Normal automorphisms of the metabelian product of free abelian Lie algebras
Let M be the metabelian product of free abelian Lie algebras of finite rank. In this study we prove that every normal automorphism of M is an IA-automorphism and acts identically on M′.
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Öğüşlü, N.Ş. 2023-03-06T14:39:38Z 2023-03-06T14:39:38Z 2020 Normal automorphisms of the metabelian product of free abelian Lie algebras / N.Ş. Öğüşlü // Algebra and Discrete Mathematics. — 2020. — Vol. 30, № 2. — С. 230–234. — Бібліогр.: 5 назв. — англ. 1726-3255 DOI:10.12958/adm1258 2010 MSC: 17B01, 17B40. https://nasplib.isofts.kiev.ua/handle/123456789/188565 Let M be the metabelian product of free abelian Lie algebras of finite rank. In this study we prove that every normal automorphism of M is an IA-automorphism and acts identically on M′. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Normal automorphisms of the metabelian product of free abelian Lie algebras Article published earlier |
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Normal automorphisms of the metabelian product of free abelian Lie algebras |
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Normal automorphisms of the metabelian product of free abelian Lie algebras Öğüşlü, N.Ş. |
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Normal automorphisms of the metabelian product of free abelian Lie algebras |
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Normal automorphisms of the metabelian product of free abelian Lie algebras |
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Normal automorphisms of the metabelian product of free abelian Lie algebras |
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Normal automorphisms of the metabelian product of free abelian Lie algebras |
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normal automorphisms of the metabelian product of free abelian lie algebras |
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Öğüşlü, N.Ş. |
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Öğüşlü, N.Ş. |
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2020 |
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English |
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Algebra and Discrete Mathematics |
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Інститут прикладної математики і механіки НАН України |
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Let M be the metabelian product of free abelian Lie algebras of finite rank. In this study we prove that every normal automorphism of M is an IA-automorphism and acts identically on M′.
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Normal automorphisms of the metabelian product of free abelian Lie algebras / N.Ş. Öğüşlü // Algebra and Discrete Mathematics. — 2020. — Vol. 30, № 2. — С. 230–234. — Бібліогр.: 5 назв. — англ. |
| work_keys_str_mv |
AT ogusluns normalautomorphismsofthemetabelianproductoffreeabelianliealgebras |
| first_indexed |
2025-11-25T21:10:19Z |
| last_indexed |
2025-11-25T21:10:19Z |
| _version_ |
1850551762558124032 |
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© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 30 (2020). Number 2, pp. 230–234
DOI:10.12958/adm1258
Normal automorphisms of the metabelian
product of free abelian Lie algebras
N. Ş. Öğüşlü
Communicated by I. Ya. Subbotin
Abstract. Let M be the metabelian product of free abelian
Lie algebras of finite rank. In this study we prove that every normal
automorphism of M is an IA-automorphism and acts identically
on M ′.
1. Introduction
Let L be a Lie algebra over a field K. An automorphism ϕ of L is
called a normal automorphism if ϕ (I) = I for every ideal I of L. The set
of normal automorphisms of L is a normal subgroup of the automorphism
group of L.
Automorphisms and more particularly normal automorphisms have a
very important place in groups and Lie algebras. Let G be a soluble product
of class n > 2 of nontrivial free abelian groups. In [5] it is shown that the
subgroup of all normal automorphisms of G coincides with the subgroup
of all inner automophisms. In [4] Romankov showed that if S is a free
non-abelian soluble group, then the subgroup of normal automorphisms
of S is the subgroup of inner automorphisms of S. In [1] it is studied
normal automorphisms of a free metabelian nilpotent group. Let Lm,c be
the free m-generated metabelian nilpotent of class c Lie algebra over a
field of characteristic zero. In [2] it is shown that the group of normal
2010 MSC: 17B01, 17B40.
Key words and phrases: free abelian Lie algebras, metabelian product, auto-
morphisms.
https://doi.org/10.12958/adm1258
N. Ş. Öğüşlü 231
automorphisms of Lm,c is contained by the group of IA-automorphisms of
Lm,c for m > 3, c > 2.
