Leibniz algebras with absolute maximal Lie subalgebras
A Lie subalgebra of a given Leibniz algebra is said to be an absolute maximal Lie subalgebra if it has codimension one. In this paper, we study some properties of non-Lie Leibniz algebras containing absolute maximal Lie subalgebras. When the dimension and codimension of their Lie-center are greater...
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nasplib_isofts_kiev_ua-123456789-1885012025-02-09T18:18:34Z Leibniz algebras with absolute maximal Lie subalgebras Biyogmam, G.R. Tcheka, C. A Lie subalgebra of a given Leibniz algebra is said to be an absolute maximal Lie subalgebra if it has codimension one. In this paper, we study some properties of non-Lie Leibniz algebras containing absolute maximal Lie subalgebras. When the dimension and codimension of their Lie-center are greater than two, we refer to these Leibniz algebras as s-Leibniz algebras (strong Leibniz algebras). We provide a classification of nilpotent Leibniz s-algebras of dimension up to five. 2020 Article Leibniz algebras with absolute maximal Lie subalgebras / G.R. Biyogmam, C. Tcheka // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 1. — С. 52–65. — Бібліогр.: 18 назв. — англ. 1726-3255 DOI:10.12958/adm1165 2010 MSC: 17A32, 17B55, 18B99. https://nasplib.isofts.kiev.ua/handle/123456789/188501 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України |
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A Lie subalgebra of a given Leibniz algebra is said to be an absolute maximal Lie subalgebra if it has codimension one. In this paper, we study some properties of non-Lie Leibniz algebras containing absolute maximal Lie subalgebras. When the dimension and codimension of their Lie-center are greater than two, we refer to these Leibniz algebras as s-Leibniz algebras (strong Leibniz algebras). We provide a classification of nilpotent Leibniz s-algebras of dimension up to five. |
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Biyogmam, G.R. Tcheka, C. |
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Biyogmam, G.R. Tcheka, C. Leibniz algebras with absolute maximal Lie subalgebras Algebra and Discrete Mathematics |
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Biyogmam, G.R. Tcheka, C. |
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Leibniz algebras with absolute maximal Lie subalgebras |
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Leibniz algebras with absolute maximal Lie subalgebras |
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Leibniz algebras with absolute maximal Lie subalgebras |
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Leibniz algebras with absolute maximal Lie subalgebras |
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Leibniz algebras with absolute maximal Lie subalgebras |
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leibniz algebras with absolute maximal lie subalgebras |
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Leibniz algebras with absolute maximal Lie subalgebras / G.R. Biyogmam, C. Tcheka // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 1. — С. 52–65. — Бібліогр.: 18 назв. — англ. |
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Algebra and Discrete Mathematics |
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“adm-n1” — 2020/5/14 — 19:35 — page 52 — #60
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 29 (2020). Number 1, pp. 52–65
DOI:10.12958/adm1165
Leibniz algebras with absolute maximal Lie
subalgebras
G. R. Biyogmam and C. Tcheka
Communicated by L. A. Kurdachenko
Abstract. A Lie subalgebra of a given Leibniz algebra is
said to be an absolute maximal Lie subalgebra if it has codimension
one. In this paper, we study some properties of non-Lie Leibniz
algebras containing absolute maximal Lie subalgebras. When the
dimension and codimension of their Lie-center are greater than two,
we refer to these Leibniz algebras as s-Leibniz algebras (strong
Leibniz algebras). We provide a classification of nilpotent Leibniz
s-algebras of dimension up to five.
1. Introduction
Leibniz algebras were introduced by Jean-Louis Loday (see [16]) as
noncommutative versions of Lie algebras. Several authors (see [2,3,5,6,11,
12] for examples) have investigated whether the results on Lie algebras can
be extended to Leibniz algebras. In [5] for instance, D. Barnes proved an
analogue of Levi’s decomposition for Leibniz algebras, which states that
every finite dimensional Leibniz algebra g over a field of characteristic zero
can be written as a direct sum of its solvable radical and a semisimple Lie
subalgebra. Note that in this theorem, the Lie algebra s is not necessarily
a maximal Lie subalgebra of g. In this paper, we examine non-Lie Leibniz
algebras that can be written as semi-direct sum in which one summand is
a maximal Lie subalgebra of codimension one. We will refer to these Lie
subalgebras to as absolute maximal Lie subalgebras. Maximal subalgebras
2010 MSC: 17A32, 17B55, 18B99.
Key words and phrases: Leibniz algebras, s-Leibniz algebras, Lie-center.
https://doi.org/10.12958/adm1165
“adm-n1” — 2020/5/14 — 19:35 — page 53 — #61
G. R. Biyogmam, C. Tcheka 53
of codimension one in Lie algebras have been studied in [1,17,18]. We will
particularly focus on the Leibniz algebras containing Lie subalgebras of
codimension one, for which the Lie-center has dimension and codimension
greater than two. We call them s-Leibniz algebras. We investigate their
properties in section 3. This work opens the interesting debate as to
evaluate how much properties does an s-Leibniz algebra inherit from its
absolute maximal Lie subalgebra structure. For instance, we prove that an
s-Leibniz algebra g with absolute maximal Lie subalgebra s is nilpotent
and solvable if s is a nilpotent ideal. Also, we provide certain conditions
under which a Leibniz algebra is an s-Leibniz algebra. We also show that
all absolute maximal Lie subalgebras of an s-Leibniz algebra meet at its
Lie-center. In section 4, we provide a classification of nilpotent s-Leibniz
algebra of dimension up to five.
