Automorphisms and derivations of Leibniz algebras
We extend some general properties of automorphisms and derivations known for the Lie algebras to finite-dimensional complex Leibniz algebras. The analogs of the Jordan – Chevalley decomposition for derivations and the multiplicative decomposition for automorphisms of finite-dimensional complex Leibn...
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| author | Ladra, M. Rikhsiboev, I. M. Turdibaev, R. M. Ладра, М. Ріхсибоєв, І. М. Турдібаєв, Р. М. |
| author_facet | Ladra, M. Rikhsiboev, I. M. Turdibaev, R. M. Ладра, М. Ріхсибоєв, І. М. Турдібаєв, Р. М. |
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| description | We extend some general properties of automorphisms and derivations known for the Lie algebras to finite-dimensional
complex Leibniz algebras. The analogs of the Jordan – Chevalley decomposition for derivations and the multiplicative
decomposition for automorphisms of finite-dimensional complex Leibniz algebras are obtained. |
| first_indexed | 2026-03-24T02:14:43Z |
| format | Article |
| fulltext |
UDC 512.5
M. Ladra* (Univ. Santiago de Compostela, Spain),
I. M. Rikhsiboev (Malaysian Ins. Industr. Technology, Univ. Kuala Lumpur, Malaysia),
R. M. Turdibaev* (Univ. Santiago de Compostela, Spain)
AUTOMORPHISMS AND DERIVATIONS OF LEIBNIZ ALGEBRAS
АВТОМОРФIЗМИ ТА ПОХIДНI АЛГЕБР ЛЕЙБНIЦА
We extend some general properties of automorphisms and derivations known for the Lie algebras to finite-dimensional
complex Leibniz algebras. The analogs of the Jordan – Chevalley decomposition for derivations and the multiplicative
decomposition for automorphisms of finite-dimensional complex Leibniz algebras are obtained.
Деякi загальнi властивостi автоморфiзмiв та похiдних, що вiдомi для алгебр Лi, розширeно на випадок комплексних
алгебр Лейбнiца. Встановлено аналоги розкладу Джордана – Шевальє для похiдних та мультиплiкативного розкладу
для автоморфiзмiв скiнченновимiрних комплексних алгебр Лейбнiца.
1. Introduction. Leibniz algebras were first introduced by A. Bloh [2] as D-algebras. Later they
were rediscovered and given another impulse of investigation due to works of Loday [11, 12] as a
nonantisymmetric version of Lie algebras. Many results of Lie algebras are also established in Leibniz
algebras. Since the study of the properties of derivations and automorphisms of Lie algebras play an
essential role in the theory of Lie algebras, the question naturally arises whether the corresponding
results can be extended to the more general framework of the Leibniz algebras.
In this work we consider some general properties of derivations and automorphisms of Leibniz
algebras. We extend some results obtained for derivations and automorphisms of Lie algebras in
[5, 8] to the case of Leibniz algebras. Among them we prove the analogue of the Jordan – Chevalley
decomposition, which expresses a derivation of a Leibniz algebra as the sum of its commuting
semisimple and nilpotent parts. Similar results were established in [5] and [7] for Lie algebras. If
the linear operator is invertible, then the Jordan – Chevalley decomposition expresses it as a product
of commuting semisimple and unipotent operators. Gantmacher [5] proved that any automorphism
of a Lie algebra decomposes into the product of commuting semisimple automorphism and exponent
of a nilpotent derivation. In this work we verify that the same results hold in Leibniz algebras.
In 1955, Jacobson [8] proved that every Lie algebra over a field of characteristic zero admitting
a nonsingular derivation is nilpotent. The problem whether the converse of this statement is correct
remained open until an example of a nilpotent Lie algebra in which every derivation is nilpotent
(and hence, singular) was constructed in [4]. Nilpotent Lie algebras with this property were named
characteristically nilpotent Lie algebras. In [10] it was proved that every irreducible component of
the variety of complex filiform Lie algebras of dimension greater than 7 contains a Zariski open
set consisting of characteristically nilpotent Lie algebras. Note that among nilpotent Lie algebras of
dimension less than 7, characteristically nilpotent Lie algebras do not occur due to the classification
given in [6].
In this paper we prove that a finite-dimensional complex Leibniz algebra admitting a nondege-
nerate derivation is nilpotent. Similar to the Lie case, the inverse of this statement does not hold.
* The first and the last authors were supported by Ministerio de Economı́a y Competitividad (Spain), grant MTM2013-
43687-P and Xunta de Galicia, grant GRC2013-045 (European FEDER support included).
c\bigcirc M. LADRA, I. M. RIKHSIBOEV, R. M. TURDIBAEV, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7 933
934 M. LADRA, I. M. RIKHSIBOEV, R. M. TURDIBAEV
The notion of characteristically nilpotent Leibniz algebra is defined similarly as in the Lie case. It is
established in [14] that a characteristically nilpotent non-Lie filiform Leibniz algebra occurs starting
with dimension 5.
Some versions of Theorems 3.3 and 3.4 are presented in [15]. We also have found that some of
the presented results of our work are also proved with another techniques in [3].
In the present paper, all vector spaces and algebras are considered over the field of the complex
numbers \BbbC . We will denote by Ci
k the binomial coefficient
\biggl(
k
i
\biggr)
.
2. Preliminaries. In this section we present some known notions and results concerning Leibniz
algebras that we use further in this work.
Definition 2.1. An algebra L over a field \BbbF is called a Leibniz algebra if for any x, y, z \in L,
the Leibniz identity \bigl[
[x, y], z
\bigr]
=
\bigl[
[x, z], y] + [x, [y, z]
\bigr]
is satisfied, where [ - , - ] is the multiplication in L.
