This paper classifies all right solvable indecomposable extensions of a specific naturally graded quasi-filiform Leibniz algebra of second type, expanding understanding of its algebraic structure.
Contribution
It provides a complete classification of solvable extensions of the algebra $ ext{L}^4$, a previously studied algebraic structure, revealing new algebraic configurations.
Findings
01
All possible right solvable indecomposable extensions are constructed.
02
The classification enhances understanding of the algebra's extension properties.
03
Results contribute to the theory of Leibniz algebra extensions.
Abstract
For a sequence of the naturally graded quasi-filiform Leibniz algebra of second type L4 introduced by Camacho, G\'{o}mez, Gonz\'{a}lez and Omirov, all the possible right solvable indecomposable extensions over the field C are constructed.
Tables6
Table 1. Table 1. Right Leibniz identities in case ( 1 ) 1 (1) in Theorem 5.1.1 , ( n ≥ 5 𝑛 5 n\geq 5 ).
Steps
Ordered triple
Result
where is fixed
where is fixed.
Combining with
Table 2. Table 2. Right Leibniz identities in case (1) (a) with a nilradical ℒ 4 , ( n = 4 ) superscript ℒ 4 𝑛 4 \mathscr{L}^{4},(n=4) .
Steps
Ordered triple
Result
Table 3. Table 3. Right Leibniz identities in the codimension two nilradical ℒ 4 , ( n ≥ 5 ) superscript ℒ 4 𝑛 5 \mathscr{L}^{4},(n\geq 5) .
Steps
Ordered triple
Result
where
where
Altogether with
where
where
Combining with
.
Table 4. Table 4. Left Leibniz identities in case ( 1 ) 1 (1) in Theorem 5.2.1 , ( n ≥ 5 𝑛 5 n\geq 5 ).
Steps
Ordered triple
Result
Table 5. Table 5. Left Leibniz identities in case (1) (a) with a nilradical ℒ 4 , ( n = 4 ) superscript ℒ 4 𝑛 4 \mathscr{L}^{4},(n=4) .
Steps
Ordered triple
Result
Table 6. Table 6. Left Leibniz identities in the codimension two nilradical ℒ 4 , ( n ≥ 5 ) superscript ℒ 4 𝑛 5 \mathscr{L}^{4},(n\geq 5) .
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TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Nonlinear Waves and Solitons
Full text
Solvable extensions of the naturally graded quasi-filiform Leibniz algebra
of second type L4
Abstract.
For a sequence of the naturally graded quasi-filiform Leibniz algebra
of second type L4 introduced by Camacho, Gómez, González and Omirov, all the possible right solvable indecomposable
extensions over the field C are constructed.
A. Shabanskaya
Department of Mathematics, Adrian College, 110 S Madison St.,
Adrian, MI 49221, USA
Leibniz algebras were discovered by Bloch in 1965 [4] who called them D− algebras. Later on they were considered by Loday and Cuvier [10, 15, 16, 17] as
a non-antisymmetric analogue of Lie algebras. It makes every Lie algebra be a Leibniz algebra, but the converse is not true.
Exactly Loday named them Leibniz algebras after Gottfried Wilhelm Leibniz.
Since then many analogs of important theorems in Lie theory were found to be true for Leibniz algebras, such as
the analogue of Levi’s theorem which was proved by Barnes [3]. He showed that any finite-dimensional complex Leibniz algebra is decomposed into a semidirect sum of the solvable radical and a semisimple Lie algebra.
Therefore the biggest challenge in the classification problem of finite-dimensional complex Leibniz algebras is to study the solvable part. And to classify solvable Leibniz algebras, we need nilpotent Leibniz algebras as their nilradicals, same as in the case of Lie algebras [18].
Every Leibniz algebra satisfies a generalized version of the Jacobi identity called the Leibniz identity.
There are two Leibniz identities: the left and the right.
We call Leibniz algebras right Leibniz algebras if they satisfy the right Leibniz identity
and left, if they satisfy the left. A left Leibniz algebra is not necessarily
a right Leibniz algebra [11].
Leibniz algebras inherit an important property of Lie algebras which is that the right (left)
multiplication operator of a right (left) Leibniz algebra is a derivation [9]. Besides the algebra of right (left) multiplication operators
is endowed with a structure of a Lie algebra by means of the commutator [9].
Also the quotient
algebra by the two-sided ideal generated by the square elements of a Leibniz algebra is a Lie algebra [19],
where such ideal is the minimal, abelian and in the case of
non-Lie Leibniz algebras it is always non trivial.
It is possible to find solvable Leibniz algebras in any finite dimension working with the sequence of nilpotent Leibniz algebras in any finite dimension and their “nil-independent”
derivations. This method for Lie algebras is based on what was shown by
Mubarakzyanov in [18]: the dimension of the complimentary vector space to the nilradical does not exceed the number of
nil-independent derivations of the nilradical. This result
was extended to Leibniz algebras by Casas, Ladra, Omirov and Karimjanov [8] with the help
of [2]. Besides, similarly to the case of Lie algebras, for a solvable Leibniz
algebra L we also have the inequality dimnil(L)≥21dimL [18].
There is the following work performed over the field of characteristic zero using
this method: Casas, Ladra, Omirov and Karimjanov classified solvable Leibniz algebras with null-filiform nilradical [8];
Omirov and his colleagues Casas, Khudoyberdiyev, Ladra, Karimjanov, Camacho and Masutova classified solvable Leibniz algebras whose nilradicals are a direct sum of null-filiform algebras [13], naturally graded filiform [9, 14], triangular [12] and finally filiform [7].
Bosko-Dunbar, Dunbar, Hird and Stagg attempted to classify left solvable Leibniz algebras with Heisenberg nilradical [5].
Left and right solvable extensions of R18 [1], L1,L2 and L3 [6] over the field of real numbers were
found by Shabanskaya in [22, 23, 24].
The starting point of the present article is a naturally graded quasi-filiform non Lie Leibniz algebra of the second type
L4,(n≥4) in the notation of [6].
This algebra is left and right at the same time and an associative when n=4.
Naturally graded quasi-filiform Leibniz algebras in any finite dimension over C were studied by Camacho, Gómez, González,
Omirov [6]. They found six such algebras of the first type, where two of them depend on a parameter and eight algebras of the second type with one of them
depending on a parameter.
This paper continues a work on finding all solvable extensions of quasi-filiform Leibniz algebras over the field of complex numbers. For
a sequence L4 such extensions of codimension at most two are
possible.
The paper is organized as follows: in Section 2 we give some basic definitions,
in Section 3 we show what involves constructing solvable Leibniz algebras with a given nilradical.
In Section 4 we describe the nilpotent sequence L4 and give the summary of the results stated in theorems, which could be found in
the remaining Section 5.
As regards notation, we use ⟨e1,e2,...,er⟩ to
denote the r-dimensional subspace generated by
e1,e2,...,er, where r\in\mbox{\mathbb{N}}. Besides g
and l
are used to denote solvable right and left Leibniz algebras, respectively.
Throughout the paper all the algebras are finite dimensional over the field of complex numbers and
if the bracket is not given, then it is assumed to be zero, except the brackets for the
nilradical, which most of the time are not given (see Remark 5.1) to save space.
In the tables an ordered triple is a shorthand notation
for a derivation property of the multiplication operators, which is either Rz([x,y])=[Rz(x),y]+[x,Rz(y)] or Lz([x,y])=[Lz(x),y]+[x,Lz(y)]. We also assign Ren+1:=R and Len+1:=L.
We use Maple software to compute the Leibniz identity, the “absorption” (see [20, 21] and
Section 3.2), the change
of basis for solvable Leibniz algebras in some particular dimensions, which are generalized and proved in an arbitrary finite dimension.
2. Preliminaries
We give some Basic definitions encountered working with Leibniz algebras.
Definition 2.1**.**
A vector space L over a field F with a bilinear operation
[−,−]:L→L*
is called a Leibniz algebra if for any x,y,z∈L the Leibniz identity*
[TABLE]
holds. This Leibniz identity is known as the right Leibniz identity and we call L
in this case a right Leibniz algebra.
2. 2.
There exists the version corresponding to the left Leibniz identity
[TABLE]
and a Leibniz algebra L is called a left Leibniz algebra.
Remark 2.1**.**
In addition, if L satisfies [x,x]=0 for every x∈L, then it is a Lie algebra.
Therefore every Lie algebra is a Leibniz algebra, but the converse is not true.
Definition 2.2**.**
The two-sided ideal C(L)={x∈L:[x,y]=[y,x]=0} is said to be the center of L.
Definition 2.3**.**
A linear map d:L→L of a Leibniz algebra L is a derivation
if for all x,y∈L
[TABLE]
If L is a right Leibniz algebra and x∈L, then the right multiplication operator Rx:L→L defined as Rx(y)=[y,x],y∈L
is a derivation (for a left Leibniz algebra L with x∈L, the left multiplication operator Lx:L→L,Lx(y)=[x,y],y∈L is a derivation).
Any right Leibniz algebra L is associated with the algebra of right multiplications R(L)={Rx∣x∈L}
endowed with the structure of a Lie algebra by means of the commutator [Rx,Ry]=RxRy−RyRx=R[y,x],
which defines an antihomomorphism between L and R(L).
For a left Leibniz algebra L, the corresponding algebra of left multiplications L(L)={Lx∣x∈L}
is endowed with the structure of a Lie algebra by means of the commutator as well [Lx,Ly]=LxLy−LyLx=L[x,y].
In this case we have a homomorphism between L and L(L).
Definition 2.4**.**
Let d1,d2,...,dn be derivations of a Leibniz algebra L. The derivations d1,d2,...,dn are said to be “nil-independent”
if α1d1+α2d2+α3d3+…+αndn is not nilpotent for any scalars α1,α2,...,αn∈F,
otherwise they are “nil-dependent”.
Definition 2.5**.**
For a given Leibniz algebra L, we define the sequence of two-sided ideals as follows:
[TABLE]
which are the lower central series and the derived series of L, respectively.
A Leibniz algebra L is said to be nilpotent (solvable) if there exists m\in\mbox{\mathbb{N}} such that Lm=0 (L(m)=0).
The minimal such number m is said to be the index of nilpotency (solvability).
Definition 2.6**.**
A Leibniz algebra L is called a quasi-filiform if Ln−3=0 and Ln−2=0, where dim(L)=n.
3. Constructing solvable Leibniz algebras with a given nilradical
Every solvable Leibniz algebra L contains a unique maximal nilpotent
ideal called the nilradical and denoted nil(L) such that
dimnil(L)≥21dim(L) [18]. Let us consider
the problem of constructing solvable Leibniz algebras
L with a given nilradical N=nil(L). Suppose
{e1,e2,e3,e4,...,en} is a basis for the nilradical and
{en+1,...,ep} is a basis for a subspace complementary to
the nilradical.
Calculations show that satisfying the right Leibniz identity is equivalent to the
following conditions:
[TABLE]
[TABLE]
Then the entries of the matrices
Aa, which are (Aik)a, must satisfy the equations (\refRLeibniz) obtained from
all the possible Leibniz identities between the triples {ea,ei,ej}.
Since N is the nilradical of L, no nontrivial linear
combination of the matrices Aa,(n+1≤a≤p) is nilpotent
i.e. the matrices Aa must be “nil-independent” [8, 18].
Let us now consider the right multiplication operator Rea and restrict it
to N, (n+1≤a≤p). We shall
get outer derivations of the nilradical N=nil(L)
[8]. Then finding the matrices Aa is the same as
finding outer derivations Rea of N. Further the commutators [Reb,Rea]=R[ea,eb],(n+1≤a,b≤p) due to (\refNil) consist of inner derivations of
N. So those commutators give the structure constants Babk as shown in the last equation of (\refRRLeibniz) but only up to the elements in the center of the nilradical
N, because if ei,(1≤i≤n) is in the center of N, then (Rei)∣N=0, where (Rei)∣N
is an inner derivation of the nilradical.