2. Leibniz algebras
We fix K as a ground field such that 1
2 ∈ K. All vector spaces and
tensor products are considered over K. A (left) Leibniz algebra [15, 16] is
a vector space g equipped with a bilinear map [−,−] : g⊗ g → g, usually
called the Leibniz bracket of g, satisfying the Leibniz identity :
[x, [y, z]] = [[x, y], z] + [y, [x, z]], x, y, z ∈ g.
Leibniz algebras form a semi-abelian category [7, 14], denoted by Leib,
whose morphisms are linear maps that preserve the Leibniz bracket. The
Leibniz identity above generalizes the Jacobi identity since g becomes a Lie
algebra when this bracket is antisymmetric. This defines the inclusion
functor Lie →֒ Leib. For a Leibniz algebra g, we denote by gann the subspace
of g spanned by all elements of the form [x, x], x ∈ g. Given a Leibniz
algebra g, it is clear that the quotient g
Lie
= g/gann is a Lie algebra. This
defines a left adjoint functor to Lie →֒ Leib, known as the Liezation functor
(−)Lie : Leib → Lie, which assigns the Lie algebra g
Lie
to a given Leibniz
algebra g.
A subalgebra h of a Leibniz algebra g is said to be left (resp. right)
ideal of g if [h, q] ∈ h (resp. [q, h] ∈ h), for all h ∈ h, q ∈ g. If h is both
left and right ideal, then h is called two-sided ideal of g. In this case
g/h naturally inherits a Leibniz algebra structure. The Lie-center of the
Leibniz algebra g is the two-sided ideal
ZLie(g) = {z ∈ g | [g, z] + [z, g] = 0 for all g ∈ g}.
“adm-n1” — 2020/5/14 — 19:35 — page 54 — #62
54 Leibniz algebras
For a Leibniz algebra g and two-sided ideals m and n of g, the Lie-centralizer
of m and n over g is
CLie
g (m, n) = {g ∈ g | [g,m] + [m, g] ∈ n, for all m ∈ m} .
In particular, the Lie-centralizer of m is defined by
CLie
g (m, 0) = {g ∈ g/[g,m] + [m, g] = 0, for all m ∈ m}.
These notions of Lie-center and Lie-centralizer were studied in [4,6,8,9] as
a result of approaching the relative theory of Leibniz algebras with respect
to the Liezation functor. Note that ZLie(g) = {x ∈ g | CLie
g ({x}, 0) = g}.
Also, [x, y]Lie = [x+ y, x+ y]− [x, x]− [y, y] for all x, y ∈ g. Consequently,
[g, g]Lie ⊆ gann. Considering the set T = {t ∈ g | [t, t] = 0}, it is clear that
the sets T and g are equal iff g is a Lie algebra.
3. Leibniz algebras with absolute maximal Lie
subalgebras
In this section, we describe non-Lie Leibniz algebras containing absolute
maximal Lie subalgebras. We discuss the cases where dim(ZLie(g)) = n−1
or dim(ZLie(g)) = 1, and pay a particular attention to the non-Lie Leibniz
algebras g satisfying 1 < dim(ZLie(g)) < n− 1.
Definition 3.1. Let g be a Leibniz algebra. A subalgebra s of g is called
absolute maximal Lie subalgebra if s is a maximal Lie subalgebra of g of
codimension one.
Clearly, if g is a Lie algebra, then all maximal subalgebras of codi-
mension one are absolute maximal Lie subalgebras. Studies of maximal
subalgebras of codimension one in Lie algebras can be found in [1, 17, 18].
For the remaining of the paper, we are interested in non-Lie Leibniz
algebras.
Proposition 3.2. Let g be a finite dimensional non-Lie Leibniz algebra
and s a maximal Lie subalgebra of g. Then ZLie(g) is a 2-sided ideal of s.
Proof. It is enough to show that ZLie(g) ⊆ s since ZLie(g) is a 2-sided
ideal of g. Indeed, let g ∈ ZLie(g) and assume that g /∈ s. Then, L :=
span{g, s} = g since s is a maximal subalgebra of g. Moreover L is
a Lie algebra because [g, x] = −[x, g] for all x ∈ g as g ∈ ZLie(g). This
contradicts the fact that g is a non-Lie Leibniz algebra.
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G. R. Biyogmam, C. Tcheka 55
Lemma 3.3. Let g be a n-dimensional non-Lie Leibniz algebra such that
(g− ZLie(g)) ∩ T 6= ∅. The following is true:
a) ZLie(g) is not a maximal Lie subalgebra of g,
b) dim(ZLie(g)) < n− 1,
c) If T is a subalgebra of g and |(g − ZLie(g)) ∩ T | > k, then
dim(ZLie(g)) < n− k for any k < n.