In other words, the right multiplication operator [ - , z] by any element z is a derivation (see [11]).
Any Lie algebra is a Leibniz algebra, and conversely any Leibniz algebra L is a Lie algebra if
[x, x] = 0 for all x \in L. Moreover, if L\mathrm{a}\mathrm{n}\mathrm{n} = ideal \langle [x, x] | x \in L\rangle , then the factor algebra L/L\mathrm{a}\mathrm{n}\mathrm{n}
is a Lie algebra.
For a Leibniz algebra L consider the following derived and lower central series:
(i) L(1) = L, L(n+1) = [L(n), L(n)], n > 1;
(ii) L1 = L, Ln+1 = [Ln, L], n > 1.
Definition 2.2. An algebra L is called solvable (nilpotent) if there exists s \in \BbbN (k \in \BbbN ,
respectively) such that L(s) = 0 (Lk = 0, respectively).
For a linear map A of a vector space V we denote by V\lambda = \{ x \in V | (A - \lambda I)k(x) =
= 0 for some k \in \BbbN \} the generalized eigenspace for eigenvalue \lambda of A.
The following proposition provides an additive Jordan – Chevalley decomposition of an endomor-
phism.
Proposition 2.1 [7]. Let V be a finite-dimensional vector space over \BbbC , x \in \mathrm{E}\mathrm{n}\mathrm{d}(V ).
(i) There exist unique xd, xn \in \mathrm{E}\mathrm{n}\mathrm{d}(V ) satisfying the conditions: x = xd + xn, xd is diagonal-
izable, xn is nilpotent, xd and xn commute.
(ii) There exist polynomials p(t), q(t) \in \BbbC [t], without constant term such that xd = p(x) and
xn = q(x). In particular, xd and xn commute with any endomorphism commuting with x.
(iii) If A \subseteq B \subseteq V are subspaces and x maps B in A, then xd and xn also map B in A.
In Leibniz algebras a derivation is defined as usual [12].
Definition 2.3. A linear operator d : L \rightarrow L is called a derivation of L if
d
\bigl(
[x, y]
\bigr)
=
\bigl[
d(x), y] + [x, d(y)
\bigr]
for any x, y \in L.
For an arbitrary element x \in L, we consider the right multiplication operator Rx : L \rightarrow L
defined by Rx(z) = [z, x]. Right multiplication operators are derivations of the algebra L. The set
R(L) = \{ Rx | x \in L\} is a Lie algebra with respect to the commutator and the following identity
holds:
RxRy - RyRx = R[y,x]. (2.1)
Definition 2.4 [9]. A subset S of an associative algebra A over a field \BbbF is called weakly closed
if for every pair (a, b) \in S \times S there exists an element \gamma (a, b) \in \BbbF such that ab+ \gamma (a, b) ba \in S.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
AUTOMORPHISMS AND DERIVATIONS OF LEIBNIZ ALGEBRAS 935
Further, we need a result concerning the weakly closed sets.
Theorem 2.1 [9]. Let S be a weakly closed subset of the associative algebra A of linear
transformations of a finite-dimensional vector space V over \BbbF . Assume every W \in S is nilpotent,
that is, W k = 0 for some positive integer k. Then the enveloping associative algebra S\ast of S is
nilpotent.
The classical Engel’s theorem for Lie algebras has the following analogue in Leibniz algebras.
Theorem 2.2 ([1], Engel’s theorem). A Leibniz algebra L is nilpotent if and only if Rx is nilpo-
tent for any x \in L.
Definition 2.5. The set \mathrm{A}\mathrm{n}\mathrm{n}r(L) =
\bigl\{
x \in L | [L, x] = 0
\bigr\}
of a Leibniz algebra L is called the
right annihilator of L.
One can show that \mathrm{A}\mathrm{n}\mathrm{n}r(L) is an ideal of L.
For a Leibniz algebra L, let H be a maximal solvable ideal in the sense that H contains any
solvable ideal of L. Since the sum of solvable ideals is again a solvable ideal (see [1]), this implies
the existence of the unique maximal solvable ideal, which is called the radical of L.
Similarly, let K be a maximal nilpotent ideal of Leibniz algebra L. Since the sum of nilpotent
ideals is a nilpotent ideal (see [1]), this implies the existence of a unique maximal nilpotent ideal,
which is the nilradical of L.
3. Main result. This section is devoted to the extension of known results for Lie algebras on
automorphisms and derivations to Leibniz algebras.
Lemma 3.1. Let L be a finite-dimensional Leibniz algebra with a derivation d. Then for any
\alpha , \beta \in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(d) we have
[L\alpha , L\beta ] \subseteq
\left\{ L\alpha +\beta , if \alpha + \beta is an eigenvalue of d,
0, if \alpha + \beta is not an eigenvalue of d.
