Some aspects of Leibniz algebra theory
One of the key tendencies in the development of Leibniz algebra theory is the search for analogues of the basic results of Lie algebra theory. In this survey, we consider the reverse situation. Here the main attention is paid to the results reflecting the difference of the Leibniz algebras from the...
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| author | Kirichenko, V.V. Kurdachenko, L.A. Pypka, A.A. Subbotin, I.Ya. |
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| citation_txt | Some aspects of Leibniz algebra theory / V.V. Kirichenko, L.A. Kurdachenko, A.A. Pypka, I.Ya. Subbotin // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 1. — С. 1-33. — Бібліогр.: 48 назв. — англ. |
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| description | One of the key tendencies in the development of Leibniz algebra theory is the search for analogues of the basic results of Lie algebra theory. In this survey, we consider the reverse situation. Here the main attention is paid to the results reflecting the difference of the Leibniz algebras from the Lie algebras.
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Algebra and Discrete Mathematics SURVEY ARTICLE
Volume 24 (2017). Number 1, pp. 1–33
c© Journal “Algebra and Discrete Mathematics”
Some aspects of Leibniz algebra theory
Vladimir V. Kirichenko, Leonid A. Kurdachenko,
Aleksandr A. Pypka and Igor Ya. Subbotin
To Professor N. N. Semko on the occasion of his 60th birthday
Abstract. One of the key tendencies in the development
of Leibniz algebra theory is the search for analogues of the basic
results of Lie algebra theory. In this survey, we consider the reverse
situation. Here the main attention is paid to the results reflecting
the difference of the Leibniz algebras from the Lie algebras.
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 [13–15], in
which he called them the D-algebras. However, in that time these works
were not in demand, and they have not been properly developed. Only
after two decades, a real interest to Leibniz algebras rose. It happened
2010 MSC: 17A32, 17A60.
Key words and phrases: Leibniz algebra, cyclic Leibniz algebra, left (right)
center, lower (upper) central series, finite dimensional Leibniz algebra, nilpotent Leib-
niz algebra, extraspecial Leibniz algebra, bilinear form, left (right) idealizer, Frattini
subalgebra, nil-radical, nil-algebra, soluble Leibniz algebra, left (right) subideal, Leibniz
T -algebra, Baer radical.
2 Some aspects of Leibniz algebra theory
thanks to the work of J.-L. Loday [33] (see also [34, 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 differentiation of functions. Later, some authors used to call these
algebras by Loday algebras though J.-L. Loday himself, sometimes under
the nom-de-plume Guillaume William Zinbiel (here Zinbiel is the inverse
of Leibniz), in survey [48] noted that this term does not fit.
An algebra R over a field F is called right Leibniz algebra if it satisfies
the (right) Leibniz identity
[a, [b, c]] = [[a, b], c] − [[a, c], b]
for all a, b, c ∈ R.
Note at once that the classes of left Leibniz algebras and right Leibniz
algebras are different. The following simple example justifies it.
Example 1. Let F be an arbitrary field and L be a vector space over F
having a basis {a, b}. Define the operation [·, ·] on L by the following rule:
[a, a] = [a, b] = b, [b, a] = [b, b] = 0.
It is not hard to check that L becomes a left Leibniz algebra. But
0 = [[a, a], a] 6= [[a, a], a] + [a, [a, a]] = [a, b] = b.
Let R be a right Leibniz algebra, then put ⊂ a, b ⊃= [b, a]. Then we
have
⊂⊂ a, b ⊃, c ⊃ = [c, [b, a]] = [[c, b], a] − [[c, a], b]
=⊂ a, ⊂ b, c ⊃⊃ − ⊂ b, ⊂ a, c ⊃⊃ .
Thus, this substitution leads us to a left Leibniz algebra. Similarly, we
can make a transfer from a left Leibniz algebra to a right Leibniz algebra.
An algebra L over a field F is called a symmetric Leibniz algebra if it
is both a left and right Leibniz algebra.
We prefer to work with left Leibniz algebras even though many authors
prefer to consider right Leibniz algebras. The choice of left Leibniz algebras
is more suitable for us because they have more visible relationships with
the differentiation of products (in which the differential operator is written
to the right of a differentiable object).
Thus, in this article, the term a Leibniz algebra stands for a left
Leibniz algebra.
Kirichenko, Kurdachenko, Pypka, Subbotin 3
The Leibniz algebras appeared to be naturally related to several areas
such as differential geometry, homological algebra, classical algebraic
topology, algebraic K-theory, loop spaces, non-commutative geometry,
and so on. They found some applications in physics (see, for example,
[16, 23, 24]). The theory of Leibniz algebras has been developing quite
intensively but un-even. On one hand, some analogues of important results
from the theory of Lie algebras were proven. On the other hand, natural
questions about the structure of Leibniz algebras are not considered. For
example, until very recently, the cyclic subalgebras of Leibniz algebras were
not fully described. In this survey, we want to gather and to systematize
the main results that clarify to some extent the structure of Leibniz
algebras. We will not touch issues related to the study of homological
problems, we will not focus on the connections of Leibniz algebras, as
well as issues related to the applications of Leibniz algebras. Note, that
most of the results obtained to date relate to finite dimensional algebras.
We will try to focus on the overall results, i.e. the results that hold for
both finite dimensional and infinite dimensional algebras. Our goal is to
see which parts of the picture involving the general structure of Leibniz
algebras have already been drawn, and this will allow us to see which
parts of this picture should be drawn further. Many results on of Lie
algebras are practically unchanged carried over to Leibniz algebras. But
we would like to draw attention not to results of this kind, but to results
showing the differences between Leibniz algebras and Lie algebras.
Note at once that if L is a Lie algebra, then
[[a, b], c] + [[b, c], a] + [[c, a], b] = 0.
It follows that
[[a, b], c] = −[[b, c], a]−[[c, a], b] = [a, [b, c]]+[b, [c, a]] = [a, [b, c]]−[b, [a, c]],
which shows that every Lie algebra is a Leibniz algebra.
Conversely, suppose that [a, a] = 0 for each element a ∈ L. Then for
arbitrary elements a, b ∈ L we have
0 = [a + b, a + b] = [a, a] + [a, b] + [b, a] + [b, b] = [a, b] + [b, a].
It follows that [a, b] = −[b, a]. Then we obtain
0 = [[a, b], c] − [a, [b, c]] + [b, [a, c]]
= [[a, b], c] + [[b, c], a] − [[a, c], b]
= [[a, b], c] + [[b, c], a] + [[c, a], b]
4 Some aspects of Leibniz algebra theory
for all a, b, c ∈ L. Thus, Lie algebras can be characterized as Leibniz
algebras in which [a, a] = 0 for every element a ∈ L.
Like Lie algebras, Leibniz algebras are also associated with associative
algebras, but this connection is a little more complicated.
Let A be an associative algebra over a field F and let f : A → A be
an endomorphism of A such that f2 = f . Define the binary operation
[·, ·] on A by the following rule:
[a, b] = f(a)b − bf(a)
for all elements a, b ∈ A. We have
[[a, b], c] = [f(a)b − bf(a), c]
= f(f(a)b − bf(a))c − cf(f(a)b − bf(a))
= f(f(a)b)c − f(bf(a))c − cf(f(a)b) + cf(bf(a))
= f(a)f(b)c − f(b)f(a)c − cf(a)f(b) + cf(b)f(a);
[a, [b, c]] = [a, f(b)c − cf(b)]
= f(a)(f(b)c − cf(b)) − (f(b)c − cf(b))f(a)
= f(a)f(b)c − f(a)cf(b) − f(b)cf(a) + cf(b)f(a);
[b, [a, c]] = [b, f(a)c − cf(a)]
= f(b)(f(a)c − cf(a)) − (f(a)c − cf(a))f(b)
= f(b)f(a)c − f(b)cf(a) − f(a)cf(b) + cf(a)f(b).
Then
[a, [b, c]] − [b, [a, c]] = f(a)f(b)c − f(a)cf(b) − f(b)cf(a) + cf(b)f(a)
− (f(b)f(a)c − f(b)cf(a) − f(a)cf(b) + cf(a)f(b))
= f(a)f(b)c − f(a)cf(b) − f(b)cf(a) + cf(b)f(a)
− f(b)f(a)c + f(b)cf(a) + f(a)cf(b) − cf(a)f(b))
= f(a)f(b)c + cf(b)f(a) − f(b)f(a)c − cf(a)f(b)
= [[a, b], c].
Thus, with respect to the operations + and [·, ·] A becomes a Leibniz
algebra.
Note that if f is the identity permutation of A, then we obtain a
standard transition from associative algebras to Lie algebras.
Note the following useful property of the elements of Leibniz algebras.
