On the structure of the algebra of derivations of cyclic Leibniz algebras
We describe the algebra of derivation of finitedimensional cyclic Leibniz algebra.
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| citation_txt | On the structure of the algebra of derivations of cyclic Leibniz algebras / L.A. Kurdachenko, M.M. Semko, V.S. Yashchuk // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 2. — С. 241-252. — Бібліогр.: 11 назв. — англ. |
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© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 32 (2021). Number 2, pp. 241ś252
DOI:10.12958/adm1898
On the structure of the algebra of derivations
of cyclic Leibniz algebras
L. A. Kurdachenko, M. M. Semko, and V. S. Yashchuk
Abstract. We describe the algebra of derivation of őnite-
dimensional cyclic Leibniz algebra.
Introduction
Let L be an algebra over őnite őeld F with the binary operations
+ and [·, ·]. Then L is called a left Leibniz algebra if it satisőes 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 were őrst in the paper of A. Bloh [1], but the term łLeibniz
algebraž appears in the book of J.-L. Loday [2] and his article [3]. In [4]
J. Loday and T. Pirashvili began the real study of properties of Leibniz
algebras. The theory of Leibniz algebras was developed very intensively in
many different directions. Some of the results of this theory were presented
in the book [5]. Note that Lie algebras are a partial case of Leibniz algebras.
Conversely, if L is a Leibniz algebra in which [a, a] = 0 for every element
a ∈ L, then it is a Lie algebra. Thus, Lie algebras can be characterized as
anticommutative Leibniz algebras.
2020 MSC: 17A32, 17A60, 17A99.
Key words and phrases: Leibniz algebra, cyclic Leibniz algebra, derivation,
ideal.
https://doi.org/10.12958/adm1898
242 On the structure of the algebra of derivations
Like the Lie algebras, the structure of Leibniz algebras is strongly
affected by their algebras of derivations.
Denote by EndF (L) the set of all linear transformations of L. Then L
is an associative algebra by the operations + 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 subalgebra of a Lie algebra EndF (L). Der(L) is called
the algebra of derivations of a Leibniz algebra L.
For a Leibniz algebra, the following result shows the inŕuence of the
algebra of derivations on its structure: 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].
Among the Leibniz algebras, it is natural to study the structure of
their algebras of derivations for cyclic Leibniz algebras. The structure
of cyclic Leibniz algebras was described in [7]. In what follows, we will
demonstrate this structure.
Let L be a cyclic Leibniz algebra, L = ⟨a⟩, and we suppose that L
has a őnite dimension over a őeld F . Then there exists a positive integer
n such that L has a basis a1, . . . , an, where a1 = a, a2 = [a1, a1], . . . ,
an = [a1, an−1], [a1, an] = α2a2 + . . . + αnan [7]. Moreover, [L,L] =
Leib(L) = Fa2 + . . .+ Fan [7]. We őx these designations.
Here appear the following types of cyclic Leibniz algebras.
First case: [a1, an] = 0. In this case, L is nilpotent, and we say that L
is a cyclic algebra of type (I).
The structures of the algebras of derivations of these Leibniz algebras
have been described in [7].
The next type of cyclic Leibniz algebras appears in the following way.
In this case, [a1, an] = α2a2 + . . .+ αnan and α2 ̸= 0. In some sense, it is
a basic case. First, we recall some deőnitions.
The left (respectively right) center ζ left(L) (respectively ζright(L)) of
a Leibniz algebra L is deőned by the rule:
ζ left(L) = {x ∈ L| [x, y] = 0 for each element y ∈ L}
(respectively,
ζright(L) = {x ∈ L| [y, x] = 0 for each element y ∈ L}).
L. A. Kurdachenko, M. M. Semko, V. S. Yashchuk 243
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, Leib(L) ⩽ ζ left(L), so that L/ζ left(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 dimensions
(see [6]).
Put c = α−1
2 (α2a1 + . . . + αnan−1 − an), then [c, c] = 0, moreover,
Fc is a right center of L,L = [L,L] ⊕ Fc and [c, b] = [a1, b] for ev-
ery element b ∈ A [7]. In particular, a3 = [c, a2], . . . , an = [c, an−1],
[c, an] = α2a2 + . . .+ αnan. In this case, we say that L is a cyclic algebra
of type (II).