3.2. Solvable left Leibniz algebras
Satisfying the left Leibniz identity, we have
[TABLE]
[TABLE]
and the entries of the matrices
Aa, which are (Aa)ik, must satisfy the equations (\refLLeibniz).
Similarly Aa=(Lea)∣N,(n+1≤a≤p) are outer derivations and
the commutators [Lea,Leb]=L[ea,eb] give the structure constants Babk, but only up to the elements in the center of the nilradical
N.
Once the left or right Leibniz identities are satisfied in the most general possible way and the outer derivations are found:
(i)
We can carry out the technique of “absorption” [20, 21], which means we can
simplify a solvable Leibniz algebra without affecting the nilradical in (\refAlgebra) applying the transformation
[TABLE]
2. (ii)
We change basis without affecting the
nilradical in (\refAlgebra) to remove all the possible parameters and simplify the algebra.
4. The nilpotent sequence L4
In L4,(n≥4) the positive
integer n denotes the dimension of the algebra. The center
of this algebra is C(L4)=⟨e2,en⟩. L4 can be described explicitly as follows: in the
basis {e1,e2,e3,e4,…,en} it has only the following
non-zero brackets:
[TABLE]
The
dimensions of the ideals in the characteristic
series are
[TABLE]
A quasi-filiform Leibniz algebra L4 was introduced by Camacho, Gómez, González and Omirov
in [6].
This algebra is served as the nilradical for the
left and right solvable indecomposable extensions we construct in this
paper.
It is shown below that solvable right (left) Leibniz algebras
with the nilradical L4 only exist for dimg=n+1 and dimg=n+2(diml=n+1 and diml=n+2).
Right solvable extensions with a codimension
one nilradical L4 are found following the steps in
Theorems 5.1.1, 5.1.2 and 5.1.3 with the main result summarized in Theorem 5.1.3, where
it is shown there are eight such algebras: gn+1,i,(1≤i≤4,n≥4),g5,5,g5,6,g5,7 and g5,8. It is noticed that gn+1,1 is left as well when a=0,g5,5 is left when b=−1 and gn+1,4,g5,7,g5,8 are right and left Leibniz algebras
at the same time.
There are four solvable indecomposable right Leibniz algebras with a codimension
two nilradical: gn+2,1,(n≥5),g6,2,g6,3 and g6,4 stated in Theorem 5.1.4, where none of them is left.
We follow the steps in
Theorems 5.2.1, 5.2.2 and 5.2.3 to find codimension one left solvable extensions.
We notice in Theorem 5.2.3, that we have eight of them as well: ln+1,1,ln+1,2,ln+1,3,gn+1,4,l5,5,l5,6,g5,7 and g5,8, such that ln+1,1 is right when a=0 and
l5,5 is right when b=−1.
We find four solvable indecomposable left Leibniz algebras with a codimension two nilradical as well: ln+2,1,(n≥5),l6,2,l6,3 and l6,4 stated in Theorem 5.2.4, where none of them is right.
5. Classification of solvable indecomposable Leibniz algebras with
a nilradical L4
Our goal in this Section is to find all possible right and left solvable indecomposable extensions
of the nilpotent Leibniz algebra L4, which serves
as the nilradical of the extended algebra.
Remark 5.1**.**
It is assumed throughout this section that
solvable indecomposable right and left Leibniz algebras have the nilradical L4; however, most of the time, the brackets of the nilradical will be omitted.
5.1. Solvable indecomposable right Leibniz algebras with a nilradical L4
5.1.1. Codimension one solvable extensions of L4
The nilpotent Leibniz algebra L4 is defined in (\refL4). Suppose {en+1}
is in the complementary subspace to the nilradical L4 and g is the corresponding solvable right Leibniz algebra.
Since [g,g]⊆L4, we have the following:
[TABLE]
Theorem 5.1.1**.**
We set a1,1:=a and a3,3:=b in (\refBRLeibniz). To satisfy the right Leibniz identity, there are the following cases based on the conditions involving parameters,
each gives a continuous family of solvable Leibniz algebras:
(1)
If a1,3=0,b=−a,a=0,b=0,(n=4) or b=(3−n)a,a=0,b=0,(n≥5), then we have the following brackets for the algebra:
[TABLE]
2. (2)
If a1,3=0,b:=−a,a=0,(n=4) or b:=(3−n)a,a=0,(n≥5),
then
[TABLE]
3. (3)
If a1,3=0,a=0,b=0,(n=4) or a=0 and b=0,(n≥5), then
For (n≥5)
the proof is given in Table 1. For (n=4) we work out the following identities: 1.,3.−5.,7.,8.,10.,12.,13.(or 15.), 16.−19.
2. (2)
For (n≥5), we apply the same identities
given in Table 1, except 17. For (n=4),
the identities are as follows: 1.,3.−5.,7.,8.,10.,12.,15.,16.−19.
3. (3)
For (n≥5), same identities except 18. and 19.
applying instead
Re3([en+1,en+1])=[Re3(en+1),en+1]+[en+1,Re3(en+1)] and
R[en+1,e1]=[R(en+1),e1]+[en+1,R(e1)].
For (n=4), same identities as in case (1), except 18. and 19.
applying two identities given above.
4. (4)
Same identities as in case (1).
5. (5)
We apply the following identities:
1.,3.−5.,7.,8.,10.,12.,13.,16.,17.,Re3([e5,e5])=[Re3(e5),e5]+[e5,Re3(e5)],18.,19.
6. (6)
We apply the same identities as in (5),
except 17. and 18.
7. (7)
Same as in (5), except 19.
8. (8)
We apply the same identities as in (5), except 17.,18. and 19.
9. (9)
Same identities as in (5), except 17. and 18.
10. (10)
We have the identities:
1.,3.−5.,7.,8.,10.,12.,13.,16.,17.,19.,Re5([e5,e1])=[Re5(e5),e1]+[e5,Re5(e1)],Re3([e5,e5])=[Re3(e5),e5]+[e5,Re3(e5)].
11. (11)
Same identities as in (5), except 19.
∎
Theorem 5.1.2**.**
Applying the technique of “absorption” (see Section 3.2), we can further simplify the algebras
in each of the cases in Theorem 5.1.1 as follows:
(1)
If a1,3=0,b=−a,a=0,b=0,(n=4) or b=(3−n)a,a=0,b=0,(n≥5), then we have the following brackets for the algebra:
[TABLE]
2. (2)
If a1,3=0,b:=−a,a=0,(n=4) or b:=(3−n)a,a=0,(n≥5),
then the brackets for the algebra are
[TABLE]
3. (3)
If a1,3=0,a=0,b=0,(n=4) or a=0 and b=0,(n≥5), then
The right (a derivation)
and left (not a derivation) multiplication operators restricted to the nilradical are given below:
[TABLE]
[TABLE]
–
The transformation ek′=ek,(1≤k≤n),en+1′=en+1−a4,3e1
removes a4,3 in Ren+1 and −a4,3 in Len+1 from the (i,i−1)st positions, where (4≤i≤n),
but it affects other entries as well,
such as
the entry in the (2,1)st position in Ren+1 and Len+1,
which we change to A2,1−a4,3 and b2,1−a4,3, respectively.
It also changes the entry in the (2,3)rd position in Len+1 to
a2,3.
At the same time, it affects the coefficient in front of e2 in the bracket [en+1,en+1], which we change back to a2,n+1.
–
Applying the transformation ei′=ei,(1≤i≤n),en+1′=en+1+∑k=3n−1ak+1,1ek,
we remove ak+1,1 in Ren+1 and −ak+1,1 in Len+1 from the entries in the (k+1,1)st
positions, where (3≤k≤n−1). It changes the entry in the (2,1)st position in
Ren+1 to A2,1+2a4,1−a4,3, the entries in the (2,3)rd positions in Ren+1
and Len+1 to a2,3+a4,1.
It also affects the coefficient in front of e2 in [en+1,en+1], which
we rename back by a2,n+1. We assign a2,3+a4,1:=a2,3 and b2,1−a4,3:=b2,1. Then
A2,1+2a4,1−a4,3:=a(2b−a)b2,1+2(a−b)a2,3.
–
The transformation ej′=ej,(1≤j≤n),en+1′=en+1−2ba2,n+1e2
removes the coefficient a2,n+1 in front of e2 in [en+1,en+1] and we prove the result.
2. (2)
The right (a derivation) and left (not a derivation) multiplication operators restricted to the nilradical are as follows:
[TABLE]
[TABLE]
–
The transformation ek′=ek,(1≤k≤n),en+1′=en+1−a4,3e1
removes a4,3 in Ren+1 and −a4,3 in Len+1 from the (i,i−1)st positions, where (4≤i≤n),
but it affects other entries as well,
such as
the entry in the (2,1)st position in Ren+1 and Len+1,
which we change to A2,1−a4,3 and b2,1−a4,3, respectively.
It also changes the entry in the (2,3)rd position in Len+1 to
a2,3.
At the same time, it affects the coefficient in front of e2 in the bracket [en+1,en+1], which we change back to a2,n+1.
–
Then we apply the transformation ei′=ei,(1≤i≤n),en+1′=en+1+∑k=3n−1ak+1,1ek
to remove ak+1,1 in Ren+1 and −ak+1,1 in Len+1 from the entries in the (k+1,1)st
positions, where (3≤k≤n−1). It changes the entry in the (2,1)st position in
Ren+1 to A2,1+2a4,1−a4,3, the entries in the (2,3)rd positions in Ren+1
and Len+1 to a2,3+a4,1.
It also affects the coefficient in front of e2 in [en+1,en+1], which
we rename back by a2,n+1. We assign a2,3+a4,1:=a2,3 and b2,1−a4,3:=b2,1. Then A2,1+2a4,1−a4,3:=(5−2n)b2,1+2(n−2)a2,3, which we set to be A2,1.
–
Applying the transformation ej′=ej,(1≤j≤n),en+1′=en+1+2(n−3)aa2,n+1e2,
we remove the coefficient a2,n+1 in front of e2 in [en+1,en+1] and prove the result.
3. (3)
The right (a derivation) and left (not a derivation) multiplication operators
restricted to the nilradical are
[TABLE]
[TABLE]
–
The transformation ek′=ek,(1≤k≤n),en+1′=en+1−a4,3e1
removes a4,3 in Ren+1 and −a4,3 in Len+1 from the (i,i−1)st positions, where (4≤i≤n),
but it affects other entries as well,
such as
the entry in the (2,1)st position in Ren+1 and Len+1,
which we change to a2,1−a4,3 and a2,3+a4,1, respectively.
It also changes the entry in the (2,3)rd position in Len+1 to
a2,3.
At the same time, it affects the coefficient in front of e2 in the bracket [en+1,en+1], which we change back to a2,n+1.
–
Applying the transformation ei′=ei,(1≤i≤n),en+1′=en+1+∑k=3n−1ak+1,1ek,
we remove ak+1,1 in Ren+1 and −ak+1,1 in Len+1 from the entries in the (k+1,1)st
positions, where (3≤k≤n−1). It changes the entry in the (2,1)st position in
Ren+1 to a2,1+2a4,1−a4,3, the entries in the (2,3)rd positions in Ren+1
and Len+1 to a2,3+a4,1.
It also affects the coefficient in front of e2 in [en+1,en+1], which
we rename back by a2,n+1. Then we assign a2,1+2a4,1−a4,3:=a2,1 and a2,3+a4,1:=a2,3.