We have
[a, [b, c]] = [[a, b], c] + [b, [a, c]],
[b, [a, c]] = [[b, a], c] + [a, [b, c]]
Kirichenko, Kurdachenko, Pypka, Subbotin 5
or
[a, [b, c]] = [b, [a, c]] − [[b, a], c].
It follows that
[[a, b], c] + [b, [a, c]] = [b, [a, c]] − [[b, a], c],
and hence
[[a, b], c] = −[[b, a], c].
A Leibniz algebra L is called abelian (or trivial) if [a, b] = 0 for every
elements a, b ∈ L. In particular, an abelian Leibniz algebra is a Lie
algebra.
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] 6 A.
Let L be a Leibniz algebra over a field F , M be non-empty subset of
L, then 〈M〉 denote 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] 6 A (respectively
[A, L] 6 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, that is [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.
Denote by Leib(L) the subspace, generated by the elements [a, a],
a ∈ L. We note that Leib(L) is an ideal of L. Indeed, for arbitrary
elements a, x ∈ L we have
[a, [a, x]] = [[a, a], x] + [a, [a, x]],
so [[a, a], x] = 0. Furthermore,
[x + [a, a], x + [a, a]] = [x, x] + [x, [a, a]] + [[a, a], x] + [[a, a], [a, a]]
= [x, x] + [x, [a, a]].
It follows that [x, [a, a]] = [x + [a, a], x + [a, a]] − [x, x] ∈ Leib(L).
Put K = Leib(L). Then in factor-algebra L/K we have
[a + K, a + K] = [a, a] + K = K
6 Some aspects of Leibniz algebra theory
for each element a ∈ L. By mentioned above we obtain that L/K is a Lie
algebra. Conversely, suppose that H is an ideal of L such that L/H is a
Lie algebra. Then
H = [a + H, a + H] = [a, a] + H,
which implies that [a, a] ∈ H for every element a ∈ L. Then Leib(L) 6 H.
The ideal Leib(L) is called the Leibniz kernel of algebra L.
We note the 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 abelian subalgebra of L.
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) for the
limit ordinals λ. 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.
Proposition 1. Let L be a Leibniz algebra over a field F .
(i) If H is an ideal of L, then [H, H] is an ideal of L.
(ii) If H is an ideal of L, then [L, H] is a subalgebra of L.
(iii) If H is an ideal of L, then [H, L] is a subalgebra of L.
(iv) If H is an ideal of L, then [L, H] + [H, L] is an ideal of L.
(v) If H is an ideal of L, then [γj(H), γk(H)] 6 γj+k(H) for every
positive integers j, k.
(vi) If H is an ideal of L, then γj(H) is an ideal of L for each positive
integer j. In particular, γj(L) is an ideal of L for each positive
integer j.
(vii) If H is an ideal of L, then γj(γk(H)) 6 γjk(H) for every positive
integers j, k.
We remark that if A, B are ideals of a Leibniz algebra L, then, in
general, [A, B] needs not be an ideal. The following example justifies it
(see [11]).
Kirichenko, Kurdachenko, Pypka, Subbotin 7
Example 2. Let L be a vector space over a field F with the basis
{e1, e2, e3, e4, e5}. Define the operation on basis vectors by the following
rule
[·, ·] e1 e2 e3 e4 e5
e1 0 e2 0 e4 e5
e2 −e2 0 e4 0 0
e3 0 e5 0 0 0
e4 e5 0 0 0 0
e5 −e5 0 0 0 0
Let A = Fe2 + Fe4 + Fe5 and B = Fe3 + Fe4 + Fe5. It is not hard to
check that A, B are ideals of L. However, [A, B] = Fe4 is not an ideal.
As usual, we say that a Leibniz algebra L is finite dimensional, if the
dimension L as a vector space over F is finite. The condition to be finite
dimensional is very strong. That is why the majority of results on Leibniz
algebras were obtained for finite dimensional Leibniz algebras.
If dimF (L) = 1, then L = Fa for some element a ∈ L. Then [a, a] =
αa where α ∈ F . We have
0 = [[a, a], a] = [αa, a] = α[a, a] = α2a.
It follows that α = 0, that is [a, a] = 0 and L is abelian.
Suppose now that dimF (L) = 2 and L is not a Lie algebra. It follows
that K = Leib(L) is non-zero. Since K is abelian, K 6= L. Hence there
exists an element a such that b = [a, a] 6= 0. By this choice, a 6∈ K. Then
L = Fa + Fb, and we have [b, a] = 0. The fact that K is an ideal of L
implies that [a, b] = βb for some β ∈ F . Suppose that β 6= 0 and put
c = β−1a. Then [c, b] = β−1[a, b] = β−1βb = b. We have
[c, c] = β−2[a, a] = β−2b = d,
and
[c, d] = [c, β−2b] = β−2[c, b] = β−2b = d.
By this choice, {c, d} is a basis of L. Thus, we obtain the following two
non-isomorphic 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.
8 Some aspects of Leibniz algebra theory
The structure of 3-dimensional Leibniz algebras is more complicated.
Investigation of Leibniz algebras, having dimensions 3 and 4 has been
conducted in the papers [1, 2, 4, 5, 17,18,20–22,39,41,46].
One of the first questions that naturally arises in the study of any
algebraic structure is the question of the structure of its cyclic substruc-
tures (that is, substructures generated by one element). In particular, for
a Leibniz algebra, the question of the structure of its cyclic subalgebras
naturally arises. Unlike Lie algebras, associative algebras, groups, etc., cy-
clic Leibniz algebras is no necessarily abelian. We now give some concepts
that will be needed farther and not only for this description.
The left (respectively right) center ζ left(L) (respectively ζright(L)) of
a Leibniz algebra L is defined by the rule:
ζ left(L) = {x ∈ L| [x, y] = 0 for each element y ∈ L}
(respectively,
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 it is not true
for the right center. Moreover, Leib(L) 6 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.
We will construct now the following examples [30].
Example 3. Let F be a field. Put L = Fe1 ⊕Fe2 ⊕Fe3 ⊕Fe4 and define
an operation [·, ·] by the following rule:
[e1, e1] = e2, [e1, e2] = −e2 − e3, [e1, e3] = e2 + e3, [e1, e4] = 0,
[e2, e1] = 0, [e3, e1] = 0, [e4, e1] = e2 + e3, [ej , ek] = 0
for all j, k ∈ {2, 3, 4}. It is possible to check that this operation defines a
Leibniz algebra. We can see that ζright(L) = Fe4 and ζright(L) is not an
ideal. Furthermore, ζ left(L) = Fe2 ⊕ Fe3, so that
ζright(L) ∩ ζ left(L) = 〈0〉.
Moreover, dimF (ζright(L)) = 1, dimF (ζ left(L)) = 2. Note also that
[L, L] = Leib(L) = ζ left(L).
The center ζ(L) of L is the intersection of the left and right centers,
that is
ζ(L) = {x ∈ L| [x, y] = 0 = [y, x] for each element y ∈ L}.
Kirichenko, Kurdachenko, Pypka, Subbotin 9
Example 4. Let F be a field. Put L = Fe1 ⊕ Fe2 ⊕ Z where a subspace
Z has a countable basis {zn| n ∈ N}. Put [zn, x] = 0 for every x ∈ L and
[e1, e1] = [e2, e2] = [e1, e2] = [e2, e1] = z1, [e1, z1] = [e2, z1] = 0.
By such definitions, we have
0 = [[ej , ek], em] and [ej , [ek, em]] − [ek, [ej , em]] = 0 − 0 = 0
for all j, k, m ∈ {1, 2}. Take into account the equalities
0 = [[e1, e2], z] = [e1, [e2, z]] − [e2, [e1, z]],
0 = [[e2, e1], z] = [e2, [e1, z]] − [e1, [e2, z]],
we obtain [e2, [e1, z]] − [e1, [e2, z]] = 0 for every z ∈ Z. Now we put
[e1, zj ] = zj , [e2, zj ] = zj+1
for all j > 1. By this definition, we have
0 = [[ej , z], ek] and [ej , [z, ek]] − [z, [ej , ek]] = [ej , 0] − 0 = 0,
0 = [[z, ej ], ek] and [z, [ej , ek]] − [ej , [z, ek]] = 0 − [ej , 0] = 0
for all j, k ∈ {1, 2} and z ∈ Z. As we have seen above
[[ej , ek], z] = [ej , [ek, z]] − [ek, [ej , z]]
for all j, k ∈ {1, 2} and z ∈ Z. Hence, L is a Leibniz algebra. By it con-
struction Z is a left center of L, the right center coincides with the center
of L and coincides with Fz1, so that, the left center has finite codimension
(and therefore, infinite dimension) and the right center and the center
have finite dimension. By the construction, [L, L] = Z. Furthermore
[e1 + z1, e1 + z1] = [e1, e1] + [z1, e1] + [e1, z1] + [z1, z1] = z1,
[e1 + zj , e1 + zj ] = [e1, e1] + [zj , e1] + [e1, zj ] + [zj , zj ] = z1 + zj
for j > 1. It follows that Leib(L) = Z.
