over a field F. If L is non-nilpotent
and Nil(L) = D is Abelian, then L = D ⊕ V, where V = Fv, [v, v] = 0, [v, d] = d = –[d, v] for every ele-
ment d ∈ Nil(L). In particular, L is a Lie algebra.
Theorem C. Let L be a hyper-Abelian Leibniz T-algebra over a field F. If char(F) ≠ 2, then
Nil(L) is Abelian.
We say that a field F is 2-closed, if the equation x2 = a has a solution in F for every element a ≠
0. We note that every locally finite (in particular, finite) field of characteristic 2 is 2-closed.
Theorem D. Let L be a hyper-Abelian Leibniz T-algebra over a field F. Suppose that L is non-
nilpotent and Nil(L) is non-Abelian. If a field F is 2-closed and char(F) = 2, then L = (Fe ⊕ Fc) ⊕ Fv,
where
[e, e] = c, [c, e] = [e, c] = [c, v] = [v, c] = 0,
[v, v] = 0, [v, e] = e + γc = [e, v], γ ∈ F.
REFERENCES
1. Bloh, A. M. (1965). On a generalization of the concept of Lie algebra. Dokl. AN SSSR, 165, pp. 471-473.
2. Bloh, A. M. (1967). Cartan — Eilenberg homology theory for a generalized class of Lie algebras. Dokl. AN
SSSR, 175, pp. 824-826.
3. Bloh, A. M. (1971). A certain generalization of the concept of Lie algebra. Algebra and number theory.
Uchenye Zapiski Moskov. Gos. Pedagog. Inst., 375, pp. 9-20 (in Russian).
4. Loday, J. L. (1993). Une version non commutative des algèbres de Lie: les algèbres de Leibniz. Enseign. Math.,
39, pp. 269-293.
5. Butterfield, J. & Pagonis, C. (1999). From Physics to Philosophy. Cambridge: Cambridge Univ. Press.
6. Dobrev, V. (Eds.) (2013). Lie theory and its applications in physics. IX International workshop. Tokyo:
Springer.
7. Duplij, S. & Wess, J. (Ed.). (2001). Noncommutative Structures in Mathematics and Physics. NATO Science
Series II. Vol. 22. Dordrecht: Kluwer.
8. Kurdachenko, L. A., Semko, N. N. & Subbotin, I. Ya. (2017). The Leibniz algebras whose subalgebras are ide-
als. Open Math., 15, pp. 92-100.
9. Chupordya, V. A., Kurdachenko, L. A. & Subbotin, I. Ya. (2016). On some minimal Leibniz algebras. J. Algebra
and Appl., 16, Iss. 5, 1750082, 16 p.
10. Stewart, I. N. (1969). Subideals of Lie algebras. Ph.D. Thesis, University of Warwick.
11. Gein, A. G. & Muhin, Yu. N. (1980). Complements to subalgebras of Lie algebras. Mat. Zapiski Ural. Gos.
Univ., 12, pp. 24-48 (in Russian).
12. Amayo, R. K. & Stewart, I. (1974). Infinite-dimensional Lie algebras. Dordrecht: Springer.
Received 11.05.2017
19ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2017. № 9
On Leibniz algebras, whose subideals are ideals
Л.А. Курдаченко 1, І.Я. Субботін 2, В.С. Ящук 1
1 Дніпровський національний університет ім. Олеся Гончара
2 Національний університет, Лос-Анджелес, США
E-mail: lkurdachenko@i.ua, isubboti@nu.edu, ViktoriiaYashchuk@mail.ua
ПРО АЛГЕБРИ ЛЕЙБНІЦА,
КОЖЕН ПІДІДЕАЛ ЯКИХ Є ІДЕАЛОМ
Отримано опис розв’язних алгебр Лейбніца, всі підідеали яких є ідеалами. Наведено теореми, що дають
опис деяких типів T-алгебр Лейбніца. Зокрема, структура T-алгебр Лейбніца істотно залежить від струк-
тури її ніль-радикала.
Ключові слова: алгебра Лейбніца, ідеал, підідеал, Т-алгебра.
Л.А. Курдаченко 1, И.Я. Субботин 2, В.С. Ящук 1
1 Днепровский национальный университет им. Олеся Гончара
2 Национальный университет, Лос-Анджелес, США
E-mail: lkurdachenko@i.ua, isubboti@nu.edu, ViktoriiaYashchuk@mail.ua
ОБ АЛГЕБРАХ ЛЕЙБНИЦА,
КАЖДЫЙ ПОДИДЕАЛ КОТОРЫХ ЯВЛЯЕТСЯ ИДЕАЛОМ
Получено описание разрешимых алгебр Лейбница, все подидеалы которых являются идеалами. Приведены
теоремы, которые дают описание некоторых типов T-алгебр Лейбница. В частности, структура T-алгебр
Лейбница существенно зависит от структуры ее ниль-радикала.
Ключевые слова: алгебра Лейбница, идеал, подидеал, Т-алгебра.
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