–
The transformation ej′=ej,(1≤j≤n),en+1′=en+1−2ba2,n+1e2
removes the coefficient a2,n+1 in front of e2 in [en+1,en+1] and we prove the result.
4. (4)
The right (a derivation) and left (not a derivation) multiplication operators
restricted to the nilradical are given below:
[TABLE]
[TABLE]
–
The transformation ek′=ek,(1≤k≤n),en+1′=en+1−a4,3e1
removes a4,3 in Ren+1 and −a4,3 in Len+1 from the entries in the (i,i−1)st positions, where (4≤i≤n),
but it affects other entries as well,
such as
the entry in the (2,1)st position in Ren+1 and Len+1,
which we change to a2,3−b2,1−a4,3+b2,3 and b2,1−a4,3, respectively.
It also changes the entry in the (2,3)rd position in Len+1 to
b2,3−2a4,3.
At the same time, it affects the coefficient in front of e2 in the bracket [en+1,en+1], which we change back to a2,n+1.
–
Applying the transformation ei′=ei,(1≤i≤n),en+1′=en+1+∑k=3n−1ak+1,1ek,
we remove ak+1,1 in Ren+1 and −ak+1,1 in Len+1 from the entries in the (k+1,1)st
positions, where (3≤k≤n−1). It changes the entry in the (2,1)st position in
Ren+1 to 2a4,1+a2,3−b2,1−a4,3+b2,3, the entries in the (2,3)rd positions in Ren+1
and Len+1 to a2,3+a4,1 and b2,3+a4,1−2a4,3, respectively.
It also affects the coefficient in front of e2 in [en+1,en+1], which
we rename back by a2,n+1. We assign a2,3+a4,1:=a2,3,b2,1−a4,3:=b2,1
and b2,3+a4,1−2a4,3:=b2,3. Then 2a4,1+a2,3−b2,1−a4,3+b2,3:=a2,3−b2,1+b2,3.
5. (5)
We apply the transformation
ei′=ei,(1≤i≤4),e5′=e5−A4,3e1+2(b+c)1(A2,1A4,3+2a4,1A4,3+b2,1A4,3−A4,32−a2,3a4,1−a4,12−a4,1b2,3−a2,5)e2+a4,1e3
and then we assign a2,3+a4,1:=a2,3 and b2,3−2b2,1+a4,1:=b2,3.
6. (6)
The transformation we apply is
ei′=ei,(1≤i≤4),e5′=e5−A4,3e1+2(c−a)1(A2,1A4,3+2a4,1A4,3+b2,1A4,3−A4,32−a2,3a4,1−a4,12−a4,1b2,3−a2,5)e2+a4,1e3.
We assign a2,3+a4,1:=a2,3 and b2,3−2b2,1+a4,1:=b2,3.
7. (7)
We apply the transformation
ei′=ei,(1≤i≤4),e5′=e5−a4,3e1+a4,1e3 and rename the coefficient in front of e2
in [e5,e5] back by a2,5. Then we assign a2,3+a4,1:=a2,3,b2,1−a4,3:=b2,1 and b2,3+a4,1−2a4,3:=b2,3.
8. (8)
We apply
ei′=ei,(1≤i≤4),e5′=e5−a4,3e1−2c1(2a2,3a4,3−a4,12+2a4,1a4,3+2a4,1b2,1+3a4,32−4a4,3b2,1+a2,5)e2+a4,1e3.
We assign a2,3+a4,1:=a2,3 and b2,1−a4,3:=b2,1.
9. (9)
The transformation is
ei′=ei,(1≤i≤4),e5′=e5−A4,3e1+4(b+c)1(2a2,1A4,3−2a2,3a4,1+a2,3A4,3−2a4,12+6a4,1A4,3−2a4,1b2,3−2A4,32+b2,3A4,3−2a2,5)e2+a4,1e3.
Then we assign a2,3+a4,1:=a2,3 and b2,3−2a2,1−3a4,1:=b2,3.
10. (10)
We apply the transformation
ei′=ei,(1≤i≤4),e5′=e5−a4,3e1+a4,1e3 and rename the coefficient in front of e2
in [e5,e5] back by a2,5. We assign a2,3+a4,1:=a2,3,a2,1+2a4,1−a4,3:=a2,1
and b2,3+a4,1−2a4,3:=b2,3.
∎
Theorem 5.1.3**.**
There are eight solvable
indecomposable right Leibniz algebras up to isomorphism with a codimension one nilradical
L4,(n≥4), which are given below:
[TABLE]
Proof.
One applies the change of basis transformations keeping the nilradical L4 given in (\refL4) unchanged.
(1)
We have the right (a derivation) and the left (not a derivation) multiplication operators restricted to the nilradical given below:
[TABLE]
[TABLE]
–
We apply the transformation e1′=e1,e2′=e2,ei′=ei−(k−i)aak−i+3,3ek,(3≤i≤n−2,i+2≤k≤n,n≥5),ej′=ej,(n−1≤j≤n+1), where k is fixed, renaming all the affected entries back.
This transformation removes a5,3,a6,3,...,an,3 in Ren+1 and −a5,3,−a6,3,...,−an,3 in Len+1.
Besides it introduces the entries in the (5,1)st,(6,1)st,...,(n,1)st positions in Ren+1 and Len+1,
which we set to be a5,1,a6,1,...,an,1 and −a5,1,−a6,1,...,−an,1, respectively.
(I) Suppose b=2a.
–
The transformation e1′=e1+a−2b1(A2,1+b(b−a)a2,3)e2,e2′=e2,e3′=e3−ba2,3e2,ei′=ei,(4≤i≤n,n≥4),en+1′=en+1−a5,1e2+∑k=4n−1ak+1,1ek
removes A2,1 and b2,1 from the (2,1)st positions in Ren+1 and Len+1,
respectively. It also removes a2,3
from the (2,3)rd positions in Ren+1 and Len+1 as well as
ak+1,1 in Ren+1 and −ak+1,1 in Len+1 from the entries in the (k+1,1)st
positions, where (4≤k≤n−1).
Remark 5.1.1**.**
If n=4, then a5,1=0 and the same is in case (II).
–
Then we scale a to unity applying the transformation ei′=ei,(1≤i≤n,n≥4),en+1′=aen+1.
Renaming ab by b, we obtain a continuous family of Leibniz algebras:
[TABLE]
(II) Suppose b:=2a. We have that A2,1=a2,3.
–
The transformation e1′=e1−ab2,1+a2,3e2,e2′=e2,e3′=e3−a2a2,3e2,ei′=ei,(4≤i≤n,n≥4),en+1′=en+1−a5,1e2+∑k=4n−1ak+1,1ek
removes a2,3 from the (2,1)st,(2,3)rd positions in Ren+1
and from the (2,3)rd position in Len+1.
It also removes b2,1 from the (2,1)st position in Len+1 as well as
ak+1,1 in Ren+1 and −ak+1,1 in Len+1 from the entries in the (k+1,1)st
positions, where (4≤k≤n−1).
–
To scale a to unity, we apply the transformation ei′=ei,(1≤i≤n,n≥4),en+1′=aen+1
and obtain a limiting case of (\refgn+1,1) with b=21 given below:
[TABLE]
2. (2)
We have the right (a derivation) and the left (not a derivation) multiplication operators restricted to the nilradical are as follows:
[TABLE]
[TABLE]
–
We apply the transformation e1′=e1,e2′=e2,ei′=ei−(k−i)aak−i+3,3ek,(3≤i≤n−2,i+2≤k≤n,n≥5),ej′=ej,(n−1≤j≤n+1), where k is fixed, renaming all the affected entries back.
This transformation removes a5,3,a6,3,...,an,3 in Ren+1 and −a5,3,−a6,3,...,−an,3 in Len+1.
Besides it introduces the entries in the (5,1)st,(6,1)st,...,(n,1)st positions in Ren+1 and Len+1,
which we set to be a5,1,a6,1,...,an,1 and −a5,1,−a6,1,...,−an,1, respectively.
–
The transformation e1′=e1+(2n−5)a1(A2,1+n−3(n−2)a2,3)e2,e2′=e2,e3′=e3+(n−3)aa2,3e2,ei′=ei,(4≤i≤n,n≥4),en+1′=en+1−a5,1e2+∑k=4n−1ak+1,1ek
removes A2,1 and b2,1 from the (2,1)st positions in Ren+1 and Len+1,
respectively. It also removes a2,3
from the (2,3)rd positions in Ren+1 and Len+1;ak+1,1 and −ak+1,1 from the entries in the (k+1,1)st
positions in Ren+1 and Len+1, respectively, where (4≤k≤n−1). (See Remark 5.1.1).
–
To scale a to unity, we apply the transformation ei′=ei,(1≤i≤n),en+1′=aen+1
renaming the coefficient a2an,n+1 in front of en in [en+1,en+1] back by an,n+1. We obtain a Leibniz algebra:
[TABLE]
If an,n+1=0, then we have a limiting case of (5.1.2) with b=3−n. If an,n+1=0,
then an,n+1=reiϕ and we apply the transformation
ej′=(reiϕ)n−2jej,(1≤j≤2),ek′=(reiϕ)n−2k−2ek,(3≤k≤n),en+1′=en+1
to scale an,n+1 to 1. We have the algebra gn+1,2 given below:
[TABLE]
3. (3)
We have the right (a derivation) and the left (not a derivation) multiplication operators restricted to the nilradical are as follows:
[TABLE]
[TABLE]
–
Applying the transformation e1′=e1−2ba2,1+a2,3e2,e2′=e2,e3′=e3−ba2,3e2,ei′=ei,(4≤i≤n+1),
we
remove a2,1 from the (2,1)st position in Ren+1 and
a2,3 from the (2,1)st, the (2,3)rd positions in
Len+1 and from the (2,3)rd position in Ren+1 keeping other entries unchanged.
–
To scale b to unity, we apply the transformation ei′=ei,(1≤i≤n),en+1′=ben+1.
Then we rename ba5,3,ba6,3,...,ban,3 by a5,3,a6,3,...,an,3, respectively.
We obtain a Leibniz algebra
[TABLE]
If a5,3=0,(n≥5), then a5,3=reiϕ and applying
the transformation ej′=(reiϕ)2jej,(1≤j≤2),ek′=(reiϕ)2k−2ek,(3≤k≤n),en+1′=en+1, we scale a5,3 to 1. We also rename all the affected entries back
and then we rename a6,3,...,an,3 by b1,...,bn−5, respectively.
We combine with the case when a5,3=0 and obtain a Leibniz algebra gn+1,3 given below:
[TABLE]
Remark 5.1.2**.**
If n=4, then ϵ=0.
4. (4)
We have the right (a derivation) and the left (not a derivation) multiplication operators restricted to the nilradical are as follows:
[TABLE]
[TABLE]
–
We apply the transformation e1′=e1,e2′=e2,ei′=ei−(k−i)aak−i+3,3ek,(3≤i≤n−2,i+2≤k≤n,n≥5),ej′=ej,(n−1≤j≤n+1), where k is fixed, renaming all the affected entries back.
This transformation removes a5,3,a6,3,...,an,3 in Ren+1 and −a5,3,−a6,3,...,−an,3 in Len+1.
Besides it introduces the entries in the (5,1)st,(6,1)st,...,(n,1)st positions in Ren+1 and Len+1,
which we set to be a5,1,a6,1,...,an,1 and −a5,1,−a6,1,...,−an,1, respectively.
–
The transformation e1′=e1+aa2,3−b2,1+b2,3e2,ei′=ei,(2≤i≤n,n≥4),en+1′=en+1+∑k=4n−1ak+1,1ek
removes a2,3−b2,1+b2,3 from the (2,1)st position in Ren+1.