Proof. First observe that
\bigl(
d - (\alpha + \beta )I
\bigr) \bigl(
[x, y]
\bigr)
=
\bigl[
d(x), y
\bigr]
+
\bigl[
x, d(y)
\bigr]
- (\alpha + \beta )[x, y] =
=
\bigl[
(d - \alpha I)(x), y
\bigr]
+
\bigl[
x, (d - \beta I)(y)
\bigr]
. Now assume that
(d - (\alpha + \beta )I)k
\bigl(
[x, y]
\bigr)
=
k\sum
i=0
Ci
k[(d - \alpha I)i(x), (d - \beta I)k - i(y)] (3.1)
for some k > 1. Then
\bigl(
d - (\alpha + \beta )I
\bigr) k+1\bigl(
[x, y]
\bigr)
=
\bigl(
d - (\alpha + \beta )I
\bigr) \Biggl( k\sum
i=0
Ci
k[(d - \alpha I)i(x), (d - \beta I)k - i(y)]
\Biggr)
=
=
k\sum
i=0
Ci
k
\bigl[
(d - \alpha I)i+1(x), (d - \beta I)k - i(y)
\bigr]
+
+
k\sum
i=0
Ci
k
\bigl[
(d - \alpha I)i(x), (d - \beta I)k - i+1(y)
\bigr]
=
=
\bigl[
(d - \alpha I)k+1(x), (y)
\bigr]
+
k - 1\sum
i=0
Ci
k
\bigl[
(d - \alpha I)i+1(x), (d - \beta I)k+1 - (i+1)(y)
\bigr]
+
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
936 M. LADRA, I. M. RIKHSIBOEV, R. M. TURDIBAEV
+
k\sum
i=1
Ci
k
\bigl[
(d - \alpha I)i(x), (d - \beta I)k+1 - i(y)
\bigr]
+
\bigl[
x, (d - \beta I)k+1(y)
\bigr]
=
=
\bigl[
(d - \alpha I)k+1(x), y
\bigr]
+
k\sum
i=1
(Ci - 1
k + Ci
k)\times
\times
\bigl[
(d - \alpha I)i(x), (d - \beta I)k+1 - i(y)
\bigr]
+
\bigl[
x, (d - \beta I)k+1(y)
\bigr]
=
=
\bigl[
(d - \alpha I)k+1(x), y
\bigr]
+
k\sum
i=1
Ci
k+1
\bigl[
(d - \alpha I)i(x), (d - \beta I)k+1 - i(y)
\bigr]
+
\bigl[
x, (d - \beta I)k+1(y)
\bigr]
=
=
k+1\sum
i=0
Ci
k+1
\bigl[
(d - \alpha I)i(x), (d - \beta I)k+1 - i(y)
\bigr]
.
Hence (3.1) holds for any k \in \BbbN .
Consider x \in L\alpha , y \in L\beta . Then there exist natural numbers p, q such that (d - \alpha I)p(x) = 0
and (d - \beta I)q(y) = 0. In (3.1) taking k = p + q we have that
\bigl(
d - (\alpha + \beta )I
\bigr) k\bigl(
[x, y]
\bigr)
= 0 which
completes the proof of the statement of the lemma.
Let d be a derivation of a Leibniz algebra L. From the definition of a derivation it is straightfor-
ward that \mathrm{k}\mathrm{e}\mathrm{r} d is a subalgebra. Moreover, by Lemma 3.1 we have [L0, L0] \subseteq L0 and hence L0 is
also a subalgebra of L.
The following theorem is a generalization of the analogous result in the theory of Lie algebras
established in [5].
Theorem 3.1. Let D be a derivation of a Leibniz algebra L. Then there exists a unique diagonal-
izable derivation D0 and a unique nilpotent derivation T such that D = D0 + T and D0T = TD0.
Proof. Let L = L\rho 1 \oplus . . .\oplus L\rho s be a decomposition of L into characteristic spaces with respect
to d. Let us define a linear operator D0 : L \rightarrow L as D0(x) = \rho ix for x \in L\rho i . Then D0 is obviously
diagonalizable and D0D = DD0.
Now we show that D0 is a derivation of L.
By Lemma 3.1 if x \in L\rho i , y \in L\rho j we obtain [x, y] \in L\rho i+\rho j if \rho i + \rho j is an eigenvalue and
[x, y] = 0 otherwise. If \rho i + \rho j is an eigenvalue of D, then we have
D0
\bigl(
[x, y]
\bigr)
= (\rho i + \rho j)[x, y],\bigl[
D0(x), y
\bigr]
+
\bigl[
x,D0(y)
\bigr]
= [\rho ix, y] + [x, \rho jy] = (\rho i + \rho j)[x, y].
So D0
\bigl(
[x, y]
\bigr)
= [D0(x), y] + [x,D0(y)].
If \rho i+\rho j is not an eigenvalue, then [x, y] = 0 and again we get D0
\bigl(
[x, y]
\bigr)
= 0 and [D0(x), y]+
+ [x,D0(y)] = (\rho i + \rho j)[x, y] = 0. Hence, D0 is a derivation.
Now denote by T = D - D0. Obviously, T is a derivation of L and T is nilpotent. Moreover,
T commutes with D0.
The uniqueness of such decomposition follows from Proposition 2.1.
In order to obtain a similar result for automorphisms of Leibniz algebras we need the following
lemma.
Lemma 3.2. Let P be a nilpotent transformation of a Leibniz algebra L such that P + I is an
automorphism. Then
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
AUTOMORPHISMS AND DERIVATIONS OF LEIBNIZ ALGEBRAS 937
P k
\bigl(
[x, y]
\bigr)
=
k\sum
i=0
i\sum
j=0
Ci
kC
j
i
\bigl[
P k - j(x), P k - i+j(y)
\bigr]
(3.2)
for all k \in \BbbN .
Proof. Let us denote Q = P + I. Since Q is an automorphism we obtain
P
\bigl(
[x, y]
\bigr)
= (Q - I)
\bigl(
[x, y]
\bigr)
=
\bigl[
Q(x), Q(y)
\bigr]
- [x, y] =
=
\bigl[
Q(x) - x,Q(y) - y
\bigr]
+
\bigl[
Q(x) - x, y
\bigr]
+
\bigl[
x,Q(y) - y
\bigr]
=
=
\bigl[
P (x), P (y)
\bigr]
+
\bigl[
P (x), y] + [x, P (y)
\bigr]
=
1\sum
i=0
i\sum
j=0
Ci
1C
j
i
\bigl[
P 1 - j(x), P 1 - i+j(y)
\bigr]
.