Put A = [L,L] and deőne the mapping lc : L → L by the rule
lc(x) = [c, x] for every element x ∈ L. The restriction of a linear transfor-
mation lc on A has the following matrices in a basis {a2, . . . , an}:
0 0 0 . . . 0 α2
1 0 0 . . . 0 α3
0 1 0 . . . 0 α4
0 0 1 . . . 0 α5
...
...
...
. . .
...
...
0 0 0 . . . 0 αn−1
0 0 0 . . . 1 αn
These matrices are non-degenerate. Hence the restriction of lc on A is an
F -automorphism of a linear space A. The őrst main result of this paper
is following.
Theorem 1. Let L be a cyclic Leibniz algebra of type (II) over a őeld F ,
and let D be the annihilator of a subspace Fc in algebra Der(L). Then the
following assertions hold:
(i) D is an Abelian ideal having dimension dimF (L) − 1; the set
{i, lc, l
2
c , . . . , l
n−2
c } is a basis of D;
(ii) D has a codimension at most 1;
(iii) if D ̸= Der(L), then char(F ) divides dimF (L)− 1.
Here i is the mapping, deőned by the rule: if x = a+σc, a ∈ A, σ ∈ F ,
is an arbitrary element of L, then put i(x) = a.
Corollary A1. Let L be a cyclic Leibniz algebra of type (II) over a őeld F .
If F has a characteristic 0, then algebra Der(L) is Abelian and has a di-
mension dimF (L)− 1.
Finally, consider the last type of őnite-dimensional cyclic Leibniz
algebra.
244 On the structure of the algebra of derivations
In this case: [a1, an] = α2a2 + . . . + αnan, but α2 = 0. Let t be the
őrst index such that αt ̸= 0. In other words, [a1, an] = αtat + . . .+ αnan.
By our condition, t > 2. Then
[a, an] = αt[a, at−1] + . . .+ αn[a, an−1] = [a, αtat−1 + . . .+ αnan−1],
which implies that αtat−1 + . . .+ αnan−1 − an ∈ AnnrightK (a1). The fact
that αt ̸= 0 implies that α−1
t ̸= 0, and then
dt−1 = α−1
t (αtat−1 + . . .+ αnan−1 − an) =
= at−1 + βtat + . . .+ βnan ∈ AnnrightK (a1).
Put
dt−2 = at−2 + βtat−1 + . . .+ βnan−1,
dt−3 = at−3 + βtat−2 + . . .+ βnan−2, . . . ,
d1 = a1 + βta2 + . . .+ βnan−t+1.
Then
[d1, d1] = [a1, d1] = d2,
[d1, d2] = [a1, d2] = d3, . . . ,
[d1, dt−2] = [a1, dt−2] = dt−1,
[d1, dt−1] = [a1, dt−1] = 0.
It follows that the subspace U = Fd1⊕Fd2⊕. . .⊕Fdt−1 is a subalgebra,
and, moreover, this subalgebra is nilpotent. Moreover, a subspace [U,U ] =
Fd2 ⊕ . . .⊕ Fdt−1 is an ideal of L. Put further dt = at, dt+1 = at+1, . . . ,
dn = an. The following matrix corresponds to this transaction:
1 βt βt+1 . . . βk 0 0 . . . 0 0
0 1 βt . . . βk−1 βk 0 . . . 0 0
...
...
...
...
...
...
...
...
...
...
0 0 0 . . . 0 1 βt . . . βk−1 βk
0 0 0 . . . 0 0 1 . . . 0 0
...
...
...
...
...
...
...
...
...
...
0 0 0 . . . 0 0 0 . . . 0 1
This matrix is non-singular, which proves that the elements {d1, . . . , dn}
present a new basis. We note that a subspace V = Fdt ⊕ . . . ⊕ Fdn is
a subalgebra. Moreover, V is an ideal of D, because [a1, dt] = dt+1, . . . ,
L. A. Kurdachenko, M. M. Semko, V. S. Yashchuk 245
[a1, dn−1] = dn, [a1, dn] = αtdt + . . .+ αndn. Moreover, [a1, dj ] = [d1, dj ]
for all j ⩾ t [7]. In this case, we say that L is a cyclic algebra of type (III).