It changes the entry in the (2,1)st position in Len+1 to a2,3+b2,3.
It also removes
ak+1,1 and −ak+1,1 from the entries in the (k+1,1)st
positions, where (4≤k≤n−1) in Ren+1 and Len+1, respectively.
–
We assign a2,3:=d and b2,3:=f
and
then we scale a to unity applying the transformation ei′=ei,(1≤i≤n,n≥4),en+1′=aen+1.
Renaming ad,af and a2a2,n+1 by d,f and a2,n+1, respectively, we obtain a right and left Leibniz algebra:
[TABLE]
which is a limiting case of (5.1.2) with b=0, when d=f=a2,n+1=0. Altogether (5.1.2) and all its limiting
cases after replacing b with a give us a Leibniz algebra gn+1,1 given below:
[TABLE]
It remains to consider a continuous family of Leibniz algebras given below
and scale any nonzero entries as much as possible.
[TABLE]
If a2,n+1=0, then a2,n+1=reiϕ and applying
the transformation ej′=(reiϕ)2jej,(1≤j≤2),ek′=(reiϕ)2k−2ek,(3≤k≤n,n≥4),en+1′=en+1, we scale it to 1. We also rename all the affected entries back.
Then we combine with the case when a2,n+1=0 and obtain a right and left Leibniz algebra gn+1,4 given below:
[TABLE]
5. (5)
Applying the transformation
e1′=e1+2a(b+c)(b+2c−2a)a2,3+(b+2c)b2,3e2,e2′=e2,e3′=e3+2a(b+c)(c−2a)a2,3+c⋅b2,3e2,e4′=e4,e5′=ce5 and renaming
ca and cb by a and b, respectively, we obtain a continuous family of Leibniz algebras given below:
[TABLE]
6. (6)
We
apply the transformation
e1′=e1+2a(a−c)(3a−2c)a2,3+(a−2c)b2,3e2,e2′=e2,e3′=e3+2a(a−c)(2a−c)a2,3−c⋅b2,3e2,e4′=e4,e5′=ce5
and rename
ca and c2a4,5 by a and a4,5, respectively,
to obtain a Leibniz algebra given below:
[TABLE]
which is a limiting case of (\refg5,5) with b:=−a if a4,5=0.
If a4,5=0, then a4,5=reiϕ. To scale a4,5 to 1,
we apply the transformation
e1′=rei2ϕe1,e2′=reiϕe2,e3′=rei2ϕe3,e4′=reiϕe4,e5′=e5
and obtain a Leibniz algebra given below:
[TABLE]
7. (7)
We apply the transformation
e1′=e1+aa2,3−b2,1+b2,3e2,ei′=ei,(2≤i≤5)
and assign d:=aa⋅b2,3+c(a2,3−b2,1+b2,3),f:=aa⋅a2,3−c(a2,3−b2,1+b2,3) to obtain a continuous family
of Leibniz algebras:
[TABLE]
Then we continue with the transformation ei′=ei,(1≤i≤4),e5′=ce5 renaming
cd,cf,ca and c2a2,5
by d,f,a and a2,5, respectively,
to obtain a Leibniz algebra:
[TABLE]
If d=f=a2,5=0, then we have a limiting case of (\refg5,5) with b=−1.
If a2,5=0, then we apply
the same transformation as we did to scale a4,5 to 1. We also rename all the affected entries back.
Then we combine with the case when a2,5=0 and obtain a Leibniz algebra given below:
[TABLE]
Remark 5.1.3**.**
We notice by applying the transformation
e1′=e1+fe2,ei′=ei,(2≤i≤5) that this algebra (\refg5,7)
is isomorphic to
[TABLE]
8. (8)
Applying the transformation
e1′=e1+ca2,3−2b2,1e2,e2′=e2,e3′=e3−cb2,1e2,e4′=e4,e5′=ce5
and renaming c2a4,5 back by a4,5, we obtain a Leibniz algebra:
[TABLE]
which is a limiting case of (\refgeneral1) with a=0 and at the same time a limiting case of (\refg5,5)
with b:=−a if a4,5=0. Therefore we only consider the case when a4,5=0.
One applies the transformation below (\refgeneral1) to scale a4,5 to 1 and we have a limiting case of (\refg5,6)
with a=0. Altogether (\refg5,6) and all its limiting cases give us the algebra g5,6 in full generality:
[TABLE]
where a=1, otherwise nilpotent.
9. (9)
If a=0,b=0,b=−c,c=0,(n=4), then we apply the transformation
e1′=e1+(b+c)2(b+2c)b2,3−(3b+2c)a2,3e2,e2′=e2,e3′=e3−b+cB2,1e2,e4′=e4,e5′=ce5. We rename cb by b and obtain a Leibniz algebra:
[TABLE]
which is a limiting case of (\refg5,5) with a=0. Altogether (\refg5,5) and all its
limiting cases give us the algebra g5,5 in full generality:
[TABLE]
where if b=−1, then a=1, otherwise nilpotent.
10. (10)
We apply the transformation
e1′=e1+ca2,3e2,ei′=ei,(2≤i≤4),e5′=ce5
and rename ca2,1,ca2,3,cb2,3 and c2a2,5 by
a2,1,a2,3,b2,3 and a2,5, respectively, to
obtain a Leibniz algebra:
[TABLE]
We scale a2,5 to 1 if nonzero.
Combining with the case when a2,5=0 and renaming all the affected entries back, we have the algebra:
[TABLE]
If a2,1=0, then we obtain a limiting case of (\refgeneral) and (\refg5,7) with a=0 according to Remark 5.1.3.
Altogether (\refg5,7) and all its limiting cases, give us the algebra g5,7 in full generality:
[TABLE]
where a=1, otherwise nilpotent.
If c:=a2,1=0 and we assign d:=a2,3+b2,3−2a2,1, then
we obtain a Leibniz algebra g5,8 given below:
[TABLE]
∎
5.1.2. Codimension two and three solvable extensions of L4
The non-zero inner derivations of L4,(n≥4) are
given by
[TABLE]
where Ei+1,1 is the n×n matrix that has 1 in the
(i+1,1)st
position and all other entries are zero.
Remark 5.1.4**.**
If we have four or more outer derivations, then they are
“nil-dependent”(see Section 2.).
Therefore the solvable algebras we are constructing might be of codimension at most three. (As Section 5.1.2.3 shows, we have at most two outer
derivations.)
*General approach to find right solvable Leibniz
algebras with a codimension two nilradical L4.222When we work with left Leibniz algebras, we first change the right multiplication operator to the left everywhere and the right Leibniz identity to the left Leibniz identity in step (iii). We also interchange
s and r in the very left in steps (i) and (ii) as well. *
(i)
We consider R[er,es]=[Res,Rer]=ResRer−RerRes,(n+1≤r≤n+2,1≤s≤n+2)
and compare with c1Re1+∑k=3n−1ckRek, because e2
and en are in the center of L4,(n≥4) defined in (\refL4) to find all the unknown commutators.
2. (ii)
We write down
[er,es],(n+1≤r≤n+2,1≤s≤n+2) including a linear combination of
e2 and en as well. We add the brackets of the nilradical L4 and outer derivations Ren+1 and Ren+2.
3. (iii)
One satisfies the right Leibniz identity: [[er,es],et]=[[er,et],es]+[er,[es,et]]
or, equivalently, Ret([er,es])=[Ret(er),es]+[er,Ret(es)],(1≤r,s,t≤n+2) for all the brackets obtained in step (ii).
4. (iv)
Then we carry out the technique of “absorption” (see Section 3.2) to remove some parameters to simplify the algebra.
5. (v)
We apply the change of basis transformations without affecting the nilradical to remove as many parameters as possible.
5.1.2.1 Codimension two solvable extensions of L4,(n=4)
There are the following cases based on the conditions involving parameters a,b and c:
(1)
b=−a,a=0,b=−c,
2. (2)
b:=−a,a=0,a=c,
3. (3)
b:=−c,a=c,a=0,
4. (4)
a=0,b=0,b=−c.
(1) (a) One could set \left(\begin{array}[]{c}a^{1}\\
b^{1}\\
c^{1}\end{array}\right)=\left(\begin{array}[]{c}1\\
a\\
0\end{array}\right) and \left(\begin{array}[]{c}a^{2}\\
b^{2}\\
c^{2}\end{array}\right)=\left(\begin{array}[]{c}1\\
b\\
1\end{array}\right),(a\neq-1,a\neq 0,b\neq-1). Therefore the vector space of outer derivations as 4×4
matrices is as follows:
[TABLE]
[TABLE]
(i) Considering R[e5,e6], we obtain that a:=1 and
α2,3:=(b+23)a2,3+2b2,3. Since b=−1, it follows that β2,3:=2b2,3−(b+21)a2,3
and R[e5,e6]=0. As a result,
[TABLE]
[TABLE]
Further, we find the following commutators:
[TABLE]
(ii) We include a linear combination of e2 and e4:
[TABLE]
Besides we have the brackets from L4 and from outer derivations Re5 and Re6 as well.
(iii) We satisfy the right Leibniz identity, which is shown in Table 2.
We obtain that Re5 and Re6 restricted to the nilradical do not change, but the remaining brackets are as follows:
[TABLE]
Altogether the nilradical L4(\refL4), the outer derivations Re5 and Re6
written in the bracket notation and the remaining brackets given above define a continuous family of Leibniz algebras
depending on the parameters.
(iv)&(v) We apply the following transformation: e1′=e1+2b2,3−a2,3e2,e2′=e2,e3′=e3−a2,3e2,e4′=e4,e5′=e5−2c2,5e2,e6′=e6−2d2,5e2+(2b+1c2,5−2c2,6)e4
and obtain a Leibniz algebra g6,2 given below:
[TABLE]
Remark 5.1.5**.**
We notice that if b=−1, then the outer derivation Re6 is nilpotent.
(1) (b) We set \left(\begin{array}[]{c}a^{1}\\
b^{1}\\
c^{1}\end{array}\right)=\left(\begin{array}[]{c}1\\
2\\
c\end{array}\right) and \left(\begin{array}[]{c}a^{2}\\
b^{2}\\
c^{2}\end{array}\right)=\left(\begin{array}[]{c}1\\
1\\
d\end{array}\right),(c\neq-2,d\neq-1). Therefore the vector space of outer derivations as 4×4
matrices is as follows:
[TABLE]
[TABLE]
(i) Considering R[e5,e6], we obtain that d=0 and we have
the system of equations:
[TABLE]
There are the following two cases:
(I)
If c=−1, then b2,3:=−23c+2α2,3−2c−2β2,3
and R[e5,e6]=0. As a result,
[TABLE]
[TABLE]
Further, we find the following:
[TABLE]
2. (II)
If c=−1, then a2,3:=2α2,3−β2,3
and R[e5,e6]=0. It follows that
[TABLE]
and we have the following commutators:
[TABLE]
(ii) We combine cases (I) and (II) together, include a linear combination of e2 and e4,
and have the following brackets:
[TABLE]
Besides we have the brackets from L4 and from outer derivations Re5 and Re6 as well.
(iii) To satisfy the right Leibniz identity, we refer to the identities given in Table 2. as much as possible.
As a result, the identities we apply are the following: 1.−6.,8.,Re6([e5,e1])=[Re6(e5),e1]+[e5,Re6(e1)],9.−11.,Re6([e6,e1])=[Re6(e6),e1]+[e6,Re6(e1)],13. and 14.
We obtain that Re5 and Re6 restricted to the nilradical do not change, but the remaining brackets are the following:
[TABLE]
Altogether the nilradical L4(\refL4), the outer derivations Re5 and Re6
written in the bracket notation and the remaining brackets given above define a continuous family of Leibniz algebras
depending on the parameters.