Now assume that (3.2) holds for some natural k > 1. Then
P k+1
\bigl(
[x, y]
\bigr)
=
k\sum
i=0
i\sum
j=0
Ci
kC
j
i P
\bigl(
[P k - j(x), P k - i+j(y)]
\bigr)
=
=
k\sum
i=0
Ci
k
i\sum
j=0
Cj
i
\Bigl( \bigl[
P k - j+1(x), P k - i+j+1(y)
\bigr]
+
+
\bigl[
P k - j+1(x), P k - i+j(y)
\bigr]
+
\bigl[
P k - j(x), P k - i+j+1(y)
\bigr] \Bigr)
.
Consider
i\sum
j=0
Cj
i
\bigl[
P k - j+1(x), P k - i+j(y)
\bigr]
+
i\sum
j=0
Cj
i
\bigl[
P k - j(x), P k - i+j+1(y)
\bigr]
=
= C0
i
\bigl[
P k+1(x), P k - i(y)
\bigr]
+
i\sum
j=1
\Bigl(
Cj
i
\bigl[
P k+1 - j(x), P k - i+j(y)
\bigr]
+
+ Cj - 1
i
\bigl[
P k+1 - j(x), P k - i+j(y)
\bigr] \Bigr)
+ Ci
i
\bigl[
P k - i(x), P k+1(y)
\bigr]
=
= C0
i+1
\bigl[
P k+1(x), P k+1 - (i+1)(y)
\bigr]
+
+
i\sum
j=1
\Bigl(
Cj
i + Cj - 1
i
\Bigr) \bigl[
P k+1 - j(x), P k+1 - (i+1)+j(y)
\bigr]
+
+ Ci+1
i+1
\bigl[
P k+1 - (i+1)(x), P k+1(y)
\bigr]
.
Using the fact Cj
i + Cj - 1
i = Cj
i+1 we have
i\sum
j=0
Cj
i
\bigl[
P k - j+1(x), P k - i+j(y)
\bigr]
+
i\sum
j=0
Cj
i
\bigl[
P k - j(x), P k - i+j+1(y)
\bigr]
=
=
i+1\sum
j=0
Cj
i+1
\bigl[
P k+1 - j(x), P k+1 - (i+1)+j(y)
\bigr]
.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
938 M. LADRA, I. M. RIKHSIBOEV, R. M. TURDIBAEV
Now
P k+1
\bigl(
[x, y]
\bigr)
=
k\sum
i=0
i\sum
j=0
Ci
kC
j
i
\bigl[
P k - j+1(x), P k - i+j+1(y)
\bigr]
+
+
k\sum
i=0
i+1\sum
j=0
Ci
kC
j
i+1
\bigl[
P k+1 - j(x), P k+1 - (i+1)+j(y)
\bigr]
=
=
\bigl[
P k+1(x), P k+1(y)
\bigr]
+
k - 1\sum
i=0
i+1\sum
j=0
Ci+1
k Cj
i+1
\bigl[
P k - j+1(x), P k+1 - (i+1)+j(y)
\bigr]
+
+
k - 1\sum
i=0
i+1\sum
j=0
Ci
kC
j
i+1
\bigl[
P k+1 - j(x), P k+1 - (i+1)+j(y)
\bigr]
+
k+1\sum
j=0
Cj
k+1
\bigl[
P k+1 - j(x), P j(y)
\bigr]
=
=
\bigl[
P k+1(x), P k+1(y)
\bigr]
+
k - 1\sum
i=0
i+1\sum
j=0
\bigl(
Ci+1
k + Ci
k
\bigr)
Cj
i+1\times
\times
\bigl[
P k - j+1(x), P k+1 - (i+1)+j(y)
\bigr]
+
k+1\sum
j=0
Cj
k+1
\bigl[
P k+1 - j(x), P j(y)
\bigr]
=
=
\bigl[
P k+1(x), P k+1(y)
\bigr]
+
k - 1\sum
i=0
i+1\sum
j=0
Ci+1
k+1C
j
i+1
\bigl[
P k - j+1(x), P k+1 - (i+1)+j(y)
\bigr]
+
+
k+1\sum
j=0
Cj
k+1
\bigl[
P k+1 - j(x), P j(y)
\bigr]
=
=
\bigl[
P k+1(x), P k+1(y)
\bigr]
+
k\sum
i=1
i\sum
j=0
Ci
k+1C
j
i
\bigl[
P k - j+1(x), P k+1 - i+j(y)
\bigr]
+
+
k+1\sum
j=0
Cj
k+1
\bigl[
P k+1 - j(x), P j(y)
\bigr]
=
=
k+1\sum
i=0
i\sum
j=0
Ci
k+1C
j
i
\bigl[
P k - j+1(x), P k+1 - i+j(y)
\bigr]
.
Thus, (3.2) is proved.
The next lemma presents the similar result for automorphisms of Leibniz algebras as Lemma 3.1
does for derivations. Notice that, it also generalizes the result for Lie algebras given in [5].
Lemma 3.3. Let L be a finite-dimensional Leibniz algebra and A be an automorphism. Then
for any \alpha , \beta \in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(A) we have
[L\alpha , L\beta ] \subseteq
\left\{ L\alpha \beta , if \alpha \beta is an eigenvalue of A,
0, if \alpha \beta is not an eigenvalue of A.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
AUTOMORPHISMS AND DERIVATIONS OF LEIBNIZ ALGEBRAS 939
Proof. First observe that
(A - \alpha \beta I)
\bigl(
[x, y]
\bigr)
=
\bigl[
A(x), A(y)
\bigr]
- \alpha \beta [x, y] =
=
\bigl[
(A - \alpha I)(x), (A - \beta I)(y)
\bigr]
+
\bigl[
(A - \alpha I)(x), \beta y
\bigr]
+
\bigl[
\alpha x, (A - \beta I)(y)
\bigr]
.