Clearly, the center ζ(L) is an ideal of L. In particular, we can consider
the factor-algebra L/ζ(L).
Now we define the upper central series
〈0〉 = ζ0(L) 6 ζ1(L) 6 . . . ζα(L) 6 ζα+1(L) 6 . . . ζγ(L) = ζ∞(L)
10 Some aspects of Leibniz algebra theory
of a Leibniz algebra L by the following rule: ζ1(L) = ζ(L) is the center of
L, and recursively, ζα+1(L)/ζα(L) = ζ(L/ζα(L)) for all ordinals α, and
ζλ(L) =
⋃
µ<λ
ζµ(L) for the limit ordinals λ. By definition, each term of
this series is an ideal of L. The last term ζ∞(L) of this series is called the
upper hypercenter of L. A Leibniz algebra L is said to be hypercentral if
it coincides with the upper hypercenter. Denote by zl(L) the length of
upper central series of L.
The introduced here concepts of the upper and lower central series
for Leibniz algebras are an analogous of others similar concepts, which
became standard in several algebraic structures. They play an important
role, for example, in Lie algebras and groups. Following this analogy, we
say that a Leibniz algebra L is called nilpotent, if there exists a positive
integer k such that γk(L) = 〈0〉. More precisely, L is said to be nilpotent
of nilpotency class c if γc+1(L) = 〈0〉, but γc(L) 6= 〈0〉. We denote the
nilpotency class of L by ncl(L).
It is a well-known fact for Lie algebras and groups that in nilpotent
Lie algebras and nilpotent groups the lower and the upper central series
have the same length.
Consider the factors γk(L)/γk+1(L), k ∈ N. By definition [L, γk(L)] =
γk+1(L). By Proposition 1, [γk(L), L] = [γk(L), γ1(L)] 6 γk+1(L).
Let
〈0〉 = C0 6 C1 6 . . . Cα 6 Cα+1 6 . . . Cγ = L
be an ascending series of ideals of Leibniz algebra L. This series is called
central if Cα+1/Cα 6 ζ(L/Cα) for each ordinal α < γ and Cλ =
⋃
µ<λ
Cµ
for the limit ordinals λ. In other words, [Cα+1, L], [L, Cα+1] 6 Cα for each
ordinal α < γ.
Proposition 2 ([30]). Let L be an Leibniz algebra over a field F , and
〈0〉 = C0 6 C1 6 . . . 6 Cn = L
be a finite central series of L. Then
(i) γj(L) 6 Cn−j+1 for every 1 6 j 6 n + 1, so that γn+1(L) = 〈0〉.
(ii) Cj 6 ζj(L) for every 0 6 j 6 n, so that ζn(L) = L.
(iii) If j, k are positive integer such that k > j, then [γj(L), ζk(L)],
[ζk(L), γj(L)] 6 ζk−j(L).
Corollary 1 ([30]). Let L be an Leibniz algebra over a field F and suppose
that L has a finite central series
〈0〉 = C0 6 C1 6 . . . 6 Cn = L.
Kirichenko, Kurdachenko, Pypka, Subbotin 11
Then L is nilpotent and ncl(L) 6 n. Furthermore, the upper central series
of L is finite, ζ∞(L) = L, zl(L) 6 n. Moreover, ncl(L) = zl(L).
This Corollary shows that a Leibniz algebra L is nilpotent if and only
if there is a positive integer k such that L = ζk(L). The least positive
integer having this property coincides with nilpotency class of L. So, as
in the cases of Lie algebras and groups, the definition of nilpotency can
be given here using the notion of the upper central series.
Here it will be appropriate to note the fact that the Leibniz algebra L
can be associative. Indeed, if [L, L] = γ2(L) 6 ζ(L), then 0 = [[x, y], z] =
[x, [y, z]] for all x, y, z ∈ L. Conversely, suppose that L is associative. Then,
taking into account the equality [[x, y], z] = [x, [y, z]], from [[x, y], z] =
[x, [y, z] − [y, [x, z]] we derive that [y, [x, z]] = 0. Since it is true for all
x, y, z ∈ L, [L, L] 6 ζright(L). Furthermore, 0 = [y, [x, z]] = [[y, x], z],
which shows that [L, L] 6 ζ left(L). So we obtain
Proposition 3. Let L be a Leibniz algebra over a field F . Then L is
associative if and only if [L, L] 6 ζ(L).
Let L be a Leibniz algebra over a field F and d be an element of L.
Put
ln1(d) = d, ln2(d) = [d, d], lnk+1(d) = [d, lnk(d)], k ∈ N.
Lemma 1 ([19]). Let L be a Leibniz algebra over a field F , a ∈ L. Then
every non-zero product of k copies of an element a with any bracketing
is coincides with lnk(a). Hence a cyclic subalgebra 〈a〉 is generated as a
subspace by the elements lnk(a), k ∈ N.
The following two natural cases appear here.
The elements dj = lnj(d), j ∈ N are linearly independent. In this case,
a subalgebra D = 〈d〉 has the lower central series
D = γ1(D) > γ2(D) > . . . > γj(D) > γj+1(D) > γω(D) = 〈0〉
of the length ω, and γj(D) =
⊕
t>j
Fdt, j ∈ N. In this case, we will say that
an element d has infinite depth.
Consider the second possibility, when elements dj = lnj(d), j ∈ N, are
not linearly independent. In this case, we have
Lemma 2 ([19]). Let L be a Leibniz algebra over a field F , a ∈ L, D = 〈a〉.
If there exists a positive integer k such that lnk+1(a) ∈ F ln1(a) + . . . +
F lnk(a), then D = F ln1(a) + . . . + F lnk(a).
12 Some aspects of Leibniz algebra theory
In particular, in this case, the subalgebra D = 〈d〉 has finite dimension
over F and we will say that an element d has finite depth. Let k be the
least positive integer such that ln1(d), . . . , lnk(d) are linearly independent,
but ln1(d), . . . , lnk(d), lnk+1(d) are not linearly independent. Then the
subset {ln1(d), . . . , lnk(d)} is a basis of D and dimF (D) = k. In this
case, we can say that an element d has depth k.
The case when an element d has finite depth turned out to be much
more diverse. The following theorem has described this case.
Theorem 1 ([19]). Let L be a Leibniz algebra over a field F , a ∈ L,
D = 〈a〉. Suppose that an element a has finite depth. Then D is an
algebra of one of the following types:
(i) D = Fa is abelian, [a, a] = 0.
(ii) There exists a positive integer k such that lnk(a) 6= 0, lnk+1(a) = 0,
that is D is a nilpotent cyclic algebra.
(iii) D = V ⊕ U where V is an abelian ideal, V 6 ζ left(D), U is a
nilpotent cyclic subalgebra, [D, D] = V ⊕ [U, U ] is an abelian ideal.
(iv) D = ζ left(D)⊕ ζright(D) where [D, D] = ζ left(D) = F ln2(a)+ . . .+
F lnk(a), ζright(D) = Fc for some element c ∈ D and [c, y] = [a, y]
for each element y ∈ ζ left(D).
For the case when F = C is a field of complex number, a description
of cyclic finite dimensional Leibniz algebras were obtained in the paper
[40]. Unlike Theorem 1, it does not show the structure of cyclic Leibniz
algebras and based on the following. Let an element a has a depth k.
Then lnk+1(d) = α2ln2(a) + . . . + αklnk(a), for some αj ∈ C, 2 6 j 6 k.
In the paper [40] a characterization for the set of coefficients (α2, . . . , αk)
was obtained.
As we already noted above, Lie algebras are a partial type of Leibniz
algebras. In this regard, it is interesting to see how the Leibniz algebras,
which are the minimal non Lie algebras with all proper subalgebras of
which are Lie algebras are organized. A description of such algebras was
obtained in [19].
Theorem 2 ([19]). Let L be a Leibniz algebra over a field F . Suppose
that every proper subalgebra of L is a Lie algebra. Then L is an algebra
of one of the following types:
(i) L is a Lie algebra.
(ii) There exists a positive integer k such that lnk(a) 6= 0, lnk+1(a) = 0,
that is L is nilpotent.