Proof. To prove a), let a /∈ ZLie(g) and a ∈ T. By definition of ZLie(g), we
have [x, a]Lie = 0 for all x ∈ ZLie(g). It follows that h := span{ZLie(g), a}
is a Lie subalgebra of g strictly containing ZLie(g). Moreover, if
dim(ZLie(g)) = n − 1, then g = h which contradicts the fact that g is
a non-Lie Leibniz algebra. This proves b).
To prove c), consider k < l < n fixed, and let a1 . . . al /∈ ZLie(g)
with [ai, ai] = 0 for all 1 6 i 6 l. Clearly [x, ai]Lie = 0 for all x ∈
ZLie(g) and 1 6 i 6 l. Also, since T is a subalgebra of g, we have
[ai, aj ]Lie = [ai + aj , ai + aj ] − [ai, ai] − [aj , aj ] = 0. It follows that h :=
span{ZLie(g), a1, . . . , al} is a Lie subalgebra of g. Since g is a non-Lie
Leibniz algebra of dimension n, it follows that dim(ZLie(g)) + l < n, and
thus dim(ZLie(g)) < n− l < n− k.
Proposition 3.4. Let g be a n-dimensional non-Lie Leibniz algebra, and
s a Lie subalgebra of g such that (g− s) ∩ T 6= ∅. If T is a subalgebra of
g, then s is not a maximal Lie subalgebra of g, and dim(s) < n− 1.
Proof. Let a /∈ s and a ∈ T, i.e. [a, a] = 0. Since T is a subalgebra of
g, we have [x, a]Lie = [x + a, x + a] − [x, x] − [a, a] = 0 for all x ∈ s. It
follows that h := span{s, a} is a Lie subalgebra of g strictly containing s.
Moreover, if dim(s) = n− 1, then g = h which contradicts the fact that g
is a non-Lie Leibniz algebra.
Corollary 3.5. Let g be a finite dimensional non-Lie Leibniz algebra such
that dim( g
ZLie(g)
) = 1. Then ZLie(g) is the unique absolute maximal Lie
subalgebra of g.
Proof. Since dim(ZLie(g)) = dim(g) − 1, one easily verifies that ZLie(g)
is a maximal Lie subalgebra of g. Now by Proposition 3.2, ZLie(g) is
contained in all maximal Lie subalgebra of g. The result follows by the
maximality of ZLie(g).
Example 3.6. Consider the Leibniz algebra g spanned by {a1, a2, a3,
a4, a5} with nonzero bracket [a1, a2] = a5, [a2, a1] = −a5, [a3, a4] =
“adm-n1” — 2020/5/14 — 19:35 — page 56 — #64
56 Leibniz algebras
a5, [a4, a3] = −a5, [a4, a4] = a5. Then one can verify that ZLie(g) =
span{a1, a2, a3, a5} is the unique maximal Lie subalgebra of g.
Proposition 3.7. Let g be a n-dimensional non-Lie left central Leibniz
algebra with one dimensional Lie-center. Then t = span{t ∈ g | [t, t] = 0}
is an absolute maximal Lie subalgebra of g.
Proof. Since g is left central, then for all t ∈ g, [t, t] ∈ ZLie(g). It follows
that g
ZLie(g)
= span{t + ZLie(g), [t, t] = 0}. Moreover, gann = ZLie(g),
so the map g → ZLie(g) defined by t 7→ [t, t] induces an isomorphism
g
t
∼= ZLie(g). Therefore the set t = span{t ∈ g | [t, t] = 0} is an absolute
maximal Lie subalgebra of g.
Proposition 3.8. Let g be a n-dimensional nilpotent non-Lie Leibniz
algebra with (n−1)-dimensional derived subalgebra. Then dim(ZLie(g)) = 1
and [g, g] is the unique absolute maximal Lie subalgebra of g.
Proof. By [2, Lemma 1], g is generated by {x1, x2, x3, . . . , xn} satisfying
the conditions [xi, x1] = xi+1 for 1 6 i 6 n− 1, [xi, xj ] = 0 for j > 2. It is
easily verified that ZLie(g) = span{xn} and [g, g] = span{x2, x3, . . . , xn}
is the only absolute maximal Lie subalgebra of g.
The following definition introduces a subclass of Leibniz algebras called
strong Leibniz algebras (s-Leibniz algebras).
Definition 3.9. A n-dimensional non-Lie Leibniz algebra g is said to be
an s-Leibniz algebra if:
(S1) dim(ZLie(g)) > 2 and codim(ZLie(g)) > 2;
(S2) g contains an absolute maximal Lie subalgebra.
Remark 3.10. Leibniz algebras of Corollary 3.5, Proposition 3.7 and
Proposition 3.8 do not satisfy the condition (S1). So they are not s-Leibniz
algebras.
Example 3.11. Consider the Leibniz algebra g spanned by {a, b, c, d, e}
with nonzero bracket [a, d] = e, [b, c] = e, [b, d] = e, [c, b] = −e, [c, c] = e,
[d, a] = −e and [d, b] = e. Then one can verify that ZLie(g) = span{a, e},
and g contains no absolute maximal Lie subalgebra, and so g is not
an s-Leibniz algebra.