Similarly to the proof of Lemma 3.2 one can establish by induction
(A - \alpha \beta I)k
\bigl(
[x, y]
\bigr)
=
k\sum
j=0
j\sum
i=0
\alpha i\beta j - iCj
kC
i
j
\bigl[
(A - \alpha I)k - i(x), (A - \beta I)k - j+i(y)
\bigr]
. (3.3)
Now let x \in L\alpha and y \in L\beta . Then there exist natural numbers p, q such that (A - \alpha I)p(x) = 0
and (A - \beta I)q(y) = 0. In (3.3) taking k = p + q we have that (A - \alpha \beta I)k
\bigl(
[x, y]
\bigr)
= 0 which
completes the proof of the lemma.
Below, we establish a technical lemma and a corollary in order to obtain a similar result to
Theorem 3.1 for automorphisms of Leibniz algebra.
Lemma 3.4. For any polynomial P of degree less than n, where n \in \BbbN , the following equality
holds:
n\sum
i=0
( - 1)iCi
nP (i) = 0.
Proof. Since \mathrm{d}\mathrm{e}\mathrm{g}P (x) < n, applying Lagrange interpolation formula to the points xk = k,
0 \leq k \leq n - 1 we get P (x) =
\sum n - 1
k=0
qk(x)P (k), where
qk(x) =
1
( - 1)n - 1 - kk!(n - k)!
x(x - 1) . . .
\bigl(
x - (k - 1)
\bigr) \bigl(
x - (k + 1)
\bigr)
. . .
\bigl(
x - (n - 1)
\bigr)
.
Now qk(n) =
n!
( - 1)n - 1 - kk!(n - k)!
=
1
( - 1)n - 1
( - 1)kCk
n.
Thus,
P (n) =
n - 1\sum
k=0
qk(n)P (k) = ( - 1)n - 1
n - 1\sum
k=0
( - 1)kCk
nP (k).
Hence,
0 =
n - 1\sum
k=0
( - 1)kCk
nP (k) + ( - 1)nCn
nP (n) =
n\sum
i=0
( - 1)iCi
nP (i).
Corollary 3.1. Let n,m be nonnegative integers such that n < m. Then
n\sum
i=0
( - 1)i
m - i
Ci
nC
n
m - i =
\left\{
1
m
, if n = 0,
0, otherwise.
Proof. Let n > 1 and consider the polynomial
P (x) =
1
n!
(m - 1 - x)(m - 2 - x) . . . (m - (n - 1) - x) =
1
m - x
Cn
m - x
of degree n - 1.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
940 M. LADRA, I. M. RIKHSIBOEV, R. M. TURDIBAEV
By Lemma 3.4 we obtain
0 =
n\sum
i=0
( - 1)iCi
nP (i) =
n\sum
i=0
( - 1)i
m - i
Ci
nC
n
m - i .
For n = 0, 1, simple calculations verify the statement of the corollary.
The following result shows that the analogous one established for Lie algebras [5] is also valid
for Leibniz algebras.
Theorem 3.2. Let A be an automorphism of a Leibniz algebra. Then there exists a unique
diagonalizable automorphism A0 and a unique nilpotent derivation T such that A = A0 \mathrm{e}\mathrm{x}\mathrm{p}(T ) and
A0T = TA0.
Proof. Let L = L\rho 1 \oplus . . . \oplus L\rho s be a decomposition of a Leibniz algebra L into generalized
eigenspaces with respect to A.
Define a linear map A0 : L \rightarrow L as A0(x) = \rho ix for x \in L\rho i . Then A0 is obviously diagonal-
izable and A0A = AA0. Notice that if x \in L\rho i , y \in L\rho j , then [A0(x), A0(y)]) = \rho i\rho j [x, y] and by
Lemma 3.3 we have [x, y] \in L\rho i\rho j . Therefore, A0
\bigl(
[x, y]
\bigr)
= \rho i\rho j [x, y], which implies that A0 is an
automorphism.
Let us denote by Q = A - 1
0 A. Then A = A0Q and A0Q = QA0. Also note that \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(Q) = \{ 1\} .
Consider P = Q - I. Obviously, P is nilpotent and hence
\mathrm{l}\mathrm{o}\mathrm{g}Q = \mathrm{l}\mathrm{o}\mathrm{g}(I + P ) = P - 1
2
P 2 + . . .+
( - 1)n - 1
n
Pn + . . .
diverges.
