Leibniz algebras of dimension 3 over finite fields
The first thing in the study of all types of algebras is the description of algebras having small dimensions. Unlike the
 simpler cases of 1- and 2-dimensional Leibniz algebras, the structure of 3-dimensional Leibniz algebras is more complicated.
 We consider the structure of Leibniz...
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| citation_txt | Leibniz algebras of dimension 3 over finite fields / V.S. Yashchuk // Доповіді Національної академії наук України. — 2018. — № 7. — С. 20-25. — Бібліогр.: 12 назв. — англ. |
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| description | The first thing in the study of all types of algebras is the description of algebras having small dimensions. Unlike the
simpler cases of 1- and 2-dimensional Leibniz algebras, the structure of 3-dimensional Leibniz algebras is more complicated.
We consider the structure of Leibniz algebras of dimension 3 over a finite field. In some cases, the structure
of the algebra essentially depends on the characteristic of the field. In others, it depends on the solvability of specific
equations in the field, and so on.
Першим кроком у вивченні всіх типів алгебр є опис таких алгебр, які мають малі вимірності. На відміну від
більш простих випадків одно- і двовимірних алгебр Лейбніца, структури тривимірних алгебр Лейбніца
складніші. У роботі розглядається структура алгебр Лейбніца вимірності 3 над скінченним полем. У деяких випадках структура алгебр суттєво залежить від характеристики поля, в інших — від можливості
розв'язання конкретних рівнянь у полі і т. п.
Первым шагом в изучении всех типов алгебр является описание таких алгебр, которые имеют малые размерности. В отличие от более простых случаев одно- и двумерных алгебр Лейбница, структуры трехмерных алгебр Лейбница сложнее. В работе рассматривается структура алгебр Лейбница размерности 3 над
конечным полем. В некоторых случаях структура алгебры зависит от характеристики поля, в других — от
разрешимости конкретных уравнений в поле и т. п.
|
| first_indexed | 2025-12-07T17:55:07Z |
| format | Article |
| fulltext |
20 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2018. № 7
doi: https://doi.org/10.15407/dopovidi2018.07.020
UDC 512.542
V.S. Yashchuk
Oles Honchar Dnipro National University
E-mail: Viktoriia.Yashchuk@i.ua
Leibniz algebras of dimension 3 over finite fields
Presented by Corresponding Member of the NAS of Ukraine V.P. Motornyi
The first thing in the study of all types of algebras is the description of algebras having small dimensions. Unlike the
simpler cases of 1- and 2-dimensional Leibniz algebras, the structure of 3-dimensional Leibniz algebras is more com-
plicated. We consider the structure of Leibniz algebras of dimension 3 over a finite field. In some cases, the structure
of the algebra essentially depends on the characteristic of the field. In others, it depends on the solvability of specific
equations in the field, and so on.
Keywords: Leibniz algebra, ideal, factor-algebra, nilpotent Leibniz algebra.
Let L be an algebra over a field F with the binary operations + and [·,·]. Then L is called a Leibniz
algebra (more precisely, 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 papers of A.M. Bloh [1—3], in which he called them the
D-algebras. However, in that time, these works were not in demand, and they have not been prop-
erly developed. Only after two decades, a real interest in Leibniz algebras rose. It happened thanks
to the work of J.-L. Loday [4] (see also [5, Section 10.6]), who “rediscovered” these algebras and
used the term Leibniz algebras, since it was Gottfried Wilhelm Leibniz who discovered and proved
the Leibniz rule for the differentiation of functions.
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 L be a Leibniz algebra over a field F, and let M be a non-empty subset of L. Then 〈M〉 de-
notes the subalgebra of L generated by M.
A subalgebra A of L 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).
© V.S. Yashchuk, 2018
21ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2018. № 7
Leibniz algebras of dimension 3 over finite fields
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., [y, x], [x, y] ∈ 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 factor-
algebra also is a Leibniz algebra.
As usual, a Leibniz algebra L is called abelian, if [x, y] = 0 for all elements x, y ∈ L. In an Abe-
lian Leibniz algebra, every subspace is a subalgebra and an 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 such that L/Leib(L) is a Lie algebra. Conversely, if H is an ideal
of L such that L/H is a Lie algebra, then Leib(L) H.
The ideal Leib(L) is called a Leibniz kernel of the algebra L.
We note a following important property of the Leibniz kernel:
[[a, a], x] = [a, [a, x]] – [a, [a, x]] = 0.
This property shows that Leib(L) is an аbelian subalgebra of L.
As usual, we say that a Leibniz algebra L is finite-dimensional, if the dimension of L as a vector
space over F is finite.