(iv)&(v) We apply the transformation: e1′=e1−2α2,3−β2,3e2,e2′=e2,e3′=e3−α2,3e2,e4′=e4,e5′=e5−2a2,6e2−2a4,6e4,e6′=e6−2b2,6e2 and obtain a Leibniz algebra g6,3:
[TABLE]
(1) (c) We set \left(\begin{array}[]{c}a^{1}\\
b^{1}\\
c^{1}\end{array}\right)=\left(\begin{array}[]{c}a\\
1\\
0\end{array}\right) and \left(\begin{array}[]{c}a^{2}\\
b^{2}\\
c^{2}\end{array}\right)=\left(\begin{array}[]{c}b\\
0\\
1\end{array}\right),(a,b\neq 0,a\neq-1). Therefore the vector space of outer derivations as 4×4
matrices is as follows:
[TABLE]
(i) Considering R[e5,e6], we obtain that a:=1 and we have
the system of equations:
[TABLE]
There are the following two cases:
(I)
If b=3, then β2,3:=2(2b−3)a2,3+(2b−1)b2,3
and R[e5,e6]=0. Consequently,
[TABLE]
Further, we find the commutators:
[TABLE]
2. (II)
If b:=3, then R[e5,e6]=0 and Re5,Re6 are as follows:
[TABLE]
We have the following commutators:
[TABLE]
(ii) We combine cases (I) and (II) together and include a linear combination of e2 and e4:
[TABLE]
We also have the brackets from L4 and from outer derivations Re5 and Re6 as well.
(iii) To satisfy the right Leibniz identity, we apply the same identities as in Table 2.
We have that Re5 and Re6 restricted to the nilradical do not change, but the remaining brackets are the following:
[TABLE]
Altogether the nilradical L4(\refL4), the outer derivations Re5 and Re6
written in the bracket notation and the remaining brackets given above define a continuous family of Leibniz algebras
depending on the parameters.
(iv)&(v) We apply the transformation: e1′=e1+2b2,3−a2,3e2,e2′=e2,e3′=e3−a2,3e2,e4′=e4,e5′=e5−2c2,5e2,e6′=e6−2d2,5e2+2c2,5−c2,6e4 and obtain a Leibniz algebra g6,4:
[TABLE]
(2) In this case we consider R[e5,e6] and compare with c1Re1+c3Re3,
where
[TABLE]
[TABLE]
We obtain that aγ−αc=0, where a,α=0, which is impossible, because \left(\begin{array}[]{c}a\\
c\end{array}\right) and \left(\begin{array}[]{c}\alpha\\
\gamma\end{array}\right) are supposed to be linearly independent.
(3) Considering R[e5,e6] and comparing with c1Re1+c3Re3,
where
[TABLE]
we have to have that aγ−αc=0, where a,α=0, which is impossible.
(4) Finally we consider R[e5,e6] and compare with c1Re1+c3Re3 as well.
We obtain that bγ−βc=0, where b,β=0, which is impossible.
5.1.2.2 Codimension two solvable extensions of L4,(n≥5)
By taking a linear combination of Ren+1 and Ren+2 and keeping in mind that no nontrivial linear combination of the matrices Ren+1 and Ren+2
can be a nilpotent matrix, one could set \left(\begin{array}[]{c}a\\
b\end{array}\right)=\left(\begin{array}[]{c}1\\
2\end{array}\right) and \left(\begin{array}[]{c}\alpha\\
\beta\end{array}\right)=\left(\begin{array}[]{c}1\\
1\end{array}\right).
Therefore the vector space of outer derivations as n×n matrices is as follows:
[TABLE]
[TABLE]
(i) Considering R[en+1,en+2], which is the same as [Ren+2,Ren+1], we deduce that
αi,3:=ai,3,(5≤i≤n),α2,3:=2a2,3 and
β2,1:=b2,1−2a2,3. As a result, the outer derivation Ren+2
changes as follows:
[TABLE]
Altogether we find the following commutators:
[TABLE]
(ii) We include a linear combination of e2 and en:
[TABLE]
Besides we have the brackets from L4 and from outer derivations Ren+1 and Ren+2 as well.
(iii) We satisfy the right Leibniz identity shown in Table 3. We notice that Ren+1 and Ren+2 restricted to the nilradical do not change, but the remaining brackets are as follows:
[TABLE]
Altogether the nilradical L4(\refL4), the outer derivations Ren+1 and Ren+2
and the remaining brackets given above define a continuous family of solvable right Leibniz algebras
depending on the parameters. Then we apply the technique of “absorption” according to step (iv).
•
First we apply the transformation ei′=ei,(1≤i≤n,n≥5),en+1′=en+1−2c2,n+2e2,en+2′=en+2−4d2,n+1+a5,3e2
to remove the coefficients c2,n+2 and 2d2,n+1+a5,3 in front of e2 in
[en+1,en+2] and [en+2,en+2], respectively.
This transformation changes the coefficient in front of e2 in [en+1,en+1]
and [en+2,en+1] to −2a5,3 and −a5,3, respectively.
•
Then we apply
ei′=ei,(1≤i≤n,n≥5),en+1′=en+1+2a5,3e4,en+2′=en+2 to remove the coefficients −a5,3 and −2a5,3
in front of e2 in [en+2,en+1] and [en+1,en+1], respectively. Besides this transformation removes
a5,3 and −a5,3 in front of e4 in [en+2,en+1] and [en+1,en+2], respectively, and
changes the coefficients in front of ek,(6≤k≤n−1)
in [en+2,en+1] and [en+1,en+2] to ak+1,3−2a5,3ak−1,3 and
2a5,3ak−1,3−ak+1,3, respectively. It also affects the coefficients in front en,(n≥6) in
[en+1,en+2] and [en+2,en+1], which we rename back by cn,n+2 and −cn,n+2,
respectively. The following entries are introduced by the transformation: −2a5,3 and 2a5,3 in the (5,1)st,(n≥5) position in Ren+1 and Len+1,
respectively.
•
Applying the transformation ej′=ej,(1≤j≤n+1,n≥6),en+2′=en+2−∑k=5n−1k−1Ak+1,3ek,
where A6,3:=a6,3 and Ak+1,3:=ak+1,3−21a5,3ak−1,3−∑i=7ki−3Ai−1,3ak−i+5,3,(6≤k≤n−1,n≥7),
we remove the coefficients a6,3 and −a6,3 in front of e5 in [en+2,en+1]
and [en+1,en+2], respectively. We also remove ak+1,3−2a5,3ak−1,3 and
2a5,3ak−1,3−ak+1,3
in front of ek,(6≤k≤n−1) in [en+2,en+1] and [en+1,en+2], respectively.
This transformation introduces 4a6,3 and −4a6,3 in the (6,1)st,(n≥6) position as well as
k−1Ak+1,3 and 1−kAk+1,3 in the (k+1,1)st,(6≤k≤n−1) in Ren+2
and Len+2, respectively. It also affects the coefficients in front en,(n≥7) in
[en+1,en+2] and [en+2,en+1], which we rename back by cn,n+2 and −cn,n+2,
respectively.
•
Finally applying the transformation ei′=ei,(1≤i≤n+1,n≥5),en+2′=en+2+n−1cn,n+2en,
we remove cn,n+2 and −cn,n+2 in front of en
in [en+1,en+2] and [en+2,en+1], respectively,
without affecting other entries. We obtain that Ren+1 and Ren+2 are as follows:
[TABLE]
[TABLE]
The remaining brackets are given below:
[TABLE]
(v) Finally we apply the following two change of basis transformations:
•
e1′=e1−(b2,1−2a2,3)e2,e2′=e2,e3′=e3−2a2,3e2,ei′=ei−∑k=i+2nk−iBk−i+3,3ek,(3≤i≤n−2),en−1′=en−1,en′=en,en+1′=en+1,en+2′=en+2,
where Bj,3:=aj,3−∑k=7jk−5Bk−2,3aj−k+5,3,(5≤j≤n).
This transformation removes a5,3,a6,3,...,an,3 in Ren+1,Ren+2
and −a5,3,−a6,3,...,−an,3 in Len+1,Len+2. It also removes −2a5,3
and 2a5,3 from the (5,1)st positions in Ren+1 and Len+1, respectively.
Besides it removes 3b2,1−2a2,3 and b2,1 from the (2,1)st positions in Ren+1
and Len+1, respectively, as well as b2,1−2a2,3 from the (2,1)st positions in Ren+2
and Len+2.
The transformation also removes a2,3 from the (2,3)rd positions in Ren+1,Len+1 and
2a2,3 from the same positions in Ren+2,Len+2.
This transformation introduces the entries in the (i,1)st positions in Ren+1 and Len+1,
which we set to be ai,1 and −ai,1,(6≤i≤n), respectively. The transformation affects the entries in the (j,1)st,(8≤j≤n)
positions in Ren+2 and Len+2, but we rename all the entries in the (i,1)st positions by i−2i−3ai,1 in Ren+2 and by i−23−iai,1,(6≤i≤n) in Len+2.
•
Finally applying the transformation ek′=ek,(1≤k≤n),en+1′=en+1+∑k=5n−1ak+1,1ek,
en+2′=en+2+∑k=5n−1k−1k−2ak+1,1ek,
we remove ai,1 in Ren+1 and −ai,1 in Len+1 as well as
i−2i−3ai,1 and i−23−iai,1,(6≤i≤n) in Ren+2 and Len+2, respectively.
We obtain a Leibniz algebra gn+2,1 given below:
[TABLE]
We summarize a result
in the following theorem:
Theorem 5.1.4**.**
There are four solvable
indecomposable right Leibniz algebras up to isomorphism with a codimension two nilradical
L4,(n≥4), which are given below:
[TABLE]
5.1.2.3 Codimension three solvable extensions of L4,(n=4)
Considering R[e5,e6],R[e5,e7] and R[e6,e7] and comparing with c1Re1+c3Re3,
where
[TABLE]
we have to have, respectively, that
[TABLE]
However, it means that
\left(\begin{array}[]{c}a_{1}\\
b_{1}\\
c_{1}\end{array}\right),\left(\begin{array}[]{c}a_{2}\\
b_{2}\\
c_{2}\end{array}\right) and \left(\begin{array}[]{c}a_{3}\\
b_{3}\\
c_{3}\end{array}\right) are linearly dependent: one could see that
considering
deta1b1c1a2b2c2a3b3c3=a1(b2c3−b3c2)−a2(b1c3−b3c1)+a3(b1c2−b2c1)=a1(a2c3−a3c2)−a2(a1c3−a3c1)+a3(a1c2−a2c1)=0.
5.2. Solvable indecomposable left Leibniz algebras with a nilradical L4
5.2.1. Codimension one solvable extensions of L4
Classification follows the same steps in theorems with the same cases per step.
Theorem 5.2.1**.**
We set a1,1:=a and a3,3:=b in (\refBRLeibniz). To satisfy the left Leibniz identity, there are the following cases based on the conditions involving parameters,
each gives a continuous family of solvable Leibniz algebras:
(1)
If a1,3=0,b=−a,a=0,b=0,(n=4) or b=(3−n)a,a=0,b=0,(n≥5), then we have the following brackets for the algebra:
[TABLE]
[TABLE]
2. (2)
If a1,3=0,b:=−a,a=0,(n=4) or b:=(3−n)a,a=0,(n≥5),
then the brackets for the algebra are
[TABLE]
[TABLE]
3. (3)
If a1,3=0,a=0,b=0,(n=4) or a=0 and b=0,(n≥5), then
The proof is off-loaded to Table 4, when (n≥5). For (n=4),
we recalculate the applicable identities, which are 1.,2.,4.,6.−8.,10.−15.