Since P is nilpotent, \mathrm{l}\mathrm{o}\mathrm{g}Q is also a nilpotent transformation. We will prove that \mathrm{l}\mathrm{o}\mathrm{g}(I + P ) is
a derivation, i.e.,
\infty \sum
k=1
( - 1)k - 1
k
P k
\bigl(
[x, y]
\bigr)
=
\Biggl[ \infty \sum
k=1
( - 1)k - 1
k
P k(x), y
\Biggr]
+
\Biggl[
x,
\infty \sum
k=1
( - 1)k - 1
k
P k(y)
\Biggr]
. (3.4)
By Lemma 3.2, terms on both sides of the formula (3.4) for k = 1 vanish. Setting Ci
k = Ck - i
k
and substituting r = k - i we get P k
\bigl(
[x, y]
\bigr)
=
\sum k
r=0
\sum k - r
j=0
Cr
kC
j
k - r
\bigl[
P k - j(x), P r+j(y)
\bigr]
. Now
denote by Bk,r =
\sum k - r
j=0
Cr
kC
j
k - r
\bigl[
P k - j(x), P j+r(y)
\bigr]
for all 0 \leq r \leq k. Then P k
\bigl(
[x, y]
\bigr)
=
= Bk,0 +Bk,1 + . . .+Bk,k. Therefore,
\infty \sum
k=1
( - 1)k - 1
k
P k
\bigl(
[x, y]
\bigr)
=
\infty \sum
k=1
( - 1)k - 1
k
(Bk,0 +Bk,1 + . . .+Bk,k) =
=
\infty \sum
m=0
\biggl(
1
2m+ 1
B2m+1,0 -
1
2m
B2m,1 + . . .+
( - 1)m
m+ 1
Bm+1,m
\biggr)
-
-
\infty \sum
m=1
\biggl(
1
2m
B2m,0 -
1
2m - 1
B2m - 1,1 + . . .+
( - 1)m
m
Bm,m
\biggr)
=
=
\infty \sum
m=0
\Biggl(
m\sum
t=0
( - 1)t
2m+ 1 - t
B2m+1 - t,t
\Biggr)
-
\infty \sum
m=1
\Biggl(
m\sum
t=0
( - 1)t
2m - t
B2m - t,t
\Biggr)
=
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
AUTOMORPHISMS AND DERIVATIONS OF LEIBNIZ ALGEBRAS 941
=
\infty \sum
m=0
\left( m\sum
t=0
( - 1)t
2m+ 1 - t
2m+1 - 2t\sum
j=0
Ct
2m+1 - tC
j
2m+1 - 2t
\bigl[
P 2m+1 - t - j(x), P j+t(y)
\bigr] \right) -
-
\infty \sum
m=1
\left( m\sum
t=0
( - 1)t
2m - t
2m - 2t\sum
j=0
Ct
2m - tC
j
2m - 2t
\bigl[
P 2m - t - j(x), P j+t(y)
\bigr] \right) =
=
\infty \sum
m=0
\Biggl(
1
2m+ 1
\bigl[
P 2m+1(x), y
\bigr]
+
2m\sum
s=1
\Biggl(
s\sum
t=0
( - 1)t
2m+ 1 - t
Ct
2m+1 - t \times Cs - t
2m+1 - 2t
\Biggr)
\times
\times
\bigl[
P 2m+1 - s(x), P s(y)
\bigr]
+
1
2m+ 1
\bigl[
x, P 2m+1(y)
\bigr] \biggr)
-
\infty \sum
m=1
\biggl(
1
2m
[P 2m(x), y] +
+
2m - 1\sum
s=1
\Biggl(
s\sum
t=0
( - 1)t
2m - t
Ct
2m - tC
s - t
2m - 2t
\Biggr) \bigl[
P 2m - s(x), P s(y)
\bigr]
+
1
2m
\bigl[
x, P 2m(y)
\bigr] \Biggr)
.
Now since Ct
2m+1 - tC
s - t
2m+1 - 2t = Ct
sC
s
2m+1 - t and Ct
2m - tC
s - t
2m - 2t = Ct
sC
s
2m - t we obtain
s\sum
t=0
( - 1)t
2m+ 1 - t
Ct
2m+1 - tC
s - t
2m+1 - 2t =
s\sum
t=0
( - 1)t
2m+ 1 - t
Ct
sC
s
2m+1 - t,
s\sum
t=0
( - 1)t
2m - t
Ct
2m - tC
s - t
2m - 2t =
s\sum
t=0
( - 1)t
2m - t
Ct
sC
s
2m - t.
However, by Corollary 3.1 the last sums are zero for all 1 \leq s \leq 2m (1 \leq s \leq 2m - 1,
respectively). Hence,
\infty \sum
k=1
( - 1)k - 1
k
P k
\bigl(
[x, y]
\bigr)
=
\infty \sum
n=1
( - 1)n - 1
n
\Bigl( \bigl[
Pn(x), y
\bigr]
+
\bigl[
x, Pn(y)
\bigr] \Bigr)
=
=
\Biggl[ \infty \sum
n=1
( - 1)n - 1
n
Pn(x), y
\Biggr]
+
\Biggl[
x,
\infty \sum
n=1
( - 1)n - 1
n
Pn(y)
\Biggr]
and (3.4) is proved.
Thus, T = \mathrm{l}\mathrm{o}\mathrm{g}Q is a nilpotent derivation of L and A = A0 \mathrm{e}\mathrm{x}\mathrm{p}(T ), A0T = TA0. Now
since \mathrm{e}\mathrm{x}\mathrm{p}(T ) - I is nilpotent, we get the additive Jordan – Chevalley decomposition A = A0 +
+ A0(\mathrm{e}\mathrm{x}\mathrm{p}(T ) - I) of A. Therefore, by Proposition 2.1, A0 and as consequence T, are determined
uniquely.
The following theorems generalize the results from the theory of Lie algebras [8] to Leibniz
algebras.
Theorem 3.3. Let L be a finite-dimensional complex Leibniz algebra which admits a derivation.
Then L is a nilpotent algebra.
Proof. Let d be a nonsingular derivation of a Leibniz algebra L and L = L\rho 1 \oplus L\rho 2 \oplus \cdot \cdot \cdot \oplus L\rho k
be a decomposition of L into generalized eigenspace spaces with respect to d.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
942 M. LADRA, I. M. RIKHSIBOEV, R. M. TURDIBAEV
Let \alpha , \beta \in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(d). Then by Lemma 3.1 we obtain [. . . [[L\alpha , L\beta ], L\beta ], . . . , L\beta ]\underbrace{} \underbrace{}
k times
\subseteq L\alpha +k\beta .
Since for sufficiently large k \in \BbbN we have \alpha + k\beta \not \in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(d), and by Lemma 3.1 we get\bigl[
. . .