Thus, L = A ⊕ Fd1, A = V ⊕ [U,U ] is a direct sum of two ideals,
U = [U,U ] ⊕ Fd1 is a nilpotent cyclic subalgebra, i.e. is an algebra of
type (I), and V ⊕ Fd1 is a cyclic subalgebra of type (II).
The second main result of this paper gives a description of the algebra
of derivations of cyclic Leibniz algebras of type (III).
Theorem 2. Let L be a cyclic Leibniz algebra of type (III) over a őeld F .
Then Der(L) is a subdirect product of the algebras D1 and D2, where D1
is the algebra of derivations of a cyclic nilpotent Leibniz algebra, D2 is the
algebra of derivations of a cyclic Leibniz algebra of type (II).
We recall that a structure of the algebra of derivations of nilpotent
cyclic Leibniz algebras was described in [8].
1. Algebra of derivation of a cyclic Leibniz algebra of
type (II)
We show here some basic elementary properties of derivations, which
have been proved in [9].
Lemma 1. Let L be a Leibniz algebra over a őeld F , and let f be
a derivation of L. Then f(ζ left(L) ⩽ ζ left(L), f(ζright(L)) ⩽ ζright(L)
and f(ζ(L)) ⩽ ζ(L).
Corollary 1. Let L be a Leibniz algebra over a őeld F and f be a derivaion
of L. Then f(ζα(L)) ⩽ ζα(L) for every ordinal α.
Lemma 2. Let L be a Leibniz algebra over a őeld F , and let f be
a derivaion of L. Then f(γα(L)) ⩽ γα(L) for all ordinals α, in particular,
f(γ∞(L)) ⩽ γ∞(L).
Corollary 2. Let L be a cyclic Leibniz algebra of type (II) over a őeld
F , L = A⊕ S, where A = [L,L] = Leib(L), S = Fc = ζright(L). If f is
an derivaion of L, then f(A) ⩽ A, f(S) ⩽ S, in particular, f(c) = σc for
some σ ∈ F .
We start from the case of a non-nilpotent cyclic Leibniz algebra L,
having dimension 2. In this case, L = Fa1 ⊕ Fa2, where [a1, a1] = a2,
[a2, a2] = [a2, a1] = 0, [a1, a2] = a2 (see, e.g., survey [10]). This algebra
has an interesting property: every its subalgebra is an ideal. Note that
the Leibniz algebras whose subalgebras are ideals were described in [11].
246 On the structure of the algebra of derivations
Let f be an arbitrary derivaion of L. We have f(a1) = γa1 + αa2 for
some elements α, γ ∈ F . Then
f(a2) = f([a1, a1]) = [f(a1), a1] + [a1, f(a1)] =
= [γa1 + αa2, a1] + [a1, γa1 + αa2] =
= γa2 + γa2 + αa2 = (2γ + α)a2.
Put c = a1 − a2, then Fc = ζright(L). We have
f(c) = f(a1 − a2) = f(a1)− f(a2) =
= γa1 + αa2 − (2γ + α)a2 = γa1 − 2γa2.
On the other hand, Lemma 1 shows that f(c) ∈ Fc. It is possible,
only if γ = 0. In this case, f(a1) = αa2 and f(a2) = αa2. In this case,
we can see that Der(L) ∼= F , in particular, Der(L) is Abelian and has
a dimension 1.
Now, we suppose that dimF (L) > 2.
Lemma 3. Let L be a cyclic Leibniz algebra of type (II) over a őeld F ,
and let D be the annihilator of a subspace Fc in algebra Der(L). Then
D is an ideal of Der(L) and a factor-algebra Der(L)/D has dimension at
most 1.
Proof. Let f be an arbitrary derivaion of L. Since
f(Fc) = f(ζright(L)) ⩽ ζright(L)
by Lemma 1, we obtain that f(c) = αc for some element α ∈ F .
Clearly, D is a subalgebra of Der(L). Let f be an arbitrary derivaion,
and let g be an element of D. Then
[f, g](c) = (f ◦ g)(c)− (g ◦ f)(c) = f(g(c))− g(f(c)) =
= f(0)− g(αc) = −αg(c) = 0,
[g, f ](c) = −[f, g](c) = 0,
so that D is an ideal of Der(L).