2. (2)
We repeat case (1), except the identity 13.
3. (3)
If (n≥5), then we apply the identities given in Table 4,
except 14. and 15. applying instead: Le3[en+1,en+1]=[Le3(en+1),en+1]+[en+1,Le3(en+1)]
and L[e1,en+1]=[L(e1),en+1]+[e1,L(en+1)].
For (n=4), we apply the same identities as in case (1) except 14. and 15.
applying two identities given above.
4. (4)
We apply the same identities as in case (1).
5. (5)
We apply the following identities: 1.,2.,4.,6.−8.,10.−14.,Le3[e5,e5]=[Le3(e5),e5]+[e5,Le3(e5)],15.
6. (6)
We apply the same identities as in case (2) for (n=4).
7. (7)
We apply the same identities as in case (1) for (n=4).
8. (8)
We apply the same identities as in case (1) for (n=4), except 13. and 15.
9. (9)
We apply the same identities as in case (2) for (n=4).
10. (10)
The same identities as in case (5).
11. (11)
The same identities as in case (5), except the identity 15.
∎
Theorem 5.2.2**.**
Applying the technique of “absorption” (see Section 3.2), we can further simplify the algebras
in each of the cases in Theorem 5.2.1 as follows:
(1)
If a1,3=0,b=−a,a=0,b=0,(n=4) or b=(3−n)a,a=0,b=0,(n≥5), then we have the following brackets for the algebra:
[TABLE]
[TABLE]
2. (2)
If a1,3=0,b:=−a,a=0,(n=4) or b:=(3−n)a,a=0,(n≥5),
then the brackets for the algebra are as follows:
[TABLE]
[TABLE]
3. (3)
If a1,3=0,a=0,b=0,(n=4) or a=0 and b=0,(n≥5), then
The right (not a derivation)
and the left (a derivation) multiplication operators restricted to the nilradical are given below:
[TABLE]
[TABLE]
–
We start with applying the transformation ek′=ek,(1≤k≤n),en+1′=en+1−a4,3e1
to
remove a4,3 in Ren+1 and −a4,3 in Len+1 from the (i,i−1)st positions, where (4≤i≤n),
but it affects other entries as well,
such as
the entry in the (2,1)st position in Ren+1 and Len+1 that
we change to a2,1−a4,3 and B2,1−a4,3, respectively.
It also changes the entry in the (2,3)rd position in Len+1 to
a2,3.
At the same time, it affects the coefficient in front of e2 in the bracket [en+1,en+1], which we change back to a2,n+1.
–
Then we apply the transformation ei′=ei,(1≤i≤n),en+1′=en+1+∑k=3n−1ak+1,1ek
to remove ak+1,1 in Ren+1 and −ak+1,1 in Len+1 from the entries in the (k+1,1)st
positions, where (3≤k≤n−1). The transformation changes the entry in the (2,1)st position in
Ren+1 to 2a4,1+a2,1−a4,3, the entries in the (2,3)rd positions in Ren+1
and Len+1 to a2,3+a4,1.
It also affects the coefficient in front of e2 in [en+1,en+1], which
we rename back by a2,n+1. We assign 2a4,1+a2,1−a4,3:=a2,1 and a2,3+a4,1:=a2,3. Then
B2,1−a4,3:=a(2b−a)a2,1+2(a−b)a2,3.
–
Applying the transformation ej′=ej,(1≤j≤n),en+1′=en+1+2ba2,n+1e2,
we
remove the coefficient a2,n+1 in front of e2 in [en+1,en+1].
2. (2)
The right (not a derivation) and left (a derivation) multiplication operators restricted to the nilradical are as follows:
[TABLE]
[TABLE]
–
We apply the transformation ek′=ek,(1≤k≤n),en+1′=en+1−a4,3e1 to
remove a4,3 in Ren+1 and −a4,3 in Len+1 from the (i,i−1)st positions, where (4≤i≤n),
but it affects other entries as well,
such as
the entry in the (2,1)st position in Ren+1 and Len+1 that
we change to a2,1−a4,3 and B2,1−a4,3, respectively.
It also changes the entry in the (2,3)rd position in Len+1 to
a2,3.
At the same time, it affects the coefficient in front of e2 in the bracket [en+1,en+1], which we change back to a2,n+1.
–
Then we apply the transformation ei′=ei,(1≤i≤n),en+1′=en+1+∑k=3n−1ak+1,1ek
to remove ak+1,1 in Ren+1 and −ak+1,1 in Len+1 from the entries in the (k+1,1)st
positions, where (3≤k≤n−1). It changes the entry in the (2,1)st position in
Ren+1 to 2a4,1+a2,1−a4,3, the entries in the (2,3)rd positions in Ren+1
and Len+1 to a2,3+a4,1.
It also affects the coefficient in front of e2 in [en+1,en+1], which
we rename back by a2,n+1. We assign 2a4,1+a2,1−a4,3:=a2,1 and a2,3+a4,1:=a2,3. Then
B2,1−a4,3:=(5−2n)a2,1+2(n−2)a2,3.
–
Finally the transformation ej′=ej,(1≤j≤n),en+1′=en+1−2(n−3)aa2,n+1e2
removes the coefficient a2,n+1 in front of e2 in [en+1,en+1].
3. (3)
The right (not a derivation) and left (a derivation) multiplication operators
restricted to the nilradical are given below:
[TABLE]
[TABLE]
–
Applying the transformation ek′=ek,(1≤k≤n),en+1′=en+1−a4,3e1, we
remove a4,3 in Ren+1 and −a4,3 in Len+1 from the (i,i−1)st positions, where (4≤i≤n),
but the transformation affects other entries,
such as
the entry in the (2,1)st position in Ren+1 and Len+1,
that we change to a2,3−a4,1 and b2,1−a4,3, respectively.
It also changes the entry in the (2,3)rd position in Len+1 to
a2,3.
At the same time, it affects the coefficient in front of e2 in the bracket [en+1,en+1], which we change back to a2,n+1.
–
Then we apply the transformation ei′=ei,(1≤i≤n),en+1′=en+1+∑k=3n−1ak+1,1ek
to remove ak+1,1 in Ren+1 and −ak+1,1 in Len+1 from the entries in the (k+1,1)st
positions, where (3≤k≤n−1). This transformation changes the entry in the (2,1)st position in
Ren+1 as well as the entries in the (2,3)rd positions in Ren+1
and Len+1 to a2,3+a4,1.
It also affects the coefficient in front of e2 in [en+1,en+1], which
we rename back by a2,n+1. We assign a2,3+a4,1:=a2,3 and b2,1−a4,3:=b2,1.
–
Finally applying the transformation ej′=ej,(1≤j≤n),en+1′=en+1+2ba2,n+1e2,
we
remove the coefficient a2,n+1 in front of e2 in [en+1,en+1].
4. (4)
The right (not a derivation) and left (a derivation) multiplication operators
restricted to the nilradical are as follows:
[TABLE]
[TABLE]
–
We continue with the transformation ek′=ek,(1≤k≤n),en+1′=en+1−a4,3e1 to
remove a4,3 in Ren+1 and −a4,3 in Len+1 from the (i,i−1)st positions, where (4≤i≤n),
but other entries are affected as well,
such as
the entry in the (2,1)st position in Ren+1 and Len+1,
that we change to a2,1−a4,3 and a2,3−a2,1−a4,3+b2,3, respectively.
The transformation also changes the entry in the (2,3)rd position in Len+1 to
b2,3−2a4,3
and affects the coefficient in front of e2 in the bracket [en+1,en+1], which we change back to a2,n+1.
–
Applying the transformation ei′=ei,(1≤i≤n),en+1′=en+1+∑k=3n−1ak+1,1ek,
we remove ak+1,1 in Ren+1 and −ak+1,1 in Len+1 from the entries in the (k+1,1)st
positions, where (3≤k≤n−1). This transformation changes the entry in the (2,1)st position in
Ren+1 to 2a4,1+a2,1−a4,3, the entries in the (2,3)rd positions in Ren+1
and Len+1 to a2,3+a4,1 and a4,1+b2,3−2a4,3, respectively.
It also affects the coefficient in front of e2 in [en+1,en+1], that
we rename back by a2,n+1. We assign 2a4,1+a2,1−a4,3:=a2,1,a2,3+a4,1:=a2,3
and a4,1+b2,3−2a4,3:=b2,3.
5. (5)
We apply the transformation
ei′=ei,(1≤i≤4),e5′=e5−A4,3e1−2(b+c)1(B2,1A4,3+a2,1A4,3+2a4,1A4,3−A4,32−a2,3a4,1−a4,12−a4,1b2,3−a2,5)e2+a4,1e3.
Then we assign a2,3+a4,1:=a2,3 and b2,3−2a2,1−3a4,1:=b2,3.
6. (6)
One applies the transformation
ei′=ei,(1≤i≤4),e5′=e5−A4,3e1+2(a−c)1(B2,1A4,3+a2,1A4,3+2a4,1A4,3−A4,32−a2,3a4,1−a4,12−a4,1b2,3−a2,5)e2+a4,1e3.
Then we assign a2,3+a4,1:=a2,3 and b2,3−2a2,1−3a4,1:=b2,3.
7. (7)
We apply the transformation
ei′=ei,(1≤i≤4),e5′=e5−a4,3e1+a4,1e3 and rename the coefficient in front of e2
in [e5,e5] back by a2,5. Then we assign a2,1+2a4,1−a4,3:=a2,1,a2,3+a4,1:=a2,3 and b2,3+a4,1−2a4,3:=b2,3.
8. (8)
The transformation is as follows:
ei′=ei,(1≤i≤4),e5′=e5−a4,3e1+2c1(2a2,1a4,1−4a2,1a4,3+3a4,12−6a4,1a4,3−a4,32+2a4,3b2,3+a2,5)e2+a4,1e3.
We assign a2,1+2a4,1−a4,3:=a2,1 and b2,3+a4,1−2a4,3:=b2,3.
9. (9)
The transformation is
ei′=ei,(1≤i≤4),e5′=e5−A4,3e1+4(b+c)1(2A4,32−a2,3A4,3−2a4,1A4,3−2b2,1A4,3−b2,3A4,3+2a2,3a4,1+2a4,12+2a4,1b2,3+2a2,5)e2+a4,1e3.
We assign a2,3+a4,1:=a2,3 and b2,3+a4,1−2b2,1:=b2,3.
10. (10)
We apply the transformation
ei′=ei,(1≤i≤4),e5′=e5−a4,3e1+a4,1e3 and rename the coefficient in front of e2
in [e5,e5] back by a2,5. We assign a2,3+a4,1:=a2,3,b2,1−a4,3:=b2,1
and b2,3+a4,1−2a4,3:=b2,3.
∎
Theorem 5.2.3**.**
There are eight solvable
indecomposable left Leibniz algebras up to isomorphism with a codimension one nilradical
L4,(n≥4), which are given below:
[TABLE]
Proof.
One applies the change of basis transformations keeping the nilradical L4 given in (\refL4) unchanged.
(1)
We have the right (not a derivation) and the left (a derivation) multiplication operators restricted to the nilradical are as follows:
We apply the transformation e1′=e1,e2′=e2,ei′=ei−(k−i)aak−i+3,3ek,(3≤i≤n−2,i+2≤k≤n,n≥5),ej′=ej,(n−1≤j≤n+1), where k is fixed, renaming all the affected entries back.
This transformation removes a5,3,a6,3,...,an,3 in Ren+1 and −a5,3,−a6,3,...,−an,3 in Len+1.