For an arbitrary variety of Lie algebras, the metabelian product of Lie
algebras Fi, i = 1, . . . ,m is defined as
(
∗∏
Fi
)
/
(
D ∩ F ′′
)
,
where F =
∏
∗ Fi is the free product of the Lie algebras Fi and D is
the cartesian subalgebra of
∏
∗ Fi. If the algebras Fi are non-trivial free
abelian Lie algebras then the metabelian product of them is isomorphic
to F/F ′′, where F ′′ = [F ′, F ′] and F ′ = [F, F ] is the derived subalgebra.
Let M be the metabelian product of free abelian Lie algebras of finite
rank. In this study it is shown that every normal automorphism of M is
an IA-automorphism and acts identically on M ′. In proving this result we
inspired by the result of Timoshenko in the case of groups [5].
Let L be a Lie algebra and B any subset of L. We show that by 〈B〉
the ideal of L generated by the set B.
2. Normal automorphisms of metabelian product
Let Ai, i = 1, . . . ,m, be free abelian Lie algebras of finite rank over
a field K of characteristic zero and F =
∏
∗Ai is the free product of the
abelian Lie algebras Ai, i = 1, . . . ,m. If M is the metabelian product of
the algebras Ai, M is isomorphic to F/F ′′.
Definition 1. Let L be a Lie algebra. An automorphism ϕ of L is called
a normal automorphism If ϕ (I) = I for evey ideal I of L.
Theorem 1. Let Ai, i = 1, . . . ,m, be free abelian Lie algebras of finite
rank and let M be their metabelian product. If ϕ is a normal automorphism
of M then ϕ is an IA-automorphism.
Proof. Let ϕ be a normal automorphism of M . The algebra M can be
considered as M = F/F ′′. Let denote by v̂ = v + F ′′, where v ∈ F. Then
by [3] there exist ui ∈ F ′, 1 6 i 6 m, such that
ϕ (âi) = αâi + ûi,
where ai ∈ Ai and 0 6= α ∈ K. Consider the ideal 〈â1〉 of M . Since ϕ is
normal we have ϕ (â1) ∈ 〈â1〉 and so ũ1 ∈ 〈â1〉 and similarly, for the ideal
232 Normal automorphisms of metabelian product
〈
̂[a2, a3]
〉
of M we have ϕ
(
̂[a2, a3]
)
∈
〈
̂[a2, a3]
〉
. Then for an element ŷ
of
〈
̂[a2, a3]
〉
we have
ϕ
(
̂[a2, a3]
)
= α2 ̂[a2, a3] + ŷ.
Now consider the ideal
〈
̂a1 + [a2, a3]
〉
of M. Since ϕ is normal we have
ϕ
(
̂a1 + [a2, a3]
)
∈
〈
̂a1 + [a2, a3]
〉
and for an element ẑ of
〈
̂a1 + [a2, a3]
〉
ϕ
(
̂a1 + [a2, a3]
)
= c
(
̂a1 + [a2, a3]
)
+ ẑ
where c ∈ K. From the last equality we have
(α− c) â1 +
(
α2 − c
)
̂[a2, a3] = 0̂.
Then we get c = α and c = α2, that is, α2 = α. Hence α = 1 and ϕ is an
IA-automorphism.
Theorem 2. Every normal automorphism of M acts identically on M ′.
Proof. The algebra M can be considered as M = F/F ′′. Let denote by
v̂ = v + F ′′, where v ∈ F. Let ϕ be a normal automorphism of M . By
theorem 1 we have that ϕ is an IA-automorphism. Then there is an
element v̂ of M ′ such that
ϕ
(
̂[a1, a2]
)
= ̂[a1, a2] + v̂,
where a1 ∈ A1, a2 ∈ A2. Let H be the ideal of M ′ generated by the
element ̂[a1, a2]. It is clear that
H =
{
c ̂[a1, a2] : c ∈ K
}
.
Now suppose that v̂ 6= 0̂. Consider the homomorphism θ : M ′ → M ′/H
which is defined θ (û) = ϕ (û) + H for every element û ∈ M ′. Since ϕ
is a normal automorphism of M it is clear that θ is an epimorphism.