The first step in the study of all types of algebras is the description of algebras having small
dimensions.
If dimF(L) = 1, then L is an abelian Lie algebra, i. e., is L = Fa for some element a and
[a, a] = 0.
If dimF(L) = 2 and L is not a Lie algebra, then there are the following two non-isomorphic
Leibniz algebras:
L1 = Fa + Fb, [a, a] = b, [b, a] = [a, b] = [b, b] = 0,
and
L2 = Fc + Fd, [c, c] = [c, d] =d, [d, c] = [d, d] = 0
(see, e. g., survey [6]). The structure of 3-dimensional Leibniz algebras is more complicated. The
study of Leibniz algebras, having dimension 3, has been conducted in papers [7—10] for the fields
of characteristic 0, moreover for a field C of complex numbers or an algebraically closed field of
characteristic 0. We consider the opposite situation, where the structure of Leibniz algebras of
dimension 3 over a finite field should be described. As we will see later, the situation here is much
more diverse. In some cases, the structure of the algebra essentially depends on the characteristic
of the field, in others on the solvability of specific equations in the field, and so on. We will see
that the Leibniz algebras of dimension 3 are soluble. Therefore, a first natural step of our study is
a consideration of nilpotent algebras.
Let L be a Leibniz algebra. Define the lower central series
L = γ1(L) γ2(L) . . . γα(L) γα + 1(L) . . . γδ(L)
of L 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 < λ γμ(L). It is possible to show that every term of 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, …]…]]. Note the following useful properties
of subalgebras and ideals.
22 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2018. № 7
V.S. Yashchuk
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 the nilpotency class c, if γc + 1(L) = 〈0〉, but γc(L) ≠ 〈0〉. We
denote, by ncl(L), the nilpotency class of L.
The left (respectively, right) center ζleft(L) (respectively, ζright(L)) of a Leibniz algebra L is
defined by the rule:
ζleft(L) = {x ∈ L | [x, y] = 0 for each element y ∈ L}
(respectively,
ζright(L) = {x ∈ L | [y, x] = 0 for each element y ∈ L}).
It is not hard to prove that the left center of L is an ideal, but this is not true for the right
center. Moreover, the last equality shows that Leib(L) ζleft(L), so that L/ζleft(L) is a Lie algebra.
The right center is an subalgebra of L. In general, the left and right centers are different; they even
may have different dimensions. We will construct now a following examples [11].
Of course, we will consider a case where L is not a Lie algebra.
Nilpotent Leibniz algebra of dimension 3. In this section, we will suppose that L is nilpotent.
Since ncl(L) dimF(L), we have ncl(L) 3.
Let L be a Leibniz algebra. The intersection of a maximal subgroup of L is called the Frattini
subalgebra of L and denoted by Frat(L). If L does not include maximal subalgebras, then we put
L = Frat(L).
We will need the following important property of Frattini subalgebras.
Proposition 1. Let L be a finite-dimensional Leibniz algebra over a field F. If L is nilpotent, then
[L, L] = Frat(L).
Indeed, since L is nilpotent, every maximal subalgebra of L is an ideal [12, Lemma 2.2], so we
can apply Proposition 7 of paper [6].
Theorem 1. Let L be a nilpotent Leibniz algebra over a field F. If L is not a Lie algebra and
ncl(L) = 3 = dimF(L), then L has a basis {a, b, c} such that [a, a] = b, [a, b] = c, [c, a] = [a, c] =
= [c, b] = [b, c] = [b, b] = [c, c] = 0. Moreover, Leib(L) = ζleft(L) = [L, L] = Fb ⊕ Fc, ζright(L) =
= ζ(L) = γ3(L) = Fc. In particular, L is a nilpotent cyclic Leibniz algebra.
Further, the relation L = A ⊕ B means that L is a direct sum of the subspaces A and B or the
subalgebras A and B. If L = A ⊕ B, A is an ideal of L, and B is a subalgebra of L, then we will say
that L is a semidirect sum of A and B and use the symbol L = A B.
Theorem 2. Let L be a nilpotent Leibniz algebra over a field F. Suppose that L is not a Lie alge-
bra, dimF(L) = 3, ncl(L) = 2 and L has an element b ∉ γ2(L) such that [b, b] = 0. Then L is an algebra
of one of the following types:
I. L = A ⊕ B, where A, B are the ideals, B = Fb, [b, b] = 0, A = Fa ⊕ Fc is a cyclic nilpotent sub-
algebra, [a, a] = c, [c, a] = [a, c] = [c, c] = 0. Moreover, Leib(L) = [L, L] = Fc, ζleft(L) = ζright(L) =
= ζ(L) = Fb ⊕ Fc.