Besides it introduces the entries in the (5,1)st,(6,1)st,...,(n,1)st positions in Ren+1 and Len+1,
which we set to be a5,1,a6,1,...,an,1 and −a5,1,−a6,1,...,−an,1, respectively.
(I) Suppose b=2a.
–
Applying the transformation e1′=e1+2b−a1(B2,1+b(b−a)a2,3)e2,e2′=e2,e3′=e3+ba2,3e2,ei′=ei,(4≤i≤n,n≥4),en+1′=en+1−a5,1e2+∑k=4n−1ak+1,1ek, we
remove a2,1 and B2,1 from the (2,1)st positions in Ren+1 and Len+1,
respectively. This transformation also removes a2,3
from the (2,3)rd positions in Ren+1 and Len+1 as well as
ak+1,1 in Ren+1 and −ak+1,1 in Len+1 from the entries in the (k+1,1)st
positions, where (4≤k≤n−1).
(For n=4, we direct to the Remark 5.1.1.)
–
Then we scale a to unity applying the transformation ei′=ei,(1≤i≤n,n≥4),en+1′=aen+1.
Renaming ab by b, we obtain a continuous family of Leibniz algebras:
[TABLE]
(II) Suppose b:=2a. We have that B2,1=a2,3.
–
We apply the transformation e1′=e1+aa2,1+a2,3e2,e2′=e2,e3′=e3+a2a2,3e2,ei′=ei,(4≤i≤n,n≥4),en+1′=en+1−a5,1e2+∑k=4n−1ak+1,1ek
to remove a2,3 from the (2,1)st,(2,3)rd positions in Len+1
and from the (2,3)rd position in Ren+1.
This transformation also removes a2,1 from the (2,1)st position in Ren+1 as well as
ak+1,1 and −ak+1,1 from the entries in the (k+1,1)st
positions in Ren+1 and Len+1, respectively, where (4≤k≤n−1). (For n=4, we refer to Remark 5.1.1.)
–
To scale a to unity, we apply the transformation ei′=ei,(1≤i≤n,n≥4),en+1′=aen+1
and obtain a limiting case of (\refln+1,1) with b=21 given below:
[TABLE]
2. (2)
We have the right (not a derivation) and the left (a derivation) multiplication operators restricted to the nilradical are as follows:
[TABLE]
[TABLE]
–
We apply the transformation e1′=e1,e2′=e2,ei′=ei−(k−i)aak−i+3,3ek,(3≤i≤n−2,i+2≤k≤n,n≥5),ej′=ej,(n−1≤j≤n+1), where k is fixed, renaming all the affected entries back.
This transformation removes a5,3,a6,3,...,an,3 in Ren+1 and −a5,3,−a6,3,...,−an,3 in Len+1.
Moreover it introduces the entries in the (5,1)st,(6,1)st,...,(n,1)st positions in Ren+1 and Len+1,
which we set to be a5,1,a6,1,...,an,1 and −a5,1,−a6,1,...,−an,1, respectively.
–
Applying the transformation e1′=e1+(5−2n)a1(B2,1+3−n(2−n)a2,3)e2,e2′=e2,e3′=e3+(3−n)aa2,3e2,ei′=ei,(4≤i≤n,n≥4),en+1′=en+1−a5,1e2+∑k=4n−1ak+1,1ek, we
remove a2,1 and B2,1 from the (2,1)st positions in Ren+1 and Len+1,
respectively. We also remove a2,3
from the (2,3)rd positions in Ren+1 and Len+1 as well as
ak+1,1 in Ren+1 and −ak+1,1 in Len+1 from the entries in the (k+1,1)st
positions, where (4≤k≤n−1). (See Remark 5.1.1.)
–
To scale a to unity, we apply the transformation ei′=ei,(1≤i≤n),en+1′=aen+1
renaming the coefficient a2an,n+1 in front of en in [en+1,en+1] back by an,n+1. We obtain a Leibniz algebra given below:
[TABLE]
If an,n+1=0, then we have a limiting case of (5.2.1) with b=3−n. If an,n+1=0,
then we scale it to 1. It gives us the algebra ln+1,2.
3. (3)
We have the right (not a derivation) and the left (a derivation) multiplication operators restricted to the nilradical are as follows:
[TABLE]
[TABLE]
–
Applying the transformation e1′=e1+2bb2,1+a2,3e2,e2′=e2,e3′=e3+ba2,3e2,ei′=ei,(4≤i≤n+1),
we
remove b2,1 from the (2,1)st position in Len+1 and
a2,3 from the (2,1)st position in Ren+1 and from the (2,3)rd positions in
Ren+1 and Len+1 keeping other entries unchanged.
–
To scale b to unity, we apply the transformation ei′=ei,(1≤i≤n),en+1′=ben+1.
Then we rename ba5,3,ba6,3,...,ban,3 by a5,3,a6,3,...,an,3, respectively.
We obtain a Leibniz algebra
[TABLE]
If a5,3=0,(n≥5), then we scale it to 1. We also rename all the affected entries back
and then we rename a6,3,...,an,3 by b1,...,bn−5, respectively.
We combine with the case when a5,3=0 and obtain a Leibniz algebra ln+1,3.
Remark 5.2.1**.**
If n=4, then ϵ=0.
4. (4)
We have the right (not a derivation) and the left (a derivation) multiplication operators restricted to the nilradical are as follows:
[TABLE]
[TABLE]
–
We apply the transformation e1′=e1,e2′=e2,ei′=ei−(k−i)aak−i+3,3ek,(3≤i≤n−2,i+2≤k≤n,n≥5),ej′=ej,(n−1≤j≤n+1), where k is fixed renaming all the affected entries back.
This transformation removes a5,3,a6,3,...,an,3 in Ren+1 and −a5,3,−a6,3,...,−an,3 in Len+1.
Besides it introduces the entries in the (5,1)st,(6,1)st,...,(n,1)st positions in Ren+1 and Len+1,
which we set to be a5,1,a6,1,...,an,1 and −a5,1,−a6,1,...,−an,1, respectively.
–
Applying the transformation e1′=e1+aa2,1e2,ei′=ei,(2≤i≤n,n≥4),en+1′=en+1+∑k=4n−1ak+1,1ek, we
remove a2,1 from the (2,1)st position in Ren+1.
This transformation changes the entry in the (2,1)st position in Len+1 to a2,3+b2,3.
It also removes
ak+1,1 and −ak+1,1 from the entries in the (k+1,1)st
positions, where (4≤k≤n−1) in Ren+1 and Len+1, respectively.444Except this transformation it is the same as case (4)
for the right Leibniz algebras.
–
We assign a2,3:=d and b2,3:=f
and
then we scale a to unity applying the transformation ei′=ei,(1≤i≤n,n≥4),en+1′=aen+1.
Renaming ad,af and a2a2,n+1 by d,f and a2,n+1, respectively, we obtain a Leibniz algebra, which is right and left at the same time
and a limiting case of (5.2.1) with b=0, when d=f=a2,n+1=0:
[TABLE]
Altogether (5.2.1) and all its limiting
cases after replacing b with a give us a Leibniz algebra ln+1,1.
It remains to consider a continuous family of Leibniz algebras given below
and scale any nonzero entries as much as possible.
[TABLE]
If a2,n+1=0, then we scale it to 1. We also rename all the affected entries back.
Then we combine with the case when a2,n+1=0 and obtain a right and left Leibniz algebra gn+1,4.
5. (5)
Applying the transformation
e1′=e1+2a(b+c)(2a−b)a2,3−b⋅b2,3e2,e2′=e2,e3′=e3+2a(b+c)(2a+c)a2,3+c⋅b2,3e2,
e4′=e4,e5′=ce5 and renaming
ca and cb by a and b, respectively, we obtain a continuous family of Leibniz algebras given below:
[TABLE]
6. (6)
We
apply the transformation
e1′=e1+2(c−a)3a2,3+b2,3e2,e2′=e2,e3′=e3+2a(c−a)(2a+c)a2,3+c⋅b2,3e2,
e4′=e4,e5′=ce5
and rename
ca and c2a4,5 by a and a4,5, respectively,
to obtain a Leibniz algebra given below:
[TABLE]
which is a limiting case of (\refl5,5) with b:=−a when a4,5=0.
If a4,5=0, then
we scale it to 1
and obtain a continuous family of Leibniz algebras:
[TABLE]
7. (7)
We apply the transformation
e1′=e1+aa2,1e2,ei′=ei,(2≤i≤5)
and assign d:=aa⋅b2,3+c⋅a2,1,f:=aa⋅a2,3−c⋅a2,1. Then this case becomes the same as case (7) of Theorem 5.1.3.
8. (8)
Applying the transformation
e1′=e1+c2a2,1−b2,3e2,e2′=e2,e3′=e3+ca2,1e2,e4′=e4,e5′=ce5
and renaming c2a4,5 back by a4,5, we obtain a Leibniz algebra:
[TABLE]
which is a limiting case of (\refg1) with a=0. If a4,5=0, then
we scale a4,5 to 1 and obtain a limiting case of (\refl5,6)
with a=0. Altogether (\refl5,6) and all its limiting cases give us the algebra l5,6.
9. (9)
One applies the transformation
e1′=e1+4(b+c)2(3b+4c)a2,3−b⋅b2,3e2,e2′=e2,e3′=e3+4(b+c)2(4b+5c)a2,3+c⋅b2,3e2,e4′=e4,e5′=ce5. Renaming cb by b, we obtain a Leibniz algebra given below:
[TABLE]
which is a limiting case of (\refl5,5) with a=0. Altogether (\refl5,5) and all its
limiting cases give us the algebra l5,5.
10. (10)
We apply the transformation
e1′=e1+ca2,3e2,ei′=ei,(2≤i≤4),e5′=ce5
and rename cb2,1,cb2,3,ca2,3 and c2a2,5 by
b2,1,b2,3,a2,3 and a2,5, respectively. Then we assign a2,1:=a2,3+b2,3−b2,1
and this case becomes the same as case (10) of Theorem 5.1.3.
∎
5.2.2. Codimension two and three solvable extensions of L4
The non-zero inner derivations of L4,(n≥4) are
given by
[TABLE]
where Ei+1,1 is the n×n matrix that has 1 in the
(i+1,1)st
position and all other entries are zero. According to Remark 5.1.4, we have at most two outer derivations.
5.2.2.1 Codimension two solvable extensions of L4,(n=4)
We consider the same cases as in Section 5.1.2.1 and follow the General approach given in Section 5.1.2.
(1) (a) One could set \left(\begin{array}[]{c}a^{1}\\
b^{1}\\
c^{1}\end{array}\right)=\left(\begin{array}[]{c}1\\
a\\
0\end{array}\right) and \left(\begin{array}[]{c}a^{2}\\
b^{2}\\
c^{2}\end{array}\right)=\left(\begin{array}[]{c}1\\
b\\
1\end{array}\right),(a\neq-1,a\neq 0,b\neq-1). Therefore the vector space of outer derivations as 4×4
matrices is as follows:
[TABLE]
[TABLE]
(i) Considering L[e5,e6], we obtain that a:=1. Since b=−1, we have that
α2,3:=(b+21)a2,3−2b2,3 and it follows that β2,3:=(21−b)a2,3+23b2,3.
It implies that L[e5,e6]=0. As a result,
[TABLE]
[TABLE]
Further, we find the following commutators:
[TABLE]
(ii) We include a linear combination of e2 and e4:
[TABLE]
Besides we have the brackets from L4 and from outer derivations Le5 and Le6 as well.
(iii) We satisfy the right Leibniz identity shown in Table 5.