\bigl[
[L\alpha , L\beta ], L\beta
\bigr]
, . . . , L\beta
\bigr]
= 0.
Thus, for x \in L\beta any right multiplication operator Rx is nilpotent, and due to the fact that \alpha , \beta
were taken arbitrarily, it follows that every operator from
\bigcup k
i=1R(L\rho i) is nilpotent.
Now from identity (2.1) and Lemma 3.1 it follows that
\bigcup k
i=1R(L\rho i) is a weakly closed set of
an associative algebra R(L). Hence, by Theorem 2.1 it follows that every operator from R(L) is
nilpotent.
Now by Theorem 2.2 we obtain the result, i.e., L is nilpotent.
Remark 3.1. The following family L(\beta ) = \langle e1, . . . , en\rangle of characteristically nilpotent Leibniz
algebras, i.e., algebras with all derivations being nilpotent, with the following multiplication:
[e0, e0] = e2, [ei, e0] = ei+1, 1 \leq i \leq n - 1,
[e0, e1] = \alpha 3e3 + \alpha 4e4 + . . .+ \alpha n - 1en - 1 + \theta en,
[ei, e1] = \alpha 3ei+2 + \alpha 4ei+3 + . . .+ \alpha n+1 - ien, 1 \leq i \leq n - 2,
where (\alpha 3, . . . , \alpha n, \theta \in \BbbC ) and \alpha i\alpha j \not = 0 for some 3 \leq i \not = j \leq n, was constructed in [14]. This
implies that the statement of Theorem 3.3 in the opposite direction does not hold.
Theorem 3.4. Let L be a finite-dimensional complex Leibniz such that it admits an automorphism
of a prime order with no fixed points. Then L is a nilpotent algebra.
Proof. Let A be an automorphism of a Leibniz algebra L with the properties given in the
statement of the theorem. Since A has no fixed points then 1 is not an eigenvalue of A.
Let L = L\rho 1 \oplus L\rho 2 \oplus . . .\oplus L\rho k be a decomposition of L into generalized eigenspaces with respect
to A. From the condition that A is an automorphism of prime order we obtain that the spectrum of
A consists of primitive pth roots of unity. Therefore, for any \alpha , \beta \in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(A) there exists k \in \BbbN
such that \alpha \beta k = 1 \not \in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(A). Hence, by Lemma 3.3 we have\bigl[
. . . [[L\alpha , L\beta ], L\beta ], . . . , L\beta
\bigr] \underbrace{} \underbrace{}
k times
\subseteq L\alpha \beta k = 0.
Thus, for x \in L\beta any right multiplication operator Rx is nilpotent, and similarly as in the proof
of Theorem 3.3 we obtain that L is nilpotent.
Let D be a derivation of a Leibniz algebra L such that D commutes with any inner derivation.
Then D(L) \subseteq \mathrm{A}\mathrm{n}\mathrm{n}r(L). Indeed, since D commutes with any right multiplication operator we
get [D(x), y] = (Ry \circ D)(x) = (D \circ Ry)(x) = D
\bigl(
[x, y]
\bigr)
= [D(x), y] + [x,D(y)] which implies
[x,D(y)] = 0 for any x, y \in L. Thus, [L,D(L)] = 0 and D(L) \subseteq \mathrm{A}\mathrm{n}\mathrm{n}r(L).
Lemma 3.5. Let J be an ideal of a Leibniz algebra L and D be a derivation given on L. Then
J +D(J) is also an ideal of L.
Proof. Since for any x \in J, y \in L we have
[y,D(x)] = D
\bigl(
[x, y]
\bigr)
- [D(x), y] \in D
\bigl(
[J, L]
\bigr)
+ [J, L] \subseteq D(J) + J,
and so
\bigl[
L,D(J)
\bigr]
\subseteq D(J) + J. Therefore,
\bigl[
L, J +D(J)
\bigr]
\subseteq J +D(J).
Similarly, since for any x \in J, y \in L we obtain
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
AUTOMORPHISMS AND DERIVATIONS OF LEIBNIZ ALGEBRAS 943\bigl[
D(x), y
\bigr]
= D
\bigl(
[x, y]
\bigr)
-
\bigl[
x,D(y)
\bigr]
\in D
\bigl(
[J, L]
\bigr)
+ [J, L] \subseteq D(J) + J,
and so [D(J), L] \subseteq D(J) + J. Therefore, [J +D(J), L] \subseteq J +D(J). This implies that J +D(J)
is an ideal of L.
Theorem 3.5. Let J be the solvable radical of a Leibniz algebra L and D be a derivation.
Then D(J) \subseteq J.
Proof. By Lemma 3.5 it follows that J +D(J) is an ideal of Leibniz algebra L. We get\bigl(
J +D(J)
\bigr) (2)
= [J +D(J), J +D(J)] \subseteq J + [D(J), D(J)] \subseteq J +D2(J (2)).
Now assume that \bigl(
J +D(J)
\bigr) (k) \subseteq J +D2k - 1\bigl(
J (k)
\bigr)
(3.5)
for some natural k > 1. Then\bigl(
J +D(J)
\bigr) (k+1)
=
\bigl[
(J +D(J))(k),
\bigl(
J +D(J))(k)
\bigr]
\subseteq
\subseteq
\bigl[
J +D2k - 1
(J (k)), J +D2k - 1
(J (k))
\bigr]
\subseteq J +
\bigl[
D2k - 1\bigl(
J (k)
\bigr)
, D2k - 1\bigl(
J (k)
\bigr) \bigr]
\subseteq
\subseteq J +D2k - 1+2k - 1\bigl(
[J (k), J (k)]
\bigr)
= J +D2k
\bigl(
J (k+1)
\bigr)
.
Hence, (3.5) is verified.