The factor-algebra of Der(L)/AnnDer(L)(Fc) is isomorphic with some
subalgebra of the algebra of linear transformations of a vector space Fc.
It follows that this factor-algebra has dimension 0 or 1.
Lemma 4. Let L be a cyclic Leibniz algebra of type (II) over a őeld F . If
L has a derivaion f such that f(c) ̸= 0, then char(F ) divides dimF (L)−1.
L. A. Kurdachenko, M. M. Semko, V. S. Yashchuk 247
Proof. By Corollary 2 f(c) = σc for some non-zero element σ of a őeld F .
Put g = σ−1f . Then g is a derivation, and g(c) = c. Let x be an arbitrary
element of L. Then x = a+ λc for some element λ ∈ F . We have
[c, x] = [c, a+ λc] = [c, a] + [c, λc] = [c, a]
and
g([c, x]) = g([c, a]) = [g(c), a] + [c, g(a)] =
= [c, a] + [c, g(a)] = [c, a+ g(a)] = [c, (g + IdL)(a)]
(here, IdL is an identity permutation of L).
Thus, we obtain g(lc(a)) = lc((g + IdL)(a)).
If h is a derivaion of L, then Lemma 2 shows that h(A) ⩽ A. Deőne
now the mapping h↓ : A → A by the rule: f↓(a) = f(a) for every a ∈ A.
It is not hard to prove that h↓ is a linear transformation of a vector space
A. Then we obtain
g↓ ◦ l↑c = l
↑
c ◦ (g
↑ + IdA).
Denote, by G (respectively, M), the matrix of a linear mapping g↑
(respectively, l↑c) in a basis {a2, . . . , an}. Then we obtain the matrix equality
GM = M(G+ E). As we have seen above, the matrix M is not singular.
Thus, we obtain M−1GM = G+ E.
Since trace(G) = trace(M−1GM), we obtain
trace(G) = trace(G+ E) = trace(G) + (n− 1)1F
(here, 1F is an identity element of a őeld F ). It follows that (n− 1)1F = 0.
In this case, char(F ) divides n− 1.
Lemma 5. Let L be a cyclic Leibniz algebra of type (II) over a őeld F ,
and let D be the annihilator of a subspace Fc in algebra Der(L). Then D
is generated as a vector space by the derivations i, lc, l
2
c , . . . , l
n−2
c . Moreover,
the set {i, lc, l
2
c , . . . , l
n−2
c } is a basis of D, so that D is Abelian and has
a dimension n− 1.
Proof. We note that the mapping i is a derivaion of L. Indeed, if y = b+τc,
b ∈ A, τ ∈ F , is another element of L, then put
i([x, y]) = i([a+ σc, b+ τc]) = i(σ[c, b]) = σ[c, b],
[i(x), y] + [x, i(y)] = [a, b+ τc] + [a+ σc, b] = σ[c, b].
Let f is an arbitrary derivaion of D. Then
(f ◦ lc)(x) = f(lc(x)) = f(lc(a+ σc)) =
= f([c, a+ σc]) = f([c, a]) = [f(c), a] + [c, f(a)] = [c, f(a)],
248 On the structure of the algebra of derivations
(lc ◦ f)(x) = lc(f(x)) = lc(f(a+ σc)) = lc(f(a)) = [c, f(a)].
Since it is true for every element x ∈ L, we obtain that f ◦ lc = lc ◦ f .
We have
a3 = [c, a2] = lc(a2),
a4 = [c, a3] = lc(a3) = lc(lc(a2)) = l
2
c(a2), . . . ,
an = [c, an−1] = l
n−2
c (a2).
Note that
l
n−1
c (a2) = lc(l
n−2
c (a2)) = lc(an) = α2 + α3lc(a2) + . . .+ αnl
n−2
c (a2),
so that we can deőne l
k
c (a2) (and, hence, lkc (a) for arbitrary a ∈ A) for
each positive integer k.
Let f be an arbitrary element of an ideal D. Let
f(a2) = β0a2 + β1a3 + . . .+ βn−2an
for some elements β0, . . . , βn−2 ∈ F . Then we obtain the presentation
f(a2) = β0i(a2) + β1lc(a2) + β2l
2
c(a2) + . . .+ βn−2l
n−2
c (a2) =
= (β0i+ β1lc + β2l
2
c + . . .+ βn−2l
n−2
c )(a2).