Example 3.12. Consider the Leibniz algebra g spanned by {a1, a2, a3,
a4, a5} with nonzero bracket [a1, a1] = a4, [a1, a2] = a3, [a2, a1] = −a3,
[a2, a3] = a5, [a3, a2] = −a5, [a1, a4] = a5. Then one can verify that
ZLie(g) = span{a2, a3, a5}, and s = span{a2, a3, a4, a5} is an absolute
maximal Lie subalgebra of g. So g is an s-Leibniz algebra.
“adm-n1” — 2020/5/14 — 19:35 — page 57 — #65
G. R. Biyogmam, C. Tcheka 57
Remark 3.13. Let g be a finite dimensional s-Leibniz algebra with
absolute maximal Lie subalgebra s. By Proposition 3.2, ZLie(g) is a 2-
sided ideal of s.
Lemma 3.14. Let g be a n-dimensional s-Leibniz algebra with a absolute
maximal Lie subalgebra s. If s is a two-sided ideal of g, then [g, g] ⊆ s.
Proof. Let 〈x〉 be the complement of s in g. Then [a, x], [x, a], [a, b] ∈ s
for all a, b ∈ s. It remains to show that [x, x] ∈ s. Indeed assume that
[x, x] = αx+h for some α ∈ R and h ∈ s. Then by linearity and using the
Leibniz identity, we have α[x, x] + [h, x] = [αx+ h, x] = [[x, x], x] = 0 ∈ s.
Since [h, x] ∈ s, it follows that either α = 0 in which case [x, x] = h or
[x, x] = 0.
Corollary 3.15. Let g be a finite dimensional s-Leibniz algebra with
absolute maximal Lie subalgebra s. If s is a nilpotent two-sided ideal of g,
then g is solvable and nilpotent.
Proof. Using Lemma 3.14, one can easily verify that if s is nilpotent,
then [g, g] is nilpotent, and thus g is nilpotent. Moreover, g is solvable by
[13, Corollary 6].
Proposition 3.16. Let g be s-Leibniz algebra and let S be the set of all
absolute maximal Lie subalgebras of g. If |S| 6= 1, then ZLie(g) =
⋂
s∈S s.
Proof. By Proposition 3.13, ZLie(g) is contained in every absolute maximal
Lie subalgebra of g. Conversely, let g ∈
⋂
s∈S s and assume that g /∈ ZLie(g).
Then [x, g] + [g, x] 6= 0 for some x ∈ g. This implies that x /∈
⋃
s∈S s,
otherwise, x, g ∈ s0 for some Lie subalgebra of g, which contradicts
[x, g] + [g, x] 6= 0. So we have span{x, s} = g for all s ∈ S, and thus
|S| = 1. A contradiction.
Recall the following Levi’s Theorem proven by D. Barnes for Leibniz
algebras:
Theorem 3.17. [5] Every finite-dimensional Leibniz algebra g over a field
of characteristic 0 can be written as g = s⊕ Rad(g), where s is a semi-
simple Lie subalgebra of g, and Rad(g) is the solvable radical of g.
Proposition 3.18. Let g be a Leibniz algebra satisfying (S1), and h an
absolute maximal subalgebra of g containing ZLie(g), and such that h/ZLie(g)
is a semisimple Lie subalgebra of g/ZLie(g). Then g is an s-Leibniz algebra
with absolute maximal Lie subalgebra h.
“adm-n1” — 2020/5/14 — 19:35 — page 58 — #66
58 Leibniz algebras
Proof. By Theorem 3.17, h = s⊕Rad(h) for some semisimple Lie subalge-
bra s of h. But since h/ZLie(g) is a semisimple Lie subalgebra of g/ZLie(g),
it follows that h = ZLie(g) + s, which is clearly a Lie-subalgebra of g.
Therefore h is an absolute maximal Lie subalgebra of g.
Corollary 3.19. Let g be a Leibniz algebra satisfying (S1), and such that
g/ZLie(g) contains an absolute maximal semisimple subalgebra. Then g is
an s-Leibniz algebra.
Proof. Let h/ZLie(g) is an absolute maximal semisimple subalgebra of
g/ZLie(g). Then by the proof of Proposition 3.18, h is a Lie subalgebra of
g. It is easy to show that h is an absolute maximal subalgebra of g.
4. Classification of all nilpotent s-Leibniz algebras
dimension at most 5
In this section, we use the classification of four and five dimensional
nilpotent Leibniz Algebras provided in [10] to determine all nilpotent s-
Leibniz algebras of dimension four and five. This consists for each Leibniz
algebra to find the Lie-center and the absolute maximal Lie subalgebra (if
it exists), and verify satisfaction of the condition (S1) on the Lie-center. As
pointed out in [10], these Leibniz algebras are non-split, i.e. they cannot
be written as a direct sum of two nontrivial ideals. Split Leibniz algebras
can be obtained from non split ones.