Let J (m) = 0. Then
\bigl(
J + D(J)
\bigr) (m) \subseteq J + D2m - 1\bigl(
J (m)
\bigr)
= J. Now
\bigl(
J + D(J)
\bigr) (2m - 1)
=
=
\bigl(
(J +D(J))(m)
\bigr) (m) \subseteq J (m) = 0.
Hence, J +D(J) is a solvable ideal of Leibniz algebra L. Since J is the solvable radical of L,
it follows that J +D(J) \subseteq J and therefore, D(J) \subseteq J.
Remark 3.2. In Theorem 3.5 if J is the nilradical, analogous arguments establish the invariance
of J with respect to any derivation of L.
It is not difficult to verify that a derivation in a Leibniz algebra induces a derivation in the
corresponding Lie quotient algebra. However, the following example shows that the inverse is not
necessarily true, i.e., not every derivation in the Lie quotient algebra can be extended to a derivation
of the Leibniz algebra.
Example 3.1. Consider a Leibniz algebra L = \langle e1, . . . , em, f1, . . . , fm\rangle with the following mul-
tiplication:
[ei, ei] = fi, 1 \leq i \leq m, [e1, ei] = fi, 1 \leq i \leq m, and 0 in other case.
Then L\mathrm{a}\mathrm{n}\mathrm{n} = \langle f1, . . . , fm\rangle and L/L\mathrm{a}\mathrm{n}\mathrm{n} is an Abelian Lie algebra. Therefore, any linear operator in
L/L\mathrm{a}\mathrm{n}\mathrm{n} is a derivation.
Now consider an arbitrary derivation d : L \rightarrow L. Since [ep, e1] = 0 for p > 1, we have that
0 = d
\bigl(
[ep, e1]
\bigr)
=
\bigl[
d(ep), e1
\bigr]
+
\bigl[
ep, d(e1)
\bigr]
.
If d(ep) = d1pe1 + . . .+ dmpem + c1pf1 + . . .+ cmpfm, then
\bigl[
d(ep), e1
\bigr]
= d1p[e1, e1] = d1pf1.
Now if d(e1) = d11e1+ . . .+dm1em+ c11f1+ . . .+ cm1fm, then
\bigl[
ep, d(e1)
\bigr]
= dp1[ep, ep] = dp1fp.
Hence we obtain a condition d1pf1 + dp1fp = 0 which implies d1p = dp1 = 0 for all 2 \leq p \leq m.
Therefore, not every derivation of L/L\mathrm{a}\mathrm{n}\mathrm{n} can be extended to L.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
944 M. LADRA, I. M. RIKHSIBOEV, R. M. TURDIBAEV
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Received 25.02.13,
after revision — 16.04.16
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
|
| id | umjimathkievua-article-1891 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:14:43Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/90/a7e78a6948da5eafae41576e72d97a90.pdf |
| spelling | umjimathkievua-article-18912019-12-05T09:30:55Z Automorphisms and derivations of Leibniz algebras Автоморфiзми та похiднi алгебр Лейбнiца Ladra, M. Rikhsiboev, I. M. Turdibaev, R. M. Ладра, М. Ріхсибоєв, І. М. Турдібаєв, Р. М. We extend some general properties of automorphisms and derivations known for the Lie algebras to finite-dimensional complex Leibniz algebras. The analogs of the Jordan – Chevalley decomposition for derivations and the multiplicative decomposition for automorphisms of finite-dimensional complex Leibniz algebras are obtained. Деякi загальнi властивостi автоморфiзмiв та похiдних, що вiдомi для алгебр Лi, розширeно на випадок комплексних алгебр Лейбнiца. Встановлено аналоги розкладу Джордана – Шевальє для похiдних та мультиплiкативного розкладу для автоморфiзмiв скiнченновимiрних комплексних алгебр Лейбнiца. Institute of Mathematics, NAS of Ukraine 2016-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1891 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 7 (2016); 933-944 Український математичний журнал; Том 68 № 7 (2016); 933-944 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1891/873 Copyright (c) 2016 Ladra M.; Rikhsiboev I. M.; Turdibaev R. M. |
| spellingShingle | Ladra, M. Rikhsiboev, I. M. Turdibaev, R. M. Ладра, М. Ріхсибоєв, І. М. Турдібаєв, Р. М. Automorphisms and derivations of Leibniz algebras |
| title | Automorphisms and derivations of Leibniz
algebras |
| title_alt | Автоморфiзми та похiднi алгебр Лейбнiца |
| title_full | Automorphisms and derivations of Leibniz
algebras |
| title_fullStr | Automorphisms and derivations of Leibniz
algebras |
| title_full_unstemmed | Automorphisms and derivations of Leibniz
algebras |
| title_short | Automorphisms and derivations of Leibniz
algebras |
| title_sort | automorphisms and derivations of leibniz
algebras |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1891 |
| work_keys_str_mv | AT ladram automorphismsandderivationsofleibnizalgebras AT rikhsiboevim automorphismsandderivationsofleibnizalgebras AT turdibaevrm automorphismsandderivationsofleibnizalgebras AT ladram automorphismsandderivationsofleibnizalgebras AT ríhsiboêvím automorphismsandderivationsofleibnizalgebras AT turdíbaêvrm automorphismsandderivationsofleibnizalgebras AT ladram avtomorfizmitapohidnialgebrlejbnica AT rikhsiboevim avtomorfizmitapohidnialgebrlejbnica AT turdibaevrm avtomorfizmitapohidnialgebrlejbnica AT ladram avtomorfizmitapohidnialgebrlejbnica AT ríhsiboêvím avtomorfizmitapohidnialgebrlejbnica AT turdíbaêvrm avtomorfizmitapohidnialgebrlejbnica |