Kirichenko, Kurdachenko, Pypka, Subbotin 13
(iii) L = V ⊕ U where V is an abelian ideal, V 6 ζ left(D), U = Fu and
[u, u] = 0, V = Fv + Fv1 and [u, v] = v1, [u, v1] = 0.
Since every abelian Leibniz algebra is a Lie algebra, we obtain
Corollary 2 ([19]). Let L be a Leibniz algebra over a field F . Suppose
that every proper subalgebra of L is abelian. Then L is an algebra of one
of the following types:
(i) L is a Lie algebra whose proper subalgebras are abelian.
(ii) There exists a positive integer k such that lnk(a) 6= 0, lnk+1(a) = 0,
that is L is nilpotent.
(iii) L = V ⊕ U where V is an abelian ideal, V 6 ζ left(L), U = Fu and
[u, u] = 0, V = Fv + Fv1 and [u, v] = v1, [u, v1] = 0.
This result implies that a description of Leibniz algebras, whose proper
subalgebras are abelian, can be deduced to the case of Lie algebras, whose
proper subalgebras are abelian. Such Lie algebras are either simple, or
soluble. Soluble minimal non-abelian Lie algebras (even soluble minimal
non-nilpotent Lie algebras) were described in [27], [44] and [45]. Simple
minimal non-abelian Lie algebras were studied in [25] and [26], but their
complete description remains to be an open problem.
Another natural question concerns the relationship of the subalgebras
and ideals. In particular, what is a structure of Leibniz algebras, all of
whose subalgebras are ideals? It is not hard to prove that a Lie algebra,
all of whose subalgebras are ideals, is abelian. For groups the situation
is different: there exists non-abelian groups, all of whose subgroups are
normal. Such groups have been described in [6]. In the case of associative
algebras, the situation is much more complicated. For Leibniz algebras the
situation is quite diverse. At once it is possible to specify a simple example
of non-abelian Leibniz algebra, all of whose subalgebras are ideals.
Example 5. Let L be a vector space over a field F , having dimension 2,
{a, b} be a basis of L. Define the operation [·, ·] by the following rule:
[a, a] = b, [b, b] = [b, a] = [a, b] = 0.
A direct check justifies that L becomes a Leibniz algebra. If λa + µb is
an arbitrary element of L and λ 6= 0, then [λa + µb, λa + µb] = λ2b. Since
λ2 6= 0, we obtain that the subalgebra generated by λa + µb includes Fb.
Since L/Fb is abelian, 〈λa+µb〉 is an ideal. Hence, every cyclic subalgebra
of L is an ideal. It follows that every subalgebra of L is an ideal.
14 Some aspects of Leibniz algebra theory
As we shall see later, any non-abelian Leibniz algebra, whose subalge-
bras are ideals, is constructed from such algebras as from bricks. Here
are more details.
A Leibniz algebra L is called an extraspecial algebra if it satisfies the
following condition:
(i) ζ(L) is non-trivial and has dimension 1;
(ii) L/ζ(L) is abelian.
It is important to observe that there are extraspecial Leibniz algebras
in which not every subalgebra is an ideal. The following example of an
extraspecial Leibniz algebra from the paper [31] shows this. Moreover,
the existence of subalgebras that are not ideals depends on the choice of
the field.
Example 6. Let F be a field, put L = Fa ⊕ Fb ⊕ Fc. Define on L an
operation [·, ·] by the following rule:
c = [a, a] = [b, b] = [a, b], [c, c] = [c, a] = [c, b] = [a, c] = [b, c] = [b, a] = 0.
From this definition it follows that [L, L] 6 Fc, c ∈ ζ(L), 〈c〉 = Fc. The
equality
[[x, y], z] = [x, [y, z]] − [y, [x, z]]
occurs automatically, because [x, y], [y, z], [x, z] ∈ ζ(L). Thus L is a Leibniz
algebra. Let x be an arbitrary element of L, then x = λa + µb + νc for
some λ, µ, ν ∈ F . We have
[x, x] = [λa + µb + νc, λa + µb + νc]
= λ2[a, a] + λµ[a, b] + λν[a, c] + λµ[b, a] + µ2[b, b]
+ µν[b, c] + λν[c, a] + µν[c, b] + ν2[c, c]
= λ2c + λµc + µ2c = (λ2 + λµ + µ2)c.
Let F = F2. If (λ, µ) 6= (0, 0), then λ2 + λµ + µ2 = 1, that is, [x, x] = c
whenever x 6∈ Fc. It follows that ζ(L) = Fc and 〈x〉 = Fx ⊕ Fc. It
follows that 〈x〉 is an ideal of L. Since Fc is an ideal, we obtain that every
subalgebra of L is an ideal.
Let F = F5. Suppose that λ2 +λµ+µ2 = 0. It follows that (λ+ 1
2µ)2 =
µ2(1
4 − 1). In field F5 a solution of an equation 4x = 1 is 4, so that
1
4 − 1 = 3. But the equation x2 = 3 has no solutions in F5. This shows
that the equality λ2 + λµ + µ2 = 0 is true only when λ = µ = 0. Thus if
(λ, µ) 6= (0, 0), then [x, x] 6= 0 and [x, x] ∈ Fc. Hence, in this case, every
subalgebra of L is an ideal.
Kirichenko, Kurdachenko, Pypka, Subbotin 15
If F = Q, then using the similar arguments we obtain again that every
subalgebra of L is an ideal and the center of L is Fc.
Consider now the case when F = F3. For element x = a + b we have
[a + b, a + b] = 3c = 0. It follows that 〈x〉 = Fx. But [x, a] = [a + b, a] =
c 6∈ Fx, which shows that a cyclic subalgebra 〈x〉 is not an ideal.
The following theorem concerned with Leibniz algebras whose every
subalgebra is an ideal.
Theorem 3 ([31]). Let L be a Leibniz algebra over a field F , all of
whose subalgebras are ideals. If L is non-abelian, then L = E ⊕ Z where
Z 6 ζ(L), and E is an extraspecial subalgebra such that [a, a] 6= 0 for
every element a 6∈ ζ(E).
With an extraspecial algebra we can connect a bilinear form in the
following way. Let Z = ζ(L), V = L/Z, and c be a fixed non-zero element
of Z. Define the mapping Φ : V × V → F by the following rule: if
x, y ∈ L, then [x, y] ∈ Z, so that [x, y] = αc for some element α ∈ F .
Put Φ(x + Z, y + Z) = α. This mapping is correct. Indeed, let x1, y1 be
elements of L such that x1 +Z = x+Z, y1 +Z = y +Z. Then x1 = x+c1,
y1 = y + c2 for some elements c1, c2 ∈ Z. Then
[x1, y1] = [x + c1, y + c2] = [x, y] + [x, c2] + [c1, y] + [c1, c2] = [x, y].
The mapping Φ is bilinear. In fact, let x, y, u 6∈ Z, [x, u] = λc, [y, u] =
µc. Then [x + y, u] = [x, u] + [y, u] = λc + µc = (λ + µ)c, so that
Φ(x + Z + y + Z, u + Z) = Φ(x + y + Z, u + Z) = λ + µ
= Φ(x + Z, u + Z) + Φ(y + Z, u + Z).
Similarly, we can show that
Φ(x + Z, y + Z + u + Z) = Φ(x + Z, y + Z) + Φ(x + Z, u + Z).
Let β ∈ F , then [βx, y] = β[x, y] = β(αc) = (βα)c. It follows that
Φ(β(x + Z), y + Z) = Φ(βx + Z, y + Z) = βα = βΦ(x + Z, y + Z).
Likewise we can show that
Φ(x + Z, β(y + Z)) = βΦ(x + Z, y + Z).
By the definition of an extraspecial algebra we obtain that a bilinear
form Φ is non-degenerate. Moreover, Theorem 3 shows that Φ(x, x) 6= 0
for every non-zero element x ∈ V .
16 Some aspects of Leibniz algebra theory
Conversely, let V be a vector space over a field F and Φ be a bilinear
form on V such that Φ(x, x) 6= 0 for every non-zero element x ∈ V . Put
L = V ⊕F . Define the operation [·, ·] on L by the following rule: if a, b ∈ V ,
α, β ∈ F , then [(a, α), (b, β)] = (0, Φ(a, b)). Put C = {(0, α)| α ∈ F}.
Then dimF (C) = 1. By this definition, [L, L] = [L, C] = [C, L] = [C, C] =
〈0〉. It follows from here that the constructed algebra a Leibniz algebra.
Furthermore, C 6 ζ(L). Moreover, C = ζ(L). Indeed, let (z, γ) ∈ ζ(L)
and suppose that z 6= 0. Then [(z, γ), (a, α)] = [(a, α), (z, γ)] = (0, 0), in
particular, [(z, γ), (z, γ)] = (0, 0). But [(z, γ), (z, γ)] = (0, Φ(z, z)). Since
z 6= 0, Φ(z, z) 6= 0, and we obtain a contradiction. This contradiction
proves the equality C = ζ(L).