Theorem 4.1. Every four dimensional non-split nilpotent s-Leibniz alge-
bra is isomorphic to a Leibniz algebra spanned by {x1, x2, x3, x4} where
the nonzero products are given by following:
A12: [x1, x1]=x4, [x1, x2]=x3, [x2, x1]=−x3+x4, [x1, x3]=x4=−[x3, x1].
A13: [x1, x2]=x3, [x2, x1]=−x3+x4, [x2, x2]=x4, [x1, x3]=x4=−[x3, x1].
A14: [x1, x1] = x3, [x1, x2] = x4.
A15: [x1, x1] = x3, [x2, x1] = x4.
A16: [x1, x2] = x4, [x2, x1] = x3, [x2, x2] = −x3.
A17: [x1, x1] = x3, [x1, x2] = x4, [x2, x1] = αx4, α ∈ C/{−1, 0}
Proof. The result is obtained by checking each Leibniz algebra of Theorem
3.2.3 in [10]. We kept the same numbering of these Leibniz algebras as
in [10] for the convenience of the reader.
The basis of the corresponding Lie-centers and absolute maximal Lie
subalgebras are given in the table below:
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G. R. Biyogmam, C. Tcheka 59
4-dimensional non-split nilpotent s-Leibniz algebras
s-Leibniz algebra g Lie-center ZLie(g) Absolute maximal Lie
subalgebra s
A12, A14, A15, A17 {x3, x4} {x2, x3, x4}
A13, A16 {x3, x4} {x1, x3, x4}
Theorem 4.2. Every five dimensional non-split nilpotent s-Leibniz algebra
is isomorphic to a Leibniz algebra spanned by {x1, x2, x3, x4} where the
nonzero products are given by following:
B1: [x1, x2] = x3, [x2, x1] = −x3 + x5, [x1, x3] = x4 = −[x3, x1]..
B3: [x1, x2] = x3, [x2, x1] = −x3 + x5, [x2, x2] = x5, [x1, x3] = x4 =
−[x3, x1].
C2: [x1, x2] = x3, [x2, x1] = −x3 + x4, [x2, x2] = x5, [x1, x3] = x5 =
−[x3, x1].
C4: [x1, x2] = x3, [x2, x1] = −x3 + x5, [x2, x2] = x4, [x1, x3] = x4 =
−[x3, x1].
C8: [x1, x1] = x4, [x1, x2] = x3, [x2, x1] = −x3 + x5, [x1, x3] = x4 =
−[x3, x1].
D1: [x1, x1] = x3, [x2, x1] = −x4, [x1, x3] = x5.
D8: [x1, x1] = x3, [x1, x2] = x4, [x2, x1] = αx4, [x1, x3] = x5, α ∈
C/{−1}.
D14: [x1, x1] = x4, [x1, x2] = x3, [x1, x3] = x5.
D15: [x1, x1] = x4, [x1, x2] = x3, [x2, x1] = x5, [x1, x3] = x5
D18: [x1, x1] = x4, [x1, x2] = x3, [x2, x1] = x4, [x1, x3] = x5.
E1: [x1, x2] = −x3 + x4, [x2, x1] = x3, [x2, x3] = x5, [x3, x2] = −x5,
[x1, x4] = x5.
E10: [x1, x1] = x4, [x1, x2] = x3, [x2, x1] = −x3, [x2, x3] = x5, [x3, x2] =
−x5, [x1, x4] = x5.
F1: [x1, x2] = x3, [x2, x1] = −x3 + x5, [x1, x3] = x4, [x3, x1] = −x4,
[x2, x3] = x5, [x3, x2] = −x5.
F3: [x1, x2] = x3, [x2, x1] = −x3 + x5, [x2, x2] = x5, [x1, x3] = x4,
[x3, x1] = −x4, [x2, x3] = x5, [x3, x2] = −x5.
G1: [x1, x2] = x3, [x2, x1] = −x3 + x4, [x2, x2] = αx5, [x1, x3] = x4,
[x3, x1] = −x4, [x2, x3] = x5, [x3, x2] = −x5, α ∈ C/{−1}.
G3: [x1, x1] = αx5, [x1, x2] = x3, [x2, x1] = −x3 +x4 +x5, [x1, x3] = x4,
[x3, x1] = −x4, [x2, x3] = x5, [x3, x2] = −x5, α ∈ C/{−1}.
H1: [x1, x2] = x3, [x2, x1] = −x3 + x5, [x1, x3] = x4, [x3, x1] = −x4,
[x1, x4] = x5, [x4, x1] = −x5.
H3: [x1, x2] = x3, [x2, x1] = −x3 + x5, [x2, x2] = x5, [x1, x3] = x4,
[x3, x1] = −x4, [x1, x4] = x5, [x4, x1] = −x5.
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60 Leibniz algebras
H5: [x1, x2] = x3, [x2, x1] = −x3 + x5, [x2, x2] = αx5, [x1, x3] = x4,
[x2, x3] = x5, [x3, x2] = −x5, [x3, x1] = −x4, [x1, x4] = x5, [x4, x1] =
−x5, α ∈ C.
I6: [x1, x2] = x4, [x2, x1] = −x4, [x1, x3] = x5, [x1, x4] = x5, [x4, x1] =
−x5.