II. L = A B, where A = Fa ⊕ Fc is a cyclic nilpotent subalgebra, [a, a] = c, [c, a] = [a, c] =
= [c, c] = 0, B is an аbelian subalgebra, B = Fb, [b, b] = 0, and [a, b] = c, [b, a] = 0 = [b, c] = [c, b].
Moreover, Leib(L) = [L, L] = ζright(L) = ζ(L) = Fc, ζleft(L) = Fb ⊕ Fc.
III. L = A B, where A = Fa ⊕ Fc is a cyclic nilpotent subalgebra, [a, a] = c, [c, a] = [a, c] =
= [c, c] = 0, B is an abelian subalgebra, B = Fb, [b, b] = 0, and [a, b] = c, [b, a] = γc, γ ≠ 0, [b, c] = [c,
b] = 0. Moreover, Leib(L) = [L, L] = ζleft(L) = ζright(L) = ζ(L) = Fc.
23ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2018. № 7
Leibniz algebras of dimension 3 over finite fields
Theorem 3. Let L be a nilpotent Leibniz algebra over a field F. Suppose that L is not a Lie alge-
bra, dimF(L) = 3, ncl(L) = 2, and [d, d] ≠ 0 for each element d ∉ γ2(L) such that [b, b] = 0. Then L is
an algebra of one of the following types:
I. L = A + B, where A, B are the nilpotent ideals, A = 〈a〉, B = 〈b〉, A ∩ B = ζ(L) = Fc, [a, a] =
= [b, b] = c, [c, a] = [a, c] = [c, c] = [c, b] = [b, c] = [a, b] = [b, a] = 0. Moreover, Leib(L) = [L, L] =
= ζleft(L) = ζright(L) = ζ(L) = Fc, char(F) ≠ 2, and the equation x2 + 1 = 0 has no solution in F.
II. L = A + B, where A, B are the nilpotent ideals, A = 〈a〉, B = 〈b〉, A ∩ B = ζ(L) = Fc, [a, a] = c,
[b, b] = ρc, where ρ is a primitive root of the identity of degree |F| – 1, [c, a] = [a, c] = [c, c] = [c, b] =
= [b, c] = [a, b] = [b, a] = 0. Moreover, Leib(L) = [L, L] = ζleft(L) = ζright(L) = ζ(L) = Fc, char(F) ≠ 2.
III. L = A + B, where A, B are the nilpotent ideals, A = 〈a〉, B = 〈b〉, A ∩ B = ζ(L) = Fc, [a, a] =
= c = [a, b], [b, b] = ηc, [c, a] = [a, c] = [c, c] = [c, b] = [b, c] = [b, a] = 0. Moreover, Leib(L) = [L, L] =
= ζleft(L) = ζright(L) = ζ(L) = Fc, and a polynomial X2 + X + η has no roots in a field F.
Non-nilpotent Leibniz algebra of dimension 3 with one-dimensional Leibniz kernel. The
next step is a consideration of a case where L is non-nilpotent. We will consider Leibniz alge-
bras of dimension 3, which are not Lie algebras. It follows that Leib(L) ≠ 〈0〉. Since Leib(L) is an
аbelian ideal, L ≠ Leib(L). Hence, for Leib(L), we have only two possibility: dimF(Leib(L)) = 1,
dimF(Leib(L)) = 2.
In this section, we consider the case where dimF(Leib(L)) = 1, so that dimF(L/Leib(L)) = 2.
Theorem 4. Let L be a non-nilpotent Leibniz algebra over a field F. Suppose that L is not a
Lie algebra, dimF(L) = 3 and dimF(Leib(L)) = 1. Then L is an algebra of one of the following
types:
I. L = A ⊕ B, where A, B are the ideals, B = Fb, [b, b] = 0, A is a cyclic subalgebra, A = Fa ⊕ Fc,
where [a, a] = c = [a, c], [c, a] = [c, c] = [c, b] = [b, c] = [a, b] = [b, a] = 0. Moreover, Leib(L) = [L,
L] = Fc, ζleft(L) = Fb ⊕ Fc, ζright(L) = ζ(L) = Fb.
II. L = A B, where B = Fb, [b, b] = 0, A = Fa ⊕ Fc is a cyclic subalgebra, [a, a] = c = [a, c], [a,
b] = c, [c, a] = [c, c] = [c, b] = [b, c] = [b, a] = 0. Moreover, Leib(L) = [L, L] = Fc, ζleft(L) = Fb ⊕ Fc,
ζ(L) = ζright(L) = 〈0〉.