Put df = β0i+ β1lc + β2l
2
c + . . .+ βn−2l
n−2
c , then f(a2) = df (a2).
If a is an arbitrary element of A, then a = σ0a2 + σ1a3 + . . .+ σn−2an
for some elements σ0, . . . , σn−2 ∈ F . We have
f(a3) = f(lc(a2)) = lc(f(a2)) = lc(df (a2)) = df (lc(a2)) = df (a3).
Similarly, we obtain that f(a4) = df (a4), . . . , f(an) = df (an). It follows
that
f(a) = f(σ0a2 + σ1a3 + σ2a4 + . . .+ σn−2an) =
= σ0f(a2) + σ1f(a3) + σ2f(a4) + . . .+ σn−2f(an) =
= σ0df (a2) + σ1df (a3) + σ2df (a4) + . . .+ σn−2df (an) =
= df (σ0a2 + σ1a3 + σ2a4 + . . .+ σn−2an) = df (a).
If x = a+ σc, a ∈ A, σ ∈ F , is an arbitrary element of L, then
f(x) = f(a+ σc) = f(a) + σf(c) = f(a),
and it implies that f(x) = df (x).
L. A. Kurdachenko, M. M. Semko, V. S. Yashchuk 249
Since it is true for every element x ∈ L, we obtain that f = df .
We note that the mappings i, lc, l
2
c , . . . , l
n−2
c are linearly independent.
Indeed, suppose that λ0, λ1, . . . , λn−2 are the elements of F such that
λ0i+ λ1lc + λ2l
2
c + . . .+ λn−2l
n−2
c = 0.
Then
(λ0i+ λ1lc + λ2l
2
c + . . .+ λn−2l
n−2
c )(a2) = 0.
On the other hand,
(λ0i+ λ1lc + λ2l
2
c + . . .+ λn−2l
n−2
c )(a2) =
= λ0a2 + λ1lc(a2) + λ2l
2
c(a2) + . . .+ λn−2l
n−2
c (a2) =
= λ0a2 + λ1a3 + λ2a4 + . . .+ λn−2an.
The fact that {a2, a3, . . . , an} is a basis of A shows that
λ0 = λ1 = . . . = λn−2 = 0.
2. Proof of Theorem A
Assertion (i) follows from Lemmas 3 and 5. Assertion (ii) follows from
Lemma 3. Assertion (iii) follows from Lemma 4.
The following natural question appears from Lemma 4.
Let L be a cyclic Leibniz algebra of type (ii). Is f(c) = 0 for an
arbitrary derivaion f of L?
The following example gives a negative answer on this question.
Example 1. Let L = Fc⊕ Fa2 ⊕ Fa3 ⊕ Fa4 be a cyclic Leibniz algebra
of type (II), having dimension 4 over a őeld F3 of order 3. Let
[c, a2] = a3, [c, a3] = a4, [c, a4] = a2.
Consider a linear transformation f of L, deőned by the rule
f(c) = c,
f(a2) = 2a3 + a4,
f(a3) = a2 + a3 + 2a4,
f(a4) = 2a2 + a3 + 2a4.