I8: [x1, x2] = x4, [x2, x1] = −x4, [x2, x3] = x5, [x1, x4] = x5, [x4, x1] =
−x5.
I13: [x1, x2] = x4, [x2, x1] = −x4 + x5, [x2, x3] = −x5, [x3, x2] = x5,
[x1, x4] = x5, [x4, x1] = −x5.
I14: [x1, x2] = x4, [x2, x1] = −x4, [x2, x3] = αx5, [x3, x2] = x5, [x1, x4] =
x5, [x4, x1] = −x5, α ∈ C.
I18: [x1, x2] = x4, [x2, x1] = −x4 + x5, [x2, x2] = x5, [x2, x3] = −x5,
[x3, x2] = x5, [x1, x4] = x5, [x4, x1] = −x5.
J8: [x1, x1] = x4, [x2, x3] = αx5, [x3, x2] = x5, [x1, x4] = x5.
J10: [x1, x2] = x4, [x2, x3] = αx5, [x3, x2] = x5, [x1, x4] = x5.
K2: [x1, x2] = x4, [x2, x1] = −x4, [x3, x1] = x5.
K4: [x1, x2] = x4, [x2, x1] = −x4, [x1, x3] = x5, [x3, x1] = αx5, α ∈
C/{−1}.
K5: [x1, x2] = x4, [x2, x1] = −x4 + x5, [x1, x3] = x5, [x3, x1] = −x5.
K6: [x1, x2] = x4, [x2, x1] = −x4, [x1, x3] = x5, [x3, x1] = −αx5,
[x3, x3] = x5, α ∈ C/{−1}.
K9: [x1, x2] = x4, [x2, x1] = −x4, [x1, x3] = x5, [x3, x2] = x5.
K12: [x1, x2] = x4, [x2, x1] = −x4 + x5, [x1, x3] = x5, [x3, x1] = −x5,
[x2, x3] = x5, [x3, x2] = −x5.
L14: [x1, x2] = x4, [x2, x3] = x5.
L18: [x1, x2] = αx4, [x2, x1] = x4, [x2, x3] = x5, α ∈ C/{−1}.
L35: [x1, x2] = x5, [x2, x2] = x4, [x2, x3] = x5.
L36: [x1, x2] = x4, [x2, x1] = −x4, [x2, x2] = x4, [x2, x3] = x5.
L42: [x2, x1] = x4, [x1, x3] = x5, [x2, x3] = −x5, [x3, x2] = x5.
L44: [x1, x2] = x4, [x2, x1] = αx4, [x2, x3] = βx5, [x3, x2] = x5, α, β ∈
C/{−1}.
L45: [x1, x1] = x5, [x1, x2] = x4, [x2, x1] = αx4, [x2, x3] = βx5, [x3, x2] =
x5, α ∈ C/{−1}, β ∈ C.
L46: [x1, x2] = x4, [x2, x1] = αx4 + x5, [x2, x3] = αx5, [x3, x2] = x5,
α ∈ C/{−1}.
L50: [x1, x2] = x4, [x2, x1] = αx4, [x2, x2] = x5, [x2, x3] = x5, [x3, x2] =
−x5, α ∈ C/{−1}.
L55: [x1, x1] = x4, [x2, x1] = x4, [x1, x3] = αx5, [x2, x3] = −x5, [x3, x2] =
x5, α ∈ C/{0}.
L56: [x1, x1] = x4, [x2, x1] = x4 + x5, [x2, x3] = x5, [x3, x2] = −x5.
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G. R. Biyogmam, C. Tcheka 61
L58: [x1, x1] = x4, [x1, x2] = x4, [x2, x1] = −x4, [x1, x3] = βx5, [x2, x3] =
−x5, [x3, x2] = x5, β ∈ C/{0}.
L60: [x1, x1] = x4, [x1, x2] = x4 + x5, [x2, x1] = αx4, [x2, x3] = −x5,
[x3, x2] = x5, α ∈ C.
L61: [x1, x1] = x4, [x1, x2] = x4 + x5, [x2, x1] = x4, [x1, x3] = αx5,
[x2, x3] = −x5, [x3, x2] = x5, α ∈ C/{−1, 0}.
L63: [x1, x1] = x4, [x1, x2] = x4 + x5, [x2, x1] = αx4, [x1, x3] = βx5,
[x2, x3] = −x5, [x3, x2] = x5, α ∈ C/{−1, 0}, β ∈ C/{0}, αβ 6= −1,
(α+ 1)β 6= −1
L66: [x1, x1] = x4, [x1, x2] = x4, [x2, x1] = x4 + x5, [x1, x3] = −x5,
[x2, x3] = −x5, [x3, x2] = x5.
L80: [x1, x2] = x4, [x2, x1] = x5, [x2, x3] = x4.
L82: [x1, x2] = x4, [x2, x1] = −x4, [x2, x3] = x4, [x3, x2] = x5.
L83: [x1, x2] = x4 + 2x5, [x2, x1] = −x5, [x2, x2] = x5, [x2, x3] = x4,
[x3, x2] = x5.