III. L = A B, where B = Fb, [b, b] = 0, A = Fa ⊕ Fc is a cyclic subalgebra, [a, a] = c = [a, c], [b,
a] = [b, c] = c, [c, a] = [c, c] = [c, b] = [a, b] = 0. Moreover, Leib(L) = [L, L] = ζleft(L) = Fc, ζright(L) =
= Fb, ζ(L) = 〈0〉.
IV. L = A B, where B = Fb, [b, b] = 0, A = Fa ⊕ Fc is a cyclic subalgebra, [a, a] = c = [a, c],
[a, b] = a = –[b, a], [b, c] = –2c, [c, a] = [c, c] = [c, b] = 0. Moreover, Leib(L) = [L, L] = ζleft(L) = Fc,
ζright(L) = ζ(L) = 〈0〉.
V. L = A B, where B = Fb, [b, b] = 0, A = Fa ⊕ Fc is a cyclic subalgebra, [a, a] = c, [a, c] =
= 0, [a, b] = a + γc, γ ∈ F, [b, a] = –a + γc, [b, c] = –2c, [c, a] = [c, c] = [c, b] = 0. Moreover, Leib(L) =
= [L, L] = ζleft(L) = Fc, ζright(L) = ζ(L) = 〈0〉 whenever char(F) ≠ 2 and ζright(L) = ζ(L) = Fc whenever
char(F) = 2.
Non-nilpotent cyclic Leibniz algebra of dimension 3. The next step is a consideration of a
case where L is non-nilpotent, and dimF(Leib(L)) = 2. Here, there appear two variants: L is a cy-
clic algebra and L is a non-cyclic algebra. In this section, we will consider a case where a Leibniz
algebra of dimension 3 is cyclic.
Theorem 5. Let L be a non-nilpotent cyclic Leibniz algebra of dimension 3 over a field F. Then L
is an algebra of one of the following types:
24 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2018. № 7
V.S. Yashchuk
I. L = D A, where D = Fd, [d, d] = 0, A = Fa ⊕ Fc is a cyclic nilpotent subalgebra, [a, a] = c, [a,
c] = 0, [a, d] = δd, 0 ≠ δ ∈ F, [c, a] = [c, c] = [c, d] = [d, c] = [d, a] = 0. Moreover, Leib(L) = [L, L] =
ζleft(L) = Fd ⊕ Fc, ζ(L) = ζright(L) = Fc.
II. L = D B, where B = Fb, [b, b] = 0, D = Fd ⊕ Fc is an аbelian subalgebra, [d, d] = [d, c] =
= [c, d] = [c, c] = 0, [b, c] = d, [b, d] = γd + δd, 0 ≠ γ, δ ∈ F, [c, b] = [d, b] = 0. Moreover, Leib(L) =
= [L, L] = ζleft(L) = Fd ⊕ Fc, ζright(L) = Fb, ζ(L) = 〈0〉.
Non-nilpotent Leibniz algebra of dimension 3 with two-dimensional Leibniz kernel.
The last case of our consideration is the case where L is non-nilpotent, non-cyclic, and
dimF(Leib(L)) = 2. Then dimF(L/Leib(L)) = 1. In particular, L/Leib(L) is Abelian.
Theorem 6. Let L be a non-nilpotent non-cyclic Leibniz algebra of dimension 3 over a field F.
Suppose that L is a not Lie algebra and dimF(Leib(L)) = 2. Then L is an algebra of one of the fol-
lowing types:
I. L = A D, where D = Fd, [d, d] = 0, A = Fa ⊕ Fc is a cyclic subalgebra, [a, a] = c = [a, c],
[a, d] = d, [c, a] = [c, c] = [c, d] = [d, c] = [d, a] = 0. Moreover, Leib(L) = [L, L] = ζleft(L) = Fd ⊕ Fc,
ζ(L) = ζright(L) = 〈0〉.
II. Char(F) ≠ 2, L = A D, where D = Fd, [d, d] = 0, A = Fa ⊕ Fc is a cyclic subalgebra, [a, a] =
= c = [a, c], [a, d] = c + 2d, [c, a] = [c, c] = [c, d] = [d, c] = [d, a] = 0. Moreover, Leib(L) = [L, L] =
= ζleft(L) = Fd ⊕ Fc, ζ(L) = ζright(L) = 〈0〉.
