Levi Decomposition of Frobenius Lie Algebra of Dimension 6


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 7(3) (2022), Pages 394-400 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: April 03, 2022 Reviewed: April 11, 2022 Accepted: April 14, 2022 
DOI: http://dx.doi.org/10.18860/ca.v7i3.15656      

Levi Decomposition of Frobenius Lie Algebra of Dimension 6 

Henti*, Edi Kurniadi, Ema Carnia 

Departement of Mathematics of FMIPA, Universitas Padjadjaran, Indonesia 

Email: henti17001@mail.unpad.ac.id     

ABSTRACT 

In this paper, we study notion of the Frobenius Lie algebra M2,1(ℝ) ⋊ 𝔤𝔩2(ℝ) of dimension 6. The 

finite dimensional Lie algebra can be expressed in terms of decomposition between Levi 
subalgebra and the radical (maximal solvable ideal). This form of decomposition is called Levi 
decomposition. Our main object is further denoted by 𝔞𝔣𝔣(2) ≔ M2,1(ℝ) ⋊ 𝔤𝔩2(ℝ). The work aims 

to obtain Levi decomposition of Frobenius Lie algebra 𝔞𝔣𝔣(2)  of dimension 6. To obtained Levi 
subalgebra and the radical, we apply literature reviews about Lie algebra and decomposition 
Levi in Dagli result. The main result of this paper is Frobenius Lie algebra 𝔞𝔣𝔣(2) can be 
decomposition be semisimple Levi subalgebra 𝔥 of dimension 4 and radical solvable Rad(𝔤) of 
dimension 2. Thus, the Levi decomposition form of the Frobenius Lie algebra is given. 

Keywords: frobenius lie algebra; levi decomposition; lie algebra; radical  

INTRODUCTION 
A vector space over a field that is equipped by Lie brackets which is neither 

commutative nor associative is called Lie algebra [1]. Any finite dimensional Lie algebra 
can be expressed as semidirect sum between Levi subalgebra (Lie subalgebra) and its 
radical (maximal solvable ideal) and this form is called a Levi decomposition [1]. We 
denote a finite dimensional Lie algebra by 𝔤. On the other hand, for finite dimensional 
case, the Lie algebra 𝔤 can be written in Levi decomposition form which is given in the 
following form  

𝔤 = 𝔥 ⋉ Rad(𝔤)        (1) 
where 𝔥 is a Levi subalgebra of 𝔤 and Rad(𝔤) is a radical or solvable maximal ideal of 𝔤. 
Let 𝑆 = {𝑒1, 𝑒2, … , 𝑒𝑛} be a basis of 𝔤 and we define 𝐶(𝔤) = (𝐶(𝔤)𝑖,𝑗) be a matrix whose 

Lie brackets entries of 𝔤 are given by 
𝐶(𝔤)𝑖,𝑗 ≔ [𝑒𝑖, 𝑒𝑗]𝔤

, 1 ≤ 𝑖, 𝑗 ≤ 𝑛.      (2) 

This matrix 𝐶(𝔤) ∈ 𝑀𝑎𝑡(𝑛 × 𝑛, 𝑆(𝔤)) is called a structure matrix of 𝔤 where 𝑆(𝔤) denotes 

as symmetric algebra of 𝔤 [2]. The notion of Lie algebras has been widely studied. One of 
which is the investigation of Lie algebra with dimension 8 which can be carried out by 
Levi's decomposition [3]. Rais introduced the Lie algebra notion 𝑀𝑛,𝑝(ℝ) ⋊ 𝔤𝔩𝑛(ℝ) 

where 𝑀𝑛,𝑝(ℝ) is a vector space of matrices  of size 𝑛 × 𝑝 with real number entries and 

𝔤𝔩𝑛(ℝ) is the Lie algebra of a vector space of matrices of size 𝑛 × 𝑛  equipped with Lie 
brackets [4]. Furthermore, we can see the notions of Lie algebra in [5] and [6]. 

http://dx.doi.org/10.18860/ca.v7i3.15656
mailto:henti17001@mail.unpad.ac.id*


Levi Decomposition of Frobenius Lie Algebra of Dimension 6 

Henti 395 

Let 𝔤 be a Lie algebra with 𝔤∗ is a dual vector space of 𝔤 where 𝔤∗ consisting of real 
valued all linear functional on 𝔤. The Lie algebra 𝔤 is said to be a Frobenius Lie algebra if 
there exists a linear functional 𝜑 ∈ 𝔤∗ so that the skew-symmetric bilinear form 
𝐵𝜑 (𝑥, 𝑦) ≔  𝜑([𝑥, 𝑦]) is non degenerate.  Many studies of Frobenius Lie algebras have 

been carried out over the years. For instance, the properties of principal elements on 
Frobenius Lie algebra one of them is Frobenius Lie algebra cannot be unimodular [7]. An 
example of Frobenius Lie algebra is the affine Lie algebra 𝔞𝔣𝔣(2) can be seen in the 
classification of Frobenius Lie algebra with dimension less than or equal to 6 [8]. 
Kurniadi have constructed Frobenius Lie algebra with dimension less than or equal to 6 
from non-commutative nilpotent Lie algebra with dimension less than or equal to 4 [9]. 
Other example of Frobenius Lie algebra, notation Lie algebra 𝑀𝑛,𝑝(ℝ) ⋊ 𝔤𝔩𝑛(ℝ) where 

𝑛 = 𝑝 = 3 is Frobenius Lie algebra of dimension 18 [10]. Moreover, the Lie algebra 
𝑀3(ℝ) ⋊ 𝔤𝔩3(ℝ) has quasi-associative algebra structure [11]. The Frobenius Lie algebra 
M2(ℝ) ⋊ 𝔤𝔩2(ℝ) is the left-symmetric algebra [12]. It has been proven that the affine Lie 
algebra that is denoted by 𝔞𝔣𝔣(𝑛) ≔ ℝ𝑛 ⋊ 𝔤𝔩𝑛(ℝ) is Frobenius Lie algebra where ℝ

𝑛 is 
another form of 𝑀𝑛,1(ℝ) [13].  Readers can study more about Frobenius Lie algebra in 

the following articles: [14], [15], [16], and [17]. 
In this paper, we study about decompose Frobenius Lie algebra for special case 

𝑛 = 2 of the affine Lie algebra 𝔞𝔣𝔣(𝑛). The notion M2,1(ℝ) ⋊ 𝔤𝔩2(ℝ) can be written in 

simpler formulas as ℝ2 ⋊ 𝔤𝔩2(ℝ) and we can denote it by 𝔞𝔣𝔣(2) which is known as the 
affine Lie algebra.  In the nice formula, the affine Lie algebra 𝔞𝔣𝔣(2) ≔ ℝ2 ⋊ 𝔤𝔩2(ℝ) can be 
expressed