250 On the structure of the algebra of derivations
Let x = γc + γ2a2 + γ3a3 + γ4a4, y = λc + λ2a2 + λ3a3 + λ4a4 be the
arbitrary elements of L. We have
[x, y] = [γc+ γ2a2 + γ3a3 + γ4a4, λc+ λ2a2 + λ3a3 + λ4a4] =
= [γc, λc+ λ2a2 + λ3a3 + λ4a4] =
= γλ2a3 + γλ3a4 + γλ4a2;
f([x, y]) = f(γλ4a2 + γλ3a4 + γλ2a3) =
= γλ4f(a2) + γλ3f(a4) + γλ2f(a3) =
= γλ4(2a3 + a4) + γλ3(2a2 + a3 + 2a4) + γλ2(a2 + a3 + 2a4) =
= (γλ2 + 2γλ3)a2 + (γλ2 + γλ3 + 2γλ4)a3 + (2γλ2 + 2γλ3 + γλ4)a4;
f(x) = f(γc+ γ2a2 + γ3a3 + γ4a4) =
= γf(c) + γ2f(a2) + γ3f(a3) + γ4f(a4) =
= γc+ γ2(2a3 + a4) + γ3(a2 + a3 + 2a4) + γ4(2a2 + a3 + 2a4) =
= γc+ (2γ4 + γ3)a2 + (2γ2 + γ3 + γ4)a3 + (γ2 + 2γ3 + 2γ4)a4;
f(y) = λc+ (2λ4 + λ3)a2 + (2λ2 + λ3 + λ4)a3 + (λ2 + 2λ3 + 2λ4)a4;
[f(x), y] = [γc+ (2γ4 + γ3)a2 + (2γ2 + γ3 + γ4)a3 + (γ2 + 2γ3 + 2γ4)a4,
λc+ λ2a2 + λ3a3 + λ4a4] = [γc, λc+ λ2a2 + λ3a3 + λ4a4] =
= γλ2a3 + γλ3a4 + γλ4a2;
[x, f(y)] = [γc+ γ2a2 + γ3a3 + γ4a4,
λc+ (2λ4 + λ3)a2 + (2λ2 + λ3 + λ4)a3 + (λ2 + 2λ3 + 2λ4)a4] =
= [γc, λc+ (2λ4 + λ3)a2 + (2λ2 + λ3 + λ4)a3 + (λ2 + 2λ3 + 2λ4)a4] =
= (2γλ4 + γλ3)a3 + (2γλ2 + γλ3 + γλ4)a4 + (γλ2 + 2γλ3 + 2γλ4)a2;
[f(x), y] + [x, f(y)] = γλ2a3 + γλ3a4 + γλ4a2 + (2γλ4 + γλ3)a3+
+(2γλ2 + γλ3 + γλ4)a4 + (γλ2 + 2γλ3 + 2γλ4)a2 =
= (γλ2 + 2γλ4 + γλ3)a3 + (γλ3 + 2γλ2 + γλ3 + γλ4)a4+
+(γλ4 + γλ2 + 2γλ3 + 2γλ4)a2 =
= (γλ2 + 2γλ3)a2 + (γλ2 + 2γλ4 + γλ3)a3 + (2γλ3 + 2γλ2 + γλ4)a4 =
= [f(x), f(y)].
These equalities show that f is a derivaion of L.
L. A. Kurdachenko, M. M. Semko, V. S. Yashchuk 251
3. Algebra of derivations of a cyclic Leibniz algebra of
type (III). Proof of Theorem B
We have L = A⊕Fd1, A = V ⊕ [U,U ], U = Fd1 ⊕Fd2 ⊕ . . .⊕Fdt−1
is a nilpotent cyclic subalgebra, i.e. is an algebra of type (I). Moreover,
a subspace [U,U ] = Fd2 ⊕ . . . ⊕ Fdt−1 is an ideal of L. Furthermore,
V = Fdt ⊕ . . .⊕ Fdn is an ideal of L, and [a1, dj ] = [d1, dj ] for all j ⩾ t.
In other words, V ⊕ Fd1 is a cyclic subalgebra of type (II).
Since L/V ∼= U is a cyclic nilpotent Leibniz algebra,
Der(L)/AnnDer(L)(L/V ) = D1
is an algebra of derivations of a cyclic nilpotent Leibniz algebra. Since
L/[U,U ] ∼= V ⊕ Fd1 is a cyclic Leibniz algebra of the second type,
Der(L)/AnnDer(L)(L/[U,U ]) = D2 is the algebra of derivations of a cyclic
Leibniz algebra of type (II).
Let f ∈ AnnDer(L)(L/V ) ∩AnnDer(L)(L/[U,U ]), and let x be an arbi-
trary element of L. Then f(x) ∈ V , and, on the other hand, f(x) ∈ [U,U ].
It follows that
f(x) ∈ V ∩ [U,U ] = ⟨0⟩, so that f(x) = x.
Thus AnnDer(L)(L/V ) ∩AnnDer(L)(L/[U,U ]) = ⟨0⟩, and Remak’s the-
orem yields the embedding of algebra Der(L) into the direct product
D1 ×D2.
References
[1] Bloh A.M., On a generalization of the concept of Lie algebra, Dokl. Akad. Nauk
SSSR 165 (1965), no. 3, pp. 471ś473.