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Received 05.03.2018
25ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2018. № 7
Leibniz algebras of dimension 3 over finite fields
В.С. Ящук
Дніпровський національний університет ім. Олеся Гончара
E-mail: Viktoriia.Yashchuk@i.ua
АЛГЕБРИ ЛЕЙБНІЦА ВИМІРНОСТІ 3 НАД СКІНЧЕННИМИ ПОЛЯМИ
Першим кроком у вивченні всіх типів алгебр є опис таких алгебр, які мають малі вимірності. На відміну від
більш простих випадків одно- і двовимірних алгебр Лейбніца, структури тривимірних алгебр Лейбніца
складніші. У роботі розглядається структура алгебр Лейбніца вимірності 3 над скінченним полем. У дея-
ких випадках структура алгебр суттєво залежить від характеристики поля, в інших — від можливості
розв’язання конкретних рівнянь у полі і т. п.
Ключові слова: алгебра Лейбніца, ідеал, фактор-алгебра, нільпотентна алгебра Лейбніца.
В.С. Ящук
Днипровский национальний университет им. Олеся Гончара
E-mail: Viktoriia.Yashchuk@i.ua
АЛГЕБРЫ ЛЕЙБНИЦА РАЗМЕРНОСТИ 3 НАД КОНЕЧНЫМИ ПОЛЯМИ
Первым шагом в изучении всех типов алгебр является описание таких алгебр, которые имеют малые раз-
мерности. В отличие от более простых случаев одно- и двумерных алгебр Лейбница, структуры трехмер-
ных алгебр Лейбница сложнее. В работе рассматривается структура алгебр Лейбница размерности 3 над
конечным полем. В некоторых случаях структура алгебры зависит от характеристики поля, в других — от
разрешимости конкретных уравнений в поле и т. п.
Ключевые слова: алгебра Лейбница, идеал, фактор-алгебра, нильпотентная алгебра Лейбница.
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| id | nasplib_isofts_kiev_ua-123456789-143368 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1025-6415 |
| language | English |
| last_indexed | 2025-12-07T17:55:07Z |
| publishDate | 2018 |
| publisher | Видавничий дім "Академперіодика" НАН України |
| record_format | dspace |
| spelling | Yashchuk, V.S. 2018-10-31T11:18:07Z 2018-10-31T11:18:07Z 2018 Leibniz algebras of dimension 3 over finite fields / V.S. Yashchuk // Доповіді Національної академії наук України. — 2018. — № 7. — С. 20-25. — Бібліогр.: 12 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2018.07.020 https://nasplib.isofts.kiev.ua/handle/123456789/143368 512.542 The first thing in the study of all types of algebras is the description of algebras having small dimensions. Unlike the
 simpler cases of 1- and 2-dimensional Leibniz algebras, the structure of 3-dimensional Leibniz algebras is more complicated.
 We consider the structure of Leibniz algebras of dimension 3 over a finite field. In some cases, the structure
 of the algebra essentially depends on the characteristic of the field. In others, it depends on the solvability of specific
 equations in the field, and so on. Першим кроком у вивченні всіх типів алгебр є опис таких алгебр, які мають малі вимірності. На відміну від
 більш простих випадків одно- і двовимірних алгебр Лейбніца, структури тривимірних алгебр Лейбніца
 складніші. У роботі розглядається структура алгебр Лейбніца вимірності 3 над скінченним полем. У деяких випадках структура алгебр суттєво залежить від характеристики поля, в інших — від можливості
 розв'язання конкретних рівнянь у полі і т. п. Первым шагом в изучении всех типов алгебр является описание таких алгебр, которые имеют малые размерности. В отличие от более простых случаев одно- и двумерных алгебр Лейбница, структуры трехмерных алгебр Лейбница сложнее. В работе рассматривается структура алгебр Лейбница размерности 3 над
 конечным полем. В некоторых случаях структура алгебры зависит от характеристики поля, в других — от
 разрешимости конкретных уравнений в поле и т. п. en Видавничий дім "Академперіодика" НАН України Доповіді НАН України Математика Leibniz algebras of dimension 3 over finite fields Алгебри Лейбніца вимірності 3 над скінченними полями Алгебры Лейбница размерности 3 над конечными полями Article published earlier |
| spellingShingle | Leibniz algebras of dimension 3 over finite fields Yashchuk, V.S. Математика |
| title | Leibniz algebras of dimension 3 over finite fields |
| title_alt | Алгебри Лейбніца вимірності 3 над скінченними полями Алгебры Лейбница размерности 3 над конечными полями |
| title_full | Leibniz algebras of dimension 3 over finite fields |
| title_fullStr | Leibniz algebras of dimension 3 over finite fields |
| title_full_unstemmed | Leibniz algebras of dimension 3 over finite fields |
| title_short | Leibniz algebras of dimension 3 over finite fields |
| title_sort | leibniz algebras of dimension 3 over finite fields |
| topic | Математика |
| topic_facet | Математика |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/143368 |
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