Let V be a vector space over a field F , U a subspace of V , and Φ be
a bilinear form on V . Put
⊥U = {x ∈ V | Φ(x, u) = 0 for all elements u ∈ U},
U⊥ = {x ∈ V | Φ(u, x) = 0 for all elements u ∈ U}.
Clearly ⊥U and U⊥ are subspaces of V . ⊥U is called a left orthogonal
complement of U in V , U⊥ is called a right orthogonal complement of U
in V .
Using standard linear algebra tools one can prove the following state-
ment.
Proposition 4. Let V be a finite dimensional vector space over a field
F , U a subspace of V , and Φ be a non-degenerate bilinear form on V . If
the restriction of Φ on U is non-degenerate, then
dimF (⊥U) = dimF (U⊥) = dimF (V ) − dimF (U).
Let V be a vector space over a field F having countable dimension, and
Φ be a bilinear form on V . A basis {aj | j ∈ N} is called left orthogonal, if
Φ(aj , ak) = 0 whenever j > k.
Corollary 3. Let V be a finite dimensional vector space over a field F
and Φ be a bilinear form on V . If Φ(a, a) 6= 0 for each element 0 6= a ∈ V ,
then V has a left orthogonal basis.
Indeed, let U be an arbitrary non-zero subspace of V . If we suppose
that the restriction of Φ on U is degenerate, then ⊥U∩U 6= 〈0〉. Let 0 6= a ∈
⊥U ∩U , then Φ(a, u) = 0 for all elements u ∈ U . In particular, Φ(a, a) = 0,
and we obtain a contradiction. Hence the restriction of Φ on every non-zero
subspace is non-degenerate, and we can apply Proposition 4.
Kirichenko, Kurdachenko, Pypka, Subbotin 17
We note that if V is a finite dimensional vector space and Φ be a
bilinear form on V such that V has a left orthogonal basis, then the
matrix of the form Φ in this basis is triangular.
Now we can get a more detailed description of such bilinear forms. For
clarity, we confine ourselves to the case when a vector space V has finite
dimension n. Let Φ be a bilinear form on V such that Φ(x, x) 6= 0 for
each non-zero element x ∈ V . Choose a non-zero element v1 ∈ V and put
V1 = Fv1, U1 = ⊥V1. By Proposition 4 dimF (U1) = n − 1. Suppose that
U1 has an element v2 such that Φ(v1, v2) 6= 0. Choose in the subspace U1
a left orthogonal complement U2 to a subspace Fv2, and let {v3, . . . , vn}
be a basis of U2. Put Φ(vj , vk) = γjk, 1 6 j, k 6 n, then γj1 = 0 whenever
j > 1, γj2 = 0 for j > 2. Consider the elements γ1k where k > 3. Suppose
that not these elements are zeros. Without loss of generality we may
assume that γ13 6= 0. Put a1 = v1, a2 = v2, a3 = v3, ak = vk − γ1kγ−1
13 v3 if
k > 3. Then clearly, {a1, . . . , an} is the basis of V such that Φ(ak, a1) = 0
for k > 1, Φ(ak, a2) = 0 for k > 2 and Φ(a1, ak) = 0 for k > 3. If γ1k = 0
for all k > 3, then we will not change the basis.
Suppose that Φ(a2, ak) 6= 0 for some k > 2. Without loss of generality
we can assume that Φ(a2, a3) 6= 0. Put U3 = Fa3 + . . . + Fan. Let
Φ(aj , ak) = αjk, 1 6 j, k 6 n, and consider the elements α2k where k > 4.
Suppose that not these elements are zeros. Without loss of generality
we may assume that α24 6= 0. Put b1 = a1, b2 = a2, a3 = b3, a4 = b4,
bk = ak − α2kα−1
24 a4 if k > 4. Then clearly, {b1, . . . , bn} is the basis of V
such that Φ(bk, b1) = 0 for k > 1, Φ(bk, b2) = 0 for k > 2, Φ(b1, bk) = 0
for k > 3 and Φ(b2, bk) = 0 for k > 4. If α2k = 0 for all k > 4, we will
remain to use the previous basis.
Repeating these arguments, we come to a basis {e1, . . . , en} of V such
that Φ(ej , ek) = 0 whenever j > k, and Φ(ej , ek) = 0 whenever k > j + 3.
It is also easy to see that this description can be extended to the case of
a vector space having countable dimension, so that we have
Theorem 4. Let V be a vector space over a field F having countable
dimension, and Φ be a bilinear form on V . If Φ(a, a) 6= 0 for each element
0 6= a ∈ V , then V has a basis {e1, . . . , en} such that Φ(ej , ek) = 0
whenever j > k, and Φ(ej , ek) = 0 whenever k > j + 3, j, k ∈ N.
Corollary 4. Let L be an extraspecial Leibniz algebra over a field F ,
having countable dimension. If [a, a] 6= 0 for every element a 6∈ ζ(L), then
L has a basis {c, en| n ∈ N} such that [c, en] = [en, c] = 0, 0 6= [en, en] ∈
Fc for all n ∈ N, [ej , ek] = 0 whenever j > k and [ej , ek] = 0 whenever
k > j + 3, j, k ∈ N.
18 Some aspects of Leibniz algebra theory
It will be useful to consider the properties of nilpotent Leibniz algebras.
Let L be a Leibniz algebra. As for Lie algebras, a linear transformation f
of L is called a derivation, if f([a, b]) = [f(a), b] + [a, f(b)] for all a, b ∈ L.
Denote by EndF (L) the set of all linear transformations of L, then L is
an associative algebra by the operation + and ◦. As usual, EndF (L) is a
Lie algebra by the operations + and [·, ·], where [f, g] = f ◦ g − g ◦ f for
all f, g ∈ EndF (L). Let Der(L) be the subset of all derivations of L. If
f, g ∈ Der(L), then
(f − g)([a, b]) = f([a, b]) − g([a, b])
= [f(a), b] + [a, f(b)] − [g(a), b] − [a, g(b)]
= [f(a) − g(a), b] + [a, f(b) − g(b)]
= [(f − g)(a), b] + [a, (f − g)(b)];
[f, g]([a, b]) = (f ◦ g − g ◦ f)([a, b]) = (f ◦ g)([a, b]) − (g ◦ f)([a, b])
= f(g([a, b])) − g(f([a, b]))
= f([g(a), b] + [a, g(b)]) − g([f(a), b] + [a, f(b)])
= f([g(a), b]) + f([a, g(b)]) − g([f(a), b]) − g([a, f(b)])
= [f(g(a)), b] + [g(a), f(b)] + [f(a), g(b)] + [a, f(g(b))]
− [g(f(a)), b] − [f(a), g(b)] − [g(a), f(b)] − [a, g(f(b))]
= [f(g(a)), b] + [a, f(g(b))] − [g(f(a)), b] − [a, g(f(b))]
= [f(g(a)) − g(f(a)), b] + [a, f(g(b)) − g(f(b))]
= [[f, g](a), b] + [a, [f, g](b)].
This shows 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.
Consider the mapping la : L → L, defined by the rule la(x) = [a, x].
For every x, y ∈ L and α ∈ F we have
la(x + y) = [a,x + y] = [a, x] + [a, y] = la(x) + la(y),
la(αx) = [a, αx] = α[a, x] = αla(x),
and
la([x, y]) = [a, [x, y]] = [[a, x], y] + [x, [a, y]] = [la(x), y] + [x, la(y)].
These equalities show that la is a derivation of L. Consider some properties
of the mappings la. If a, b ∈ L and β ∈ F , then
βla(x) = β[a, x] = [βa, x] = lβa(x)
Kirichenko, Kurdachenko, Pypka, Subbotin 19
for every x ∈ L, which implies that βla = lβa. Further,
(la + lb)(x) = la(x) + lb(x) = [a, x] + [b, x] = [a + b, x] = la+b(x),
which follows that la + lb = la+b. And finally,
[la, lb](x) = (la ◦ lb − lb ◦ la)(x) = (la ◦ lb)(x) − (lb ◦ la)(x)
= la(lb(x)) − lb(la(x)) = la([b, x]) − lb([a, x])
= [a, [b, x]] − [b, [a, x]] = [[a, b], x] = l[a,b](x),
which follows that [la, lb] = l[a,b]. This shows that the set {la| a ∈ L} is a
subalgebra of Der(L).