[2] Loday J.-L., Cyclic homology. Grundlehren der Mathematischen Wissenschaften,
301, 2nd ed., Springer, Verlag, Berlin, 1992.
[3] Loday J.-L., Une version non commutative des algèbres de Lie: les algèbras de
Leibniz, L’Enseignement Mathèmatique 39 (1993), pp. 269ś293.
[4] Loday J.-L., Pirashvili T., Universal enveloping algebras of Leibniz algebras and
(co)homology, Math. Annalen 296 (1993), pp. 139ś158.
[5] Ayupov Sh.A., Omirov B.A., Rakhimov I.S., Leibniz Algebras: Structure and
Classification, CRC Press, Taylor & Francis Group, (2020).
[6] Kurdachenko L.A., Otal J., Pypka A.A., Relationships between factors of canonical
central series of Leibniz algebras, European Journal of Mathematics, 2 (2016),
pp. 565ś577.
[7] Chupordya V.A., Kurdachenko L.A., Subbotin I.Ya., On some “minimal” Leibniz
algebras, J. Algebra Appl., 16 (2017), no. 2, 1750082 (16 pages).
252 On the structure of the algebra of derivations
[8] Semko M.M., Skaskiv L.V., Yarovaya O.A. On the derivations of cyclic nilpotent
Leibniz algebras. The 13 algebraic conference in Ukraine, book of abstract , July
6ś9, (2021), Taras Shevchenko National University of Kyiv, p. 73
[9] Kurdachenko L.A., Subbotin I.Ya., Yashchuk V.S. On the endomorphisms and
derivations of some Leibniz algebras. ArXiv, ArXiv: math. RA/2104.05922 (2021).
[10] Kirichenko V.V., Kurdachenko L.A., Pypka A.A., Subbotin I.Ya., Some aspects
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[11] Kurdachenko L.A., Semko N.N., Subbotin I.Ya., The Leibniz algebras whose subal-
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Contact information
Leonid
A. Kurdachenko,
Viktoriia
S. Yashchuk
Department of Geometry and Algebra,
Faculty of Mechanics and Mathematics, Oles
Honchar Dnipro National University, 72
Gagarin ave., Dnipro, 49010, Ukraine
E-Mail(s): lkurdachenko@i.ua,
Viktoriia.S.Yashchuk@gmail.com
Mykola M. Semko University of the State Fiscal Service of
Ukraine, 31 Universytetskaya str., Irpin, 08205,
Ukraine
E-Mail(s): dr.mykola.semko@gmail.com
Received by the editors: 12.10.2021.
mailto:lkurdachenko@i.ua
mailto:Viktoriia.S.Yashchuk@gmail.com
mailto:dr.mykola.semko@gmail.com
L. A. Kurdachenko, M. M. Semko, V. S. Yashchuk
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| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-01T11:11:37Z |
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| spelling | Kurdachenko, L.A. Semko, M.M. Yashchuk, V.S. 2023-03-14T17:05:08Z 2023-03-14T17:05:08Z 2021 On the structure of the algebra of derivations of cyclic Leibniz algebras / L.A. Kurdachenko, M.M. Semko, V.S. Yashchuk // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 2. — С. 241-252. — Бібліогр.: 11 назв. — англ. 1726-3255 DOI:10.12958/adm1898 2020 MSC: 17A32, 17A60, 17A99 https://nasplib.isofts.kiev.ua/handle/123456789/188751 We describe the algebra of derivation of finitedimensional cyclic Leibniz algebra. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On the structure of the algebra of derivations of cyclic Leibniz algebras Article published earlier |
| spellingShingle | On the structure of the algebra of derivations of cyclic Leibniz algebras Kurdachenko, L.A. Semko, M.M. Yashchuk, V.S. |
| title | On the structure of the algebra of derivations of cyclic Leibniz algebras |
| title_full | On the structure of the algebra of derivations of cyclic Leibniz algebras |
| title_fullStr | On the structure of the algebra of derivations of cyclic Leibniz algebras |
| title_full_unstemmed | On the structure of the algebra of derivations of cyclic Leibniz algebras |
| title_short | On the structure of the algebra of derivations of cyclic Leibniz algebras |
| title_sort | on the structure of the algebra of derivations of cyclic leibniz algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/188751 |
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