L89: [x1, x2] = x4, [x2, x1] = −x4 + x5, [x1, x3] = −x5, [x3, x1] = x5,
[x2, x3] = x4.
L89′: [x1, x2] = x4, [x2, x1] = −x4 + x5, [x1, x3] = γx5, [x3, x1] = x5,
[x2, x3] = x4, γ ∈ C/{−1}.
L89′′: [x1, x2] = x4 + αx5, [x2, x1] = βx4 + x5, [x1, x3] = −x5, [x3, x1] =
x5, [x2, x3] = x4, α ∈ C/{−1}, β ∈ C.
L92: [x1, x2] = x4, [x2, x1] = −x4, [x2, x2] = x5, [x1, x3] = −x5, [x3, x1] =
x5, [x2, x3] = x4.
L92′: [x1, x2] = x4 + αx5, [x2, x1] = βx4, [x2, x2] = x5, [x1, x3] = −x5,
[x3, x1] = x5, [x2, x3] = x4, α, β ∈ C with (α, β) 6= (0,−1).
L94: [x1, x2] = x4 − 2βx5, [x2, x1] = αx4 + βx5, [x2, x2] = x5, [x1, x3] =
−x5, [x3, x1] = x5, [x2, x3] = x4, α ∈ C, β ∈ C/{−1, 0, 1}.
L95: [x1, x2] = x4 + αx5, [x2, x1] = βx5, [x2, x2] = x5, [x1, x3] = −x5,
[x3, x1] = x5, [x2, x3] = x4, α ∈ C, β ∈ C/{0}, α 6= −2β, β2 +αβ +
1 6= 0.
M1: [x1, x2] = x4 − x5, [x2, x1] = −x4 + x5, [x1, x3] = γx5, [x3, x1] =
−γx5, [x2, x3] = x4, [x3, x3] = x5, γ ∈ C.
M1′: [x1, x2] = x4 − x5, [x2, x1] = −x4 + x5, [x1, x3] = γx5, [x3, x1] =
θx5, [x2, x3] = x4, [x3, x3] = x5, γ, θ ∈ C, γ 6= −θ.
M2: [x1, x2] = x4, [x2, x1] = −x4, [x1, x3] = θx5, [x3, x1] = −θx5,
[x2, x3] = x4, [x3, x3] = x5, θ ∈ C.
M2′: [x1, x2] = x4, [x2, x1] = −x4, [x1, x3] = θx5, [x3, x1] = δx5, [x2, x3] =
x4, [x3, x3] = x5, δ, θ ∈ C, δ 6= −θ.
M2′′: [x1, x2] = x4 + x5, [x2, x1] = −x4 − x5, [x1, x3] = θx5, [x3, x1] =
−θx5, [x2, x3] = x4, [x3, x3] = x5, θ ∈ C.
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62 Leibniz algebras
M2′′′: [x1, x2] = x4 + x5, [x2, x1] = −x4 − x5, [x1, x3] = θx5, [x3, x1] =
δx5, [x2, x3] = x4, [x3, x3] = x5, γ, θ ∈ C, δ 6= −θ.
M7: [x1, x1] = x4 + αx5, [x1, x2] = γx5, [x2, x1] = −γx5, [x1, x3] =
λx5, [x2, x3] = x4, [x3, x2] = −x4, α, γ, λ ∈ C.
M7′: [x1, x1] = x4 + αx5, [x1, x2] = βx4 + γx5, [x2, x1] = θx5, [x1, x3] =
λx5, [x2, x3] = x4, [x3, x2] = −x4, α, β, γ, λ ∈ C, β 6= 0 or γ 6= −θ.
M9: [x1, x1] = x4 + αx5, [x1, x2] = βx4 + γx5, [x3, x1] = x5, [x1, x3] =
−x5, [x2, x3] = x4, [x3, x2] = −x4,, α, β, γ ∈ C, β 6= 0 or γ 6= 0.
M9′: [x1, x1] = x4 + αx5, [x1, x2] = βx4 + γx5, [x3, x1] = x5, [x1, x3] =
δx5, [x2, x3] = x4, [x3, x2] = −x4, α, β, γ ∈ C, δ 6= −1, β 6= 0 or
γ 6= 0.
M11: [x1, x1] = x4 + αx5, [x1, x2] = βx4 + γx5, [x2, x1] = θx5, [x2, x3] =
−x4, [x3, x2] = x4, α, β, γ, θ ∈ C, β 6= 0 or γ 6= −θ.
M12: [x1, x1] = x4, [x1, x2] = αx4 + βx5, [x2, x1] = γx5, [x1, x3] = x5,
[x2, x3] = −x4, [x3, x2] = x4, α, β, γ, θ ∈ C, α 6= 0 or γ 6= −β.
M13: [x1, x1] = x4 + αx5, [x1, x2] = γx5, [x2, x1] = −γx5, [x3, x1] =
x5, [x1, x3] = λx5, [x2, x3] = −x4, [x3, x2] = x4, α, γ, λ ∈ C, λ 6= −1.
M13′: [x1, x1] = x4 + αx5, [x1, x2] = βx4 + γx5, [x2, x1] = θx5, [x3, x1] =
x5, [x1, x3] = λx5, [x2, x3] = −x4, [x3, x2] = x4, α, β, γ, θ, λ ∈ C,
λ 6= −1, and β 6= 0 or γ 6= −θ.