Let û ∈ Kerθ. Consider the ideal 〈û〉 of M . Since ϕ is normal we have
ϕ (û) ∈ 〈û〉 . Then we have ϕ (û) = βû+ ŵ, where ŵ ∈ 〈û〉 , β ∈ K. Since
û ∈ Kerθ, we have ϕ (û) ∈ H, that is,
βû+ ŵ ∈
{
c ̂[a1, a2] : c ∈ K
}
.
N. Ş. Öğüşlü 233
Thus we have û = d ̂[a1, a2], where d ∈ K. Then we get
ϕ (û) = d ̂[a1, a2] + dv̂ ∈ H.
If v̂ 6= 0̂ we get d = 0 and û = 0̂. Hence we obtain that θ is an isomorphism.
Since ϕ (M ′) = M ′ and v̂ ∈ M ′ there exist an element ĝ of M ′ such that
ϕ (ĝ) = v̂. By the definition of θ we have
θ (ĝ) = v̂ +H.
We also have that
θ
(
̂[a1, a2]
)
= v̂ +H.
Since θ is an isomorphism we get
ĝ = ̂[a1, a2].
Thus we have
ϕ
(
̂[a1, a2]
)
= ϕ (ĝ) = v̂
and
̂[a1, a2] + v̂ = v̂.
We obtaian that ̂[a1, a2] = 0̂. This is a contradiction. Thus we get v̂ = 0̂
and
ϕ
(
̂[a1, a2]
)
= ̂[a1, a2].
Similarly, we obtain that
ϕ
(
[̂ai, aj ]
)
= [̂ai, aj ],
where ai ∈ Ai, aj ∈ Aj , 1 6 i < j 6 m. Let û ∈ M ′. Then û is a linear
combinations of some elemets of M of the form
̂[. . . [[aj1 , aj2 ] , aj3 ] , . . . , ajn ],
where aj1 , aj2 , . . . , ajn ∈
m⋃
i=1
Ai, n > 2. we know that
ϕ
(
̂[aj1 , aj2 ]
)
= ̂[aj1 , aj2 ].
Since ϕ is an IA-automorphism there exist some elements uj3 , . . . , ujn ∈ F ′
such that
ϕ (âjk) = âjk + ûjk , k > 3.
234 Normal automorphisms of metabelian product
Then
ϕ
(
̂[. . . [[aj1 , aj2 ] , aj3 ] , . . . , ajn ]
)
=
[
. . .
[
ϕ
(
̂[aj1 , aj2 ]
)
, ϕ
(
âj3
)]
, . . . , ϕ(âjn)
]
=
[
. . .
[
̂[aj1 , aj2 ], âj3 + ûj3
]
, . . . , âjn + ûjn
]
=
̂
[
. . . [[aj1 , aj2 ] , aj3 ] , . . . , ajn
]
.
Hence we get ϕ (û) = û for all û ∈ M ′. Therefore ϕ acts identically
on M ′.
References
[1] G. Endimioni, Normal automorphisms of a free metabelian nilpotent group, Glasgow
Math.J., 52 (2010), 169-177.
[2] Ş. Fındık, Normal and normally outer automorphisms of free metabelian nilpotent
Lie algebras, Serdica Math. J., 36 (2010), 171–210.
[3] N. Ş. Öğüşlü, IA-automorphisms of a solvable product of abelian Lie algebras, Int.
J. of Sci. and Research Pub., 8 (2018), no.4, 84-85.
[4] V. A. Romankov, Normal automorphisms of discrete groups, Siberian Math. J., 24
(1983), no.4, 604-614.
[5] E. I. Timoshenko, Normal automorphisms of a soluble product of abelian groups,
Siberian Math. J., 56 (2015), no.1, 191-198.
Contact information
Nazar Şahin Öğüşlü Department of Mathematics, Çukurova
University, Adana,Turkey
E-Mail(s): noguslu@cu.edu.tr
Received by the editors: 19.09.2018
and in final form 23.01.2020.
mailto:noguslu@cu.edu.tr
N. Ş. Öğüşlü
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