Similarly, consider the mapping ra : L → L, defined by the rule ra(x) =
[x, a]. For every x, y ∈ L and α ∈ F we have ra(x + y) = ra(x) + ra(y),
ra(αx) = αra(x) and
ra([x, y]) = [[x, y], a] = [x, [y, a]] − [y, [x, a]] = [x, ra(y)] − [y, ra(x)].
Also we have βra = rβa and ra + rb = ra+b for all a, b ∈ L and β ∈ F .
Theorem 5 ([9]). (Engel’s theorem for Leibniz algebras). Let L be a
finite dimensional left Leibniz algebra over a field F of characteristic 0. If
the mappings la are nilpotent for each a ∈ L, then the algebra L is itself
nilpotent. In particular, all operators la possess the common eigenvector
with zero eigenvalue. Moreover, there exists a basis of L such that the
matrix of la in this basis is upper zero-triangular for every a ∈ L.
First proof of this statement (for right Leibniz algebras) was given in
[4]. In the paper [37] the following result has been obtained.
Theorem 6. Let L be a finite dimensional left Leibniz algebra over a
field F of characteristic 0. If the mappings la are nilpotent for each a ∈ L,
then all operators ra are nilpotent for each a ∈ L. Moreover, there exists
a basis of L such that the matrices of la and ra in this basis are upper
zero-triangular for every a ∈ L.
Let L be a Leibniz algebra over a field F , M be non-empty subset of
L and H be a subalgebra of L. Put
Ann
left
H (M) = {a ∈ H| [a, M ] = 〈0〉},
Ann
right
H (M) = {a ∈ H| [M, a] = 〈0〉}.
20 Some aspects of Leibniz algebra theory
The subset Ann
left
H (M) is called the left annihilator or the left centrali-
zer of M in subalgebra H. The subset Ann
right
H (M) is called the right
annihilator or the right centralizer of M in subalgebra H. The intersection
AnnH(M) = Ann
left
H (M) ∩ Ann
right
H (M)
= {a ∈ H| [a, M ] = 〈0〉 = [M, a]}
is called the annihilator or the centralizer of M in subalgebra H.
It is not hard to see that all of these subsets are subalgebras of L.
Moreover, if M is a left ideal of L, then Ann
left
L (M) is an ideal of L.
Indeed, let x be an arbitrary element of L, a ∈ Ann
left
L (M), b ∈ M . Then
[[a, x], b] = [a, [x, b]] − [x, [a, b]] = 0 − [x, 0] = 0, and
[[x, a], b] = [x, [a, b]] − [a, [x, b]] = [x, 0] − 0 = 0.
If M is an ideal of L, then AnnL(M) is an ideal of L. Indeed, let x
be an arbitrary element of L, a ∈ AnnL(M), b ∈ M . Using the above
arguments, we obtain that [[a, x], b] = [[x, a], b] = 0. Further,
[b, [a, x]] = [[b, a], x] + [a, [b, x]] = [0, x] + 0 = 0, and
[b, [x, a]] = [[b, x], a] + [x, [b, a]] = 0 + [x, 0] = 0.
Let H be a left ideal of a Leibniz algebra L, a ∈ L. Consider the
mapping la : H → H defined by the rule la(x) = [a, x]. As above, we can
show that la is a derivation of H for every a ∈ L and the set {la| a ∈ L}
is a subalgebra of Der(H).
Consider now the mapping δ : L → Der(H), defined by the rule
δ(a) = la. By proved above,
δ(βa) = lβa = βla = βδ(a);
δ(a + b) = la+b = la + lb = δ(a) + δ(b);
δ([a, b]) = l[a,b] = [la, lb] = [δ(a), δ(b)].
These equations shows that the mapping δ is a homomorphism of the
Leibniz algebra L in the Lie algebra Der(H). Then Im(δ) is a subalgebra
of Der(H), and Im(δ) ∼= L/Ker(δ). Finally,
Ker(δ) = {a ∈ L| la = δ(a) = 0}.
In turn out, la = 0 means that 0 = la(x) = [a, x] for every element x ∈ H.
In other words,
Ker(δ) 6 Ann
left
L (H) = {a ∈ L| [a, H] = 〈0〉}
Kirichenko, Kurdachenko, Pypka, Subbotin 21
the left annihilator of H in L. The converse inclusion is obvious, so that
Ker(δ) = Ann
left
L (H). As we remarked above, Ann
left
L (H) is a two-
side ideal of L, so we obtain that L/Ann
left
L (H) is isomorphic to some
subalgebra of Der(H).
Let H be a subalgebra of L. The left idealizer or the left normalizer
of H in L is defined by the following:
I
left
L (H) = {x ∈ L| [x, h] ∈ H for all h ∈ H}.
Clearly, that the term the left normalizer arose from group theory analo-
gous.
Similarly, the right idealizer of H in L is defined by the following:
I
right
L (H) = {x ∈ L| [h, x] ∈ H for all h ∈ H}.
The idealizer of H in L is defined by the following:
IL(H) = {x ∈ L| [h, x], [x, h] ∈ H for all h ∈ H}
= I
left
L (H) ∩ I
right
L (H).
The left idealizer of H is a subalgebra of L. Indeed, let x, y ∈ I
left
L (H),
h ∈ H, α ∈ F , then
[x − y, h] = [x, h] − [y, h] ∈ H;
[αx, h] = α[x, h] ∈ H; and
[[x, y], h] = [x, [y, h]] − [y, [x, h]] ∈ H.
The idealizer of H is also a subalgebra of L. Indeed, let x, y ∈ IL(H),
h ∈ H, α ∈ F . As above we can show that x − y, αx, [x, y] ∈ IL(H).
Further,
[x − y, h] = [x, h] − [y, h] ∈ H;
[αx, h] = α[x, h] ∈ H; and
[h, [x, y]] = [[h, x], y] + [x, [h, y]] ∈ H.
However the right idealizer need not be a subalgebra. This is shown
in the following example from [8].
Example 7. Let L be a vector space over F , and {a, b, c, d} be a basis
of L. Define the operation [·, ·] by the rule
[a, b] = a, [b, a] = −a + c, [b, b] = d, [a, d] = c,
[a, a] = [a, c] = 0, [b, c] = −c, [d, [d, d]] = 0,
22 Some aspects of Leibniz algebra theory
and
[c, x] = [[d, d], x] = 0 for all x ∈ L.
It is not hard to prove that L is a Leibniz algebra. Let H = 〈a〉. Since
[a, a] = 0, 〈a〉 = Fa. Clearly, I
right
L (H) = Fb + Fc. However [c, c] = d 6∈
I
right
L (H), which shows that I
right
L (H) is not a subalgebra of L.
However, if H is a left ideal of L, then its right idealizer is a subalgebra.
Indeed, let x, y ∈ I
right
L (H), h ∈ H, then [h, [x, y]] = [[h, x], y] + [x, [h, y]].
By definition, [h, x], [h, y] ∈ H, and [[h, x], y] ∈ H. Since H is a left ideal,
[x, [h, y]] ∈ H, which implies that [x, y] ∈ I
right
L (H).
Let L be a hypercentral Leibniz algebra and let
〈0〉 = Z0 6 Z1 6 . . . Zα 6 Zα+1 6 . . . Zγ = L
be the upper central series of L. Let H be a proper subalgebra of L. Then
there exists an ordinal α such that Zα 6 H but H does not include Zα+1.
Choose an element x ∈ Zα+1 \ H. For every element h ∈ H we have
[x, h], [h, x] ∈ Zα. The inclusion Zα 6 H implies that [x, h], [h, x] ∈ H.
This shows that IL(H) 6= H, in particular I
right
L (H) 6= H 6= I
left
L (H), so
we obtain
Proposition 5. Let L be a Leibniz algebra over a field F . If L is hyper-
central, then IL(H) 6= H for every proper subalgebra H of L.
Corollary 5. Let L be a nilpotent Leibniz algebra over a field F . Then
IL(H) 6= H for every proper subalgebra H of L.
Corollary 6. Let L be a nilpotent Leibniz algebra over a field F . Then
every maximal subalgebra of L is an ideal of L.
For finitely dimensional Leibniz algebras the just mentioned properties
can help to characterize nilpotent Leibniz algebras.
Theorem 7. Let L be a finite dimensional Leibniz algebra over a field F
of characteristic 0. Then the following statements are equivalent:
(i) L is nilpotent.
(ii) Every proper subalgebra of L does not coincide with its idealizer.
(iii) Every proper subalgebra of L does not coincide with its right idealizer.
(iv) Every maximal subalgebra of L is an ideal of L.
(v) Every maximal subalgebra of L is a right ideal of L.
Kirichenko, Kurdachenko, Pypka, Subbotin 23
The most significant of these characteristics were proved in [8].
In [38] the following properties of finite dimensional Leibniz algebras
have been obtained.