We obtain that Le5 and Le6 restricted to the nilradical do not change, but the remaining brackets are as follows:
[TABLE]
Altogether the nilradical L4(\refL4), the outer derivations Le5 and Le6
and the remaining brackets given above define a continuous family of Leibniz algebras
depending on the parameters.
(iv)&(v) We apply the following transformation: e1′=e1+2a2,3−b2,3e2,e2′=e2,e3′=e3+a2,3e2,e4′=e4,e5′=e5+2c2,5e2,e6′=e6+2d2,5e2+2c2,6−(b+1)c2,5e4
and obtain a Leibniz algebra l6,2 given below:
[TABLE]
Remark 5.2.2**.**
We notice that if b=−1 in the algebra, then the outer derivation Le6 is nilpotent.
(1) (b) We set \left(\begin{array}[]{c}a^{1}\\
b^{1}\\
c^{1}\end{array}\right)=\left(\begin{array}[]{c}1\\
2\\
c\end{array}\right) and \left(\begin{array}[]{c}a^{2}\\
b^{2}\\
c^{2}\end{array}\right)=\left(\begin{array}[]{c}1\\
1\\
d\end{array}\right),(c\neq-2,d\neq-1). Therefore the vector space of outer derivations as 4×4
matrices is as follows:
[TABLE]
[TABLE]
(i) Considering L[e5,e6], we obtain that d=0 and we have
the system of equations:
[TABLE]
There are the following two cases:
(I)
If c=−1, then a2,3:=2c+4α2,3−2cβ2,3
and L[e5,e6]=0. Consequently,
[TABLE]
[TABLE]
and we find the commutators given below:
[TABLE]
2. (II)
If c=−1, then b2,3:=2β2,3−α2,3
and L[e5,e6]=0. As a result,
[TABLE]
and we have the following commutators:
[TABLE]
(ii) We combine cases (I) and (II), include a linear combination of e2 and e4,
and obtain the following:
[TABLE]
Besides we have the brackets from L4 and from outer derivations Le5 and Le6 as well.
(iii) To satisfy the left Leibniz identity, we refer to the identities given in Table 5. as much as possible.
The identities we apply are the following: 1.−6.,Le3([e6,e6])=[Le3(e6),e6]+[e6,Le3(e6)],Le1([e6,e6])=[Le1(e6),e6]+[e6,Le1(e6)],Le1([e6,e5])=[Le1(e6),e5]+[e6,Le1(e5)],Le3([e6,e5])=[Le3(e6),e5]+[e6,Le3(e5)],11.
and 12.
We have that Le5 and Le6 restricted to the nilradical do not change, but the remaining brackets are as follows:
[TABLE]
Altogether the nilradical L4(\refL4), the outer derivations Le5 and Le6
and the remaining brackets given above define a continuous family of Leibniz algebras.
(iv)&(v) Applying the transformation: e1′=e1+2α2,3−β2,3e2,e2′=e2,e3′=e3+α2,3e2,e4′=e4,e5′=e5+2a2,6e2+2a4,6e4,e6′=e6+2b2,6e2, we obtain a Leibniz algebra l6,3:
[TABLE]
(1) (c) We set \left(\begin{array}[]{c}a^{1}\\
b^{1}\\
c^{1}\end{array}\right)=\left(\begin{array}[]{c}a\\
1\\
0\end{array}\right) and \left(\begin{array}[]{c}a^{2}\\
b^{2}\\
c^{2}\end{array}\right)=\left(\begin{array}[]{c}b\\
0\\
1\end{array}\right),(a,b\neq 0,a\neq-1). Therefore the vector space of outer derivations as 4×4
matrices is as follows:
[TABLE]
[TABLE]
(i) Considering L[e5,e6], we obtain that a:=1 and we have
the system of equations:
[TABLE]
There are the following two cases:
(I)
If b=3, then α2,3:=2a2,3−b2,3⟹β2,3:=2(2b−1)a2,3+(2b+1)b2,3
and L[e5,e6]=0. As a result,
[TABLE]
Further, we find the following:
[TABLE]
2. (II)
If b:=3, then β2,3:=29a2,3+3b2,3−4α2,3 and L[e5,e6]=0 as well as Le5,Le6 are as follows:
[TABLE]
We have the following commutators:
[TABLE]
(ii) We combine cases (I) and (II) and include a linear combination of e2 and e4:
[TABLE]
Besides we have the brackets from L4 and from outer derivations Le5 and Le6 as well.
(iii) To satisfy the left Leibniz identity, we mostly apply the identities given in Table 5:
1.−5.,Le6([e6,e5])=[Le6(e6),e5]+[e6,Le6(e5)],7.−12.
We have that Le5 and Le6 restricted to the nilradical do not change, but the remaining brackets are the following:
[TABLE]
Altogether the nilradical L4(\refL4), the outer derivations Le5 and Le6
and the remaining brackets given above define a continuous family of Leibniz algebras
depending on the parameters.
(iv)&(v) We apply the transformation: e1′=e1+2a2,3−b2,3e2,e2′=e2,e3′=e3+a2,3e2,e4′=e4,e5′=e5+2c2,5e2,e6′=e6+2d2,5e2+2c2,6−c2,5e4 and obtain a Leibniz algebra l6,4:
[TABLE]
5.2.2.2 Codimension two solvable extensions of L4,(n≥5)
One could set \left(\begin{array}[]{c}a\\
b\end{array}\right)=\left(\begin{array}[]{c}1\\
2\end{array}\right) and \left(\begin{array}[]{c}\alpha\\
\beta\end{array}\right)=\left(\begin{array}[]{c}1\\
1\end{array}\right).
Therefore the vector space of outer derivations as n×n matrices is as follows:
[TABLE]
[TABLE]
(i) Considering L[en+1,en+2] and comparing with
∑i=1nciLei, we deduce that
αi,3:=ai,3,(5≤i≤n),α2,3:=2a2,3 and
α2,1:=a2,1−2a2,3. As a result, the outer derivation Len+2
changes as follows:
[TABLE]
Altogether we find the following commutators:
[TABLE]
(ii) We include a linear combination of e2 and en:
[TABLE]
Besides we have the brackets from L4 and from outer derivations Len+1 and Len+2 as well.
(iii) We satisfy the right Leibniz identity shown in Table 6. We notice that Len+1 and Len+2 restricted to the nilradical do not change, but the remaining brackets are as follows:
[TABLE]
Altogether the nilradical L4(\refL4), the outer derivations Len+1 and Len+2
and the remaining brackets given above define a continuous family of the solvable left Leibniz algebras.
Then we apply the technique of “absorption” according to step (iv).
•
We start with the transformation ei′=ei,(1≤i≤n,n≥5),en+1′=en+1+2c2,n+2e2,en+2′=en+2+4d2,n+1−a5,3e2.
This transformation removes the coefficients c2,n+2 and 2d2,n+1−a5,3 in front of e2 in
[en+2,en+1] and [en+2,en+2], respectively,
and changes the coefficient in front of e2 in [en+1,en+1]
and [en+1,en+2] to 2a5,3 and a5,3, correspondingly.
•
Then we apply the transformation
ei′=ei,(1≤i≤n,n≥5),en+1′=en+1+2a5,3e4,en+2′=en+2 to remove the coefficients a5,3 and 2a5,3
in front of e2 in [en+1,en+2] and [en+1,en+1], respectively. At the same time this transformation removes
a5,3 and −a5,3 in front of e4 in [en+2,en+1] and [en+1,en+2], respectively, and changes the coefficients in front of ek,(6≤k≤n−1)
in [en+2,en+1] and [en+1,en+2] to ak+1,3−2a5,3ak−1,3 and
2a5,3ak−1,3−ak+1,3, respectively. It also affects the coefficients in front en,(n≥6) in
[en+2,en+1] and [en+1,en+2], which we rename back by cn,n+2 and −cn,n+2,
respectively. The following entries are introduced by the transformation: −2a5,3 and 2a5,3 in the (5,1)st,(n≥5) position in Ren+1 and Len+1,
respectively.
•
Applying the transformation ej′=ej,(1≤j≤n+1,n≥6),en+2′=en+2−∑k=5n−1k−1Ak+1,3ek,
where A6,3:=a6,3 and Ak+1,3:=ak+1,3−21a5,3ak−1,3−∑i=7ki−3Ai−1,3ak−i+5,3,(6≤k≤n−1,n≥7),
we remove the coefficients a6,3 and −a6,3 in front of e5 in [en+2,en+1]
and [en+1,en+2], respectively. Besides the transformation removes ak+1,3−2a5,3ak−1,3 and
2a5,3ak−1,3−ak+1,3
in front of ek,(6≤k≤n−1) in [en+2,en+1] and [en+1,en+2], respectively.
This transformation introduces 4a6,3 and −4a6,3 in the (6,1)st,(n≥6) position
as well as
k−1Ak+1,3 and 1−kAk+1,3 in the (k+1,1)st,(6≤k≤n−1) position in Ren+2
and Len+2, respectively. It also affects the coefficients in front en,(n≥7) in
[en+2,en+1] and [en+1,en+2], which we rename back by cn,n+2 and −cn,n+2,
respectively.
•
Finally applying the transformation ei′=ei,(1≤i≤n+1,n≥5),en+2′=en+2−n−1cn,n+2en,
we remove cn,n+2 and −cn,n+2 in front of en
in [en+2,en+1] and [en+1,en+2], respectively,
without affecting other entries.
We obtain that Len+1 and Len+2 are as follows:
[TABLE]
[TABLE]
The remaining brackets are given below:
[TABLE]
(v) Finally we apply the following two change of basis transformations:
•
e1′=e1+(a2,1−2a2,3)e2,e2′=e2,e3′=e3+2a2,3e2,ei′=ei−∑k=i+2nk−iBk−i+3,3ek,(3≤i≤n−2),en−1′=en−1,en′=en,en+1′=en+1,en+2′=en+2,
where Bj,3:=aj,3−∑k=7jk−5Bk−2,3aj−k+5,3,(5≤j≤n).
This transformation removes a5,3,a6,3,...,an,3 in Ren+1,Ren+2
and −a5,3,−a6,3,...,−an,3 in Len+1,Len+2. It also removes −2a5,3
and 2a5,3 from the (5,1)st positions in Ren+1 and Len+1, respectively.
Besides it removes a2,1 and 3a2,1−2a2,3 from the (2,1)st positions in Ren+1
and Len+1, respectively, as well as a2,1−2a2,3 from the (2,1)st positions in Ren+2
and Len+2.
The transformation also removes a2,3 from the (2,3)rd positions in Ren+1,Len+1 and
2a2,3 from the same positions in Ren+2,Len+2.
It introduces the entries in the (i,1)st positions in Ren+1 and Len+1,
that we set to be ai,1 and −ai,1,(6≤i≤n), respectively. The transformation affects the entries in the (j,1)st,(8≤j≤n)
positions in Ren+2 and Len+2, but we rename all the entries in the (i,1)st positions
by i−2i−3ai,1 in Ren+2 and by i−23−iai,1,(6≤i≤n) in Len+2.
•
Applying the transformation ek′=ek,(1≤k≤n),en+1′=en+1+∑k=5n−1ak+1,1ek,en+2′=en+2+∑k=5n−1k−1k−2ak+1,1ek,
we remove ai,1 in Ren+1 and −ai,1 in Len+1 as well as
i−2i−3ai,1 and i−23−iai,1,(6≤i≤n) in Ren+2 and Len+2, respectively.
We obtain a Leibniz algebra ln+2,1 given below:
[TABLE]
We summarize a result
in the following theorem:
Theorem 5.2.4**.**
There are four solvable
indecomposable left Leibniz algebras up to isomorphism with a codimension two nilradical
L4,(n≥4), which are given below:
[TABLE]
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