M15: [x1, x2] = −x4 + γx5, [x2, x1] = x4 − γx5, [x3, x1] = µx5, [x1, x3] =
x4 + λx5, [x2, x3] = ωx5, [x3, x2] = x4, γ, µ, λω ∈ C.
Proof. The result is obtained by checking each Leibniz algebra of the
following theorems in [10]. Again, for the convenience of the reader, we
kept the same numbering as in [10], and just differentiated theorems by
the Letters as follows: Theorem 4.2.3 for the Bs, Theorem 4.2.5 for the Cs,
Theorem 4.2.6 for the Ds, Theorem 4.2.7 for the Es, Theorem 4.2.9 for the
Fs, Theorem 4.2.10 for the Gs, Theorem 4.2.11 for the Hs, Theorem 4.3.1
for the Is, Theorem 4.3.2 for the Js, Theorem 4.3.3 for the Ks, Theorem
4.3.4 for the Ls and Ms.
The basis of the corresponding Lie-centers and absolute maximal Lie
subalgebras are given in the table below:
5-dimensional non-split nilpotent s-Leibniz algebras
s-Leibniz algebra g Lie-center
ZLie(g)
Maximal Lie
subalgebra s
B1, B3 {x3, x4, x5} {x1, x3, x4, x5}
C2, C4 {x3, x4, x5} {x1, x3, x4, x5}
C8 {x3, x4, x5} {x2, x3, x4, x5}
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G. R. Biyogmam, C. Tcheka 63
5-dimensional non-split nilpotent s-Leibniz algebras
s-Leibniz algebra g Lie-center
ZLie(g)
Maximal Lie
subalgebra s
D1, D8, D14, D15, D18 {x3, x4, x5} {x2, x3, x4, x5}
E1 {x4, x5} {x2, x3, x4, x5}
E10 {x2, x3, x5} {x2, x3, x4, x5}
F1 {x3, x4, x5} {x1, x3, x4, x5},
{x2, x3, x4, x5}
F3 {x3, x4, x5} {x1, x3, x4, x5}
G1 {x3, x4, x5} {x1, x3, x4, x5}
G3 {x3, x4, x5} {x2, x3, x4, x5}
H1 {x3, x4, x5} {x1, x3, x4, x5},
{x2, x3, x4, x5}
H3, H5 {x3, x4, x5} {x1, x3, x4, x5}
I6 {x2, x4, x5} {x1, x2, x4, x5},
{x2, x3, x4, x5}
I8, I14 {x1, x4, x5} {x1, x2, x4, x5},
{x1, x3, x4, x5}
I13 {x3, x4, x5} {x1, x3, x4, x5},
{x2, x3, x4, x5}
I18 {x3, x4, x5} {x1, x3, x4, x5}
J8 {x2, x3, x5} {x2, x3, x4, x5}
J10 {x3, x5} {x2, x3, x4, x5}
K2 {x2, x4, x5} {x2, x3, x4, x5}
K4 {x2, x4, x5} {x1, x2, x4, x5},
{x2, x3, x4, x5}
K5 {x3, x4, x5} {x2, x3, x4, x5}
K6 {x2, x4, x5} {x1, x2, x4, x5}
K9 {x4, x5} {x1, x2, x4, x5}
K12 {x3, x4, x5} {x1, x3, x4, x5},
{x2, x3, x4, x5}
L14, L16, L35, L44, L46,
L50, L80, L83, L89′′, L92′,
L94, L95
{x4, x5} {x1, x3, x4, x5}
L36, L89, L92 {x1, x4, x5} {x1, x3, x4, x5}
L42, L55, L60, L61, L63, L66 {x4, x5} {x2, x3, x4, x5}
L58 {x2, x4, x5} {x2, x3, x4, x5}
L45, L56 {x3, x4, x5} {x2, x3, x4, x5}
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64 Leibniz algebras
5-dimensional non-split nilpotent s-Leibniz algebras
s-Leibniz algebra g Lie-center
ZLie(g)
Maximal Lie
subalgebra s
L82, L89′, L92 {x1, x4, x5} {x1, x2, x4, x5}
M1,M2,M2′′ {x1, x4, x5} {x1, x2, x4, x5}
M1′,M2′,M2′′′,M15 {x4, x5} {x1, x2, x4, x5}
M7,M13 {x2, x4, x5} {x2, x3, x4, x5}
M8,M10,M12,M14 {x4, x5} {x2, x3, x4, x5}
M9,M11 {x3, x4, x5} {x2, x3, x4, x5}
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Contact information
G. R. Biyogmam Department of Mathematics, Georgia College &
State University, Campus Box 17 Milledgeville,
GA 31061-0490
E-Mail(s): guy.biyogmam@gcsu.edu
Calvin Tcheka Department of Mathematics, University of
Dschang, Dschang, Cameroun
E-Mail(s): jtcheka@gmail.com
Received by the editors: 15.05.2018.
G. R. Biyogmam, C. Tcheka
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