Theorem 8. Let L be a finite dimensional Leibniz algebra over a field F
and H be a nilpotent ideal of L. Then L is nilpotent if and only if L/[H, H ]
is nilpotent. Moreover, if ncl(H) = c and ncl(L/[H, H]) = d + 1, then
ncl(L) 6
(c+1
2
)
d −
(c
2
)
.
Theorem 9. Let L be a finite dimensional Leibniz algebra over a field F .
If L is nilpotent and H is a subalgebra of L such that H +[L, L] = L, then
H = L. Conversely, if for every subalgebra H of L such that H+[L, L] = L
we have H = L, then L is nilpotent.
Let L be a Leibniz algebra. The intersection of the maximal subalge-
bras of L is called the Frattini subalgebra of L and denoted by Frat(L).
If L does not include maximal subalgebras, then put L = Frat(L).
Theorem 10 ([11]). Let L be a Leibniz algebra over a field F of charac-
teristic 0. Then Frat(L) is an ideal of L.
Note that if char(F ) is prime, it is not true even for soluble Lie
algebras [12].
Combining Theorem 10 with Corollary 5.6 of the paper [8], we obtain
Theorem 11. Let L be a Leibniz algebra over a field F of characteristic
0. If dimF (L) is finite, then Frat(L) is nilpotent.
Note the following important property of Frattini subalgebras.
Proposition 6. Let L be a finite dimensional Leibniz algebra over a
field F . If M is a subset of L such that 〈M, Frat(L)〉 = L, then 〈M〉 = L.
Indeed, suppose the contrary. Let 〈M〉 is a proper subalgebra of L.
Since dimF (L) is finite, there is a maximal subalgebra H such that 〈M〉 6
H. Being maximal, H includes Frat(L), so that 〈M, Frat(L)〉 6 H 6= L.
This contradiction proves that 〈M〉 = L.
Using the Frattini subalgebra, we can obtain the following characte-
rization of nilpotent Leibniz algebras. But first we give a slightly more
general statement.
Proposition 7. Let L be a Leibniz algebra over a field F . Then every
maximal subalgebra of L is an ideal if and only if [L, L] = Frat(L).
24 Some aspects of Leibniz algebra theory
Indeed, suppose that each maximal subalgebra of L is an ideal. Let
K be an arbitrary maximal subalgebra of L. Then 〈K, x〉 = L for each
element x 6∈ K. Since K is an ideal, L/K is a cyclic algebra. If we suppose
that Leib(L/K) is non-zero, then Leib(L/K) is a proper subalgebra of
L/K, which is impossible. Hence Leib(L/K) = 〈0〉, so that L/K is a cyclic
Lie algebra. In particular, it is abelian, which follows that [L, L] 6 K. It
is valid for each maximal subalgebra, therefore their intersection Frat(L)
includes [L, L]. On the other hand, factor-algebra L/[L, L] is abelian,
so that every its subspace is a subalgebra. Since the intersection of all
maximal subspaces of L/[L, L] is zero, then Frat(L) = [L, L].
Conversely, if [L, L] = Frat(L), then Frat(L) is an ideal and the factor-
algebra L/Frat(L) is abelian. It follows that every subalgebra including
Frat(L) is an ideal of L, in particular, every maximal subalgebra of L is
an ideal.
Using this result and Theorem 7 we obtain
Corollary 7. Let L be a finite dimensional Leibniz algebra over a field
F of characteristic 0. Then L is nilpotent if and only if [L, L] = Frat(L).
In [11] the following properties of nilpotent ideals of Leibniz algebras
have been obtained.
Theorem 12. Let L be a Leibniz algebra over a field F and K1, K2
are ideals of L. Suppose that K1, K2 are nilpotent and ncl(K1) = c1,
ncl(K2) = c2. Then the ideal K1 + K2 is nilpotent, and ncl(K1 + K2) =
c1 + c2.
Leibniz algebra L is called locally nilpotent if every finitely generated
subalgebra of L is nilpotent.
If L is an arbitrary Leibniz algebra, then denote by Nil(L) the subalge-
bra, generated by all nilpotent ideals of L. Nil(L) is called the nil-radical
of L.
If L = Nil(L), then L is called a Leibniz nil-algebra. Every nilpotent
Leibniz algebra is a nil-algebra, but converse is not true even for a Lie
algebra. If L is finite dimensional, then Theorem 12 shows that Nil(L)
is the greatest nilpotent ideal of L. In general case, Nil(L) is locally
nilpotent.
Let L be a Leibniz algebra. Define the lower derived series
L = δ0(L) > δ1(L) > . . . δα(L) > δα+1(L) > . . . δν(L)
of L by the following rule: δ0(L) = L, δ1(L) = [L, L], and recursively
δα+1(L) = [δα(L), δα(L)] for all ordinals α and δλ(L) =
⋂
µ<λ
δµ(L) for the
Kirichenko, Kurdachenko, Pypka, Subbotin 25
limit ordinals λ. For the last term δν(L) we have δν(L) = [δν(L), δν(L)].
The length ν of this series is called the derived length of L and denoted
by dl(L).
If δν(L) = 〈0〉 for some ordinal ν, then L is called a hypoabelian Leibniz
algebra. If δn(L) = 〈0〉 for some positive integer n, then we say that L is
a soluble Leibniz algebra.
If K1, K2 are soluble ideals of Leibniz algebra L, then clearly their
sum K1 + K2 is a soluble ideal of L. Therefore if L is a finite dimensional
Leibniz algebra, then its subalgebra Sol(L) generated by all soluble ideals
of L is called the soluble radical of L. By above remarked, Sol(L) is a
soluble ideal of L, and a factor-algebra L/Sol(L) does not include non-zero
soluble ideals.
Note some properties of the nil-radical and the soluble radical obtained
in [29].
Theorem 13. Let L be a finite dimensional Leibniz algebra over a field
F of characteristic 0. Then [L, Sol(L)] 6 Nil(L).
Corollary 8. Let L be a finite dimensional Leibniz algebra over a field
F of characteristic 0. Then [Sol(L), Sol(L)] is nilpotent.
Corollary 9. Let L be a finite dimensional Leibniz algebra over a field
F of characteristic 0. Then L is soluble if and only if [L, L] is nilpotent.
The last two corollaries were proved earlier in [4].
For a finite dimensional Leibniz algebra the following analogue of
the Levi’s Theorem from Lie algebras takes place. It was proved by
D. Barnes [10].
Theorem 14. Let L be a finite dimensional Leibniz algebra over a field
F of characteristic 0. Then L includes a subalgebra S (being a semisimple
Lie algebra) such that L = Sol(L) + S and Sol(L) ∩ S = 〈0〉.
The examples given in [10] show that the subalgebra S is not unique.
We will not go deeper into the structure of the finite dimensional
Leibniz algebras. These questions have been adequately reflected in many
articles on Leibniz algebras. Many of the results obtained on this problem
are analogs (not always complete) of the corresponding theorems from
the theory of Lie algebras.
Let us now consider some other natural questions of the general theory
of Leibniz algebras.
26 Some aspects of Leibniz algebra theory
Note that the relation “to be a subalgebra of a Leibniz algebra” is
transitive. However, the relation “to be an ideal” is not transitive even
for Lie algebras.
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 of a Leibniz algebra L is called a left (respectively
right) subideal of L, if there is a finite series of subalgebras
A = A0 6 A1 6 . . . 6 An = L
such that Aj−1 is a left (respectively right) ideal of Aj , 1 6 j 6 n.
Similarly, a subalgebra A of a Leibniz algebra L is called a subideal
of L, if there is a finite series of subalgebras
A = A0 6 A1 6 . . . 6 An = L
such that Aj−1 is an ideal of Aj , 1 6 j 6 n.
We note the following property of nilpotent Leibniz algebras.
Proposition 8. Let L be a nilpotent Leibniz algebra over a field F . Then
every subalgebra of L is a subideal of L.
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 algebras, in which a relation “to be an ideal” is transitive have
been studied by I. Stewart [42] and A.G. Gein and Yu.N. Muhin [28]. In
particular, soluble T -algebras and finite dimensional T -algebras over a
field of characteristic 0 has been described.
As in the mentioned above cases, the situation in Leibniz algebras is
much more complex and diverse than it was in Lie algebras. Here are few
simple examples illustrating this point.
Example 8. Let F be an arbitrary field, 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, Fc is an unique its 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 and L is a Leibniz T -algebra.
Kirichenko, Kurdachenko, Pypka, Subbotin 27
Example 9. Let now F = F2 and L be a constructed above Leibniz
algebra. 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 an unique non-zero ideal of L. But
Fc = ζ(L), thus Fc is an ideal of A. Thus A is a Leibniz T -algebra.
Example 10. Let again F = F2 and L be a constructed above Leibniz
algebra. Put D = L ⊕ Fu. Let
[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 D.
As above, we can check that D is a Leibniz T -algebra.
As we will see further, these examples are typical in some sense.
The subalgebra Ba(L) of a Leibniz algebra L generated by all nilpotent
subideals 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) 6 Ba(L). If L = Ba(L), then L is
called a Leibniz Baer algebra. Every nil-algebra is a Baer algebra, but
converse is not true even for a Lie algebra (see, for example, [3, Theorem
6.4.5]).
The description of Leibniz T -algebras has been obtained in the paper
[32]. Here are the main results of this paper.
Theorem 15. 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 6 ζ(L) and E is
an extraspecial subalgebra such that [a, a] 6= 0 for every element a 6∈ ζ(E).
A Leibniz algebra L is called hyperabelian if it has an ascending series
〈0〉 = L0 6 L1 6 . . . Lα 6 Lα+1 6 . . . Lγ = L
of ideals whose factors Lα+1/Lα are abelian for all ordinals α < γ and
Lλ =
⋃
µ<λ
Lµ for the limit ordinals λ. 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.
28 Some aspects of Leibniz algebra theory
Theorem 16. Let L be a hyperabelian 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 element d ∈ Nil(L). In
particular, L is a Lie algebra.
Theorem 17. Let L be a hyperabelian Leibniz T -algebra over a field F .
If char(F ) 6= 2, then Nil(L) is abelian.
In other words, if char(F ) 6= 2, then every Leibniz T -algebra is a Lie
algebra. Thus we can see that the case when char(F ) = 2 is very specific
here. We will consider this case with the following additional restriction.
We say that a field F is 2-closed, if an equation x2 = a has a solution in
F for every element a 6= 0. We note that every locally finite (in particular,
finite) field of characteristic 2 is 2-closed.
Theorem 18. Let L be a hyperabelian 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.
As we can see in Corollary 1 the fact that γc+1(L) = 〈0〉 is equivalent
to the fact that ζc(L) = L, i.e. the lower and the upper central series in
nilpotent Leibniz algebras have the same length. The next natural step is
the consideration of the case, when the upper (respectively lower) central
series has finite length. For this case the question about the relationships
between L/ζk(L) and γk+1(L) naturally appears.
If L is a Lie algebra such that L/ζk(L) is finitely dimensional, then
γk+1(L) is also finitely dimensional, it follows from Theorem 5.2 of the
paper [43] by I. Stewart. A corresponding result for groups has been
obtained early by R. Baer [7]. In the paper [30] the following analog of
these theorems has been obtained.
Theorem 19. Let L be a Leibniz algebra over a field F . Suppose that
codimF (ζk(L)) = d is finite for some positive integer k. Then γk+1(L)
has finite dimension. Moreover dimF (γk+1(L)) 6 2k−1dk+1.
As a corollary we obtained a bound for a dimension of γk+1(L) in a
Lie algebra L.
Corollary 10. Let L be a Lie algebra over a field F . Suppose that
codimF (ζk(L)) = d is finite for some positive integer k. Then γk+1(L)
has finite dimension. Moreover dimF (γk+1(L)) 6 1
2dk−1(d − 1).
Kirichenko, Kurdachenko, Pypka, Subbotin 29
An important specific case here is the case when the center of a Leibniz
algebra L has finite codimension. For Lie algebras the following result is
well known (see, for example [47]).
Theorem 20. Let L be a Lie algebra over a field F . If a factor-algebra
L/ζ(L) has finite dimension d, then the derived subalgebra [L, L] also has
finite dimension, moreover, dimF ([L, L]) 6 1
2d(d + 1).
A corresponding result for groups was proved much earlier.
Theorem 21. Let G be a group, C a subgroup of the center ζ(G) such
that G/C is finite. Then the derived subgroup [G, G] is finite.
In this formulation, for the first time it appears in the paper of
B.H. Neumann [36]. This theorem was obtained also by R. Baer [7].
For Leibniz algebras we obtain the following analog of these results.
Theorem 22. Let L be a Leibniz algebra over a field F . Suppose that
codimF (ζ left(L)) = d and codimF (ζright(L)) = r are finite. Then [L, L]
has finite dimension, moreover, dimF ([L, L]) 6 d(d + r).
In this connection, the following question appears: suppose that only
codimF (ζ left(L)) is finite. Is dimF ([L, L]) finite? The above constructed
Example 4 gives a negative answer on this question.
Corollary 11. Let L be a Leibniz algebra over a field F . Suppose that
codimF (ζ(L)) = d is finite. Then [L, L] has finite dimension. Moreover,
dimF ([L, L]) 6 d2.
Corollary 12. Let L be a Leibniz algebra over a field F . Suppose that
codimF (ζ(L)) = d is finite. Then the Leibniz kernel of L has finite
dimension at most 1
2d(d − 1).
We did not talk about the links of Leibniz algebras with other algebraic
structures. However, in conclusion, we would like to note one such link of
Leibniz algebras with not very ordinary but interesting algebraic structures
that were introduced by J.-L. Loday (see [35]).
Let D be a vector space over a field F . Then D is called a dialgebra
if two associative binary operation ⊢ and ⊣ are defined on D and they
satisfy the following conditions:
(D1) x ⊢ (y ⊣ z) = (x ⊢ y) ⊣ z,
(D2) x ⊣ (y ⊢ z) = x ⊣ (y ⊣ z),
(D3) (x ⊣ y) ⊢ z = (x ⊢ y) ⊢ z
30 Some aspects of Leibniz algebra theory
for all x, y, z ∈ D.
Note that our use of ⊢ and ⊣ in this bracket is the opposite of that of
Loday. This convention matches our preference for left Leibniz algebras
instead of right Leibniz algebras.
For a given a dialgebra D we define the operation [·, ·] by the rule
[x, y] = x ⊢ y − y ⊣ x, x, y ∈ D.
One can check that D becomes a Leibniz algebra relatively the operations
+ and [·, ·]. This algebra is called a Leibniz algebra associated with dialgebra
D. Conversely, J.-L. Loday proved that for any Leibniz algebra L there
exists a dialgebra D(L) such that a Leibniz algebra associated with D(L)
includes a subalgebra, which is isomorphic to L.
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Contact information
V.V. Kirichenko Faculty of Mechanics and Mathematics, Taras
Shevchenko National University of Kyiv, Volo-
dymyrska str., 64, Kyiv, 01033, Ukraine
E-Mail(s): vv.kirichenko@gmail.com
Kirichenko, Kurdachenko, Pypka, Subbotin 33
L.A. Kurdachenko,
A.A. Pypka
Department of Geometry and Algebra, Faculty
of Mechanics and Mathematics, Oles Honchar
Dniprovsk National University, Gagarin ave., 72,
Dnipro, 49010, Ukraine
E-Mail(s): lkurdachenko@i.ua,
pypka@ua.fm
I.Ya. Subbotin Department of Mathematics and Natural Scien-
ces, College of Letters and Sciences, National
University, 5245 Pacific Concourse Drive, LA,
CA 90045,USA
E-Mail(s): isubboti@nu.edu
Received by the editors: 02.06.2017.
|
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| language | English |
| last_indexed | 2025-12-07T17:58:34Z |
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| spelling | Kirichenko, V.V. Kurdachenko, L.A. Pypka, A.A. Subbotin, I.Ya. 2019-06-18T10:15:35Z 2019-06-18T10:15:35Z 2017 Some aspects of Leibniz algebra theory / V.V. Kirichenko, L.A. Kurdachenko, A.A. Pypka, I.Ya. Subbotin // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 1. — С. 1-33. — Бібліогр.: 48 назв. — англ. 1726-3255 2010 MSC:17A32, 17A60. https://nasplib.isofts.kiev.ua/handle/123456789/156236 One of the key tendencies in the development of Leibniz algebra theory is the search for analogues of the basic results of Lie algebra theory. In this survey, we consider the reverse situation. Here the main attention is paid to the results reflecting the difference of the Leibniz algebras from the Lie algebras. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Some aspects of Leibniz algebra theory Article published earlier |
| spellingShingle | Some aspects of Leibniz algebra theory Kirichenko, V.V. Kurdachenko, L.A. Pypka, A.A. Subbotin, I.Ya. |
| title | Some aspects of Leibniz algebra theory |
| title_full | Some aspects of Leibniz algebra theory |
| title_fullStr | Some aspects of Leibniz algebra theory |
| title_full_unstemmed | Some aspects of Leibniz algebra theory |
| title_short | Some aspects of Leibniz algebra theory |
| title_sort | some aspects of leibniz algebra theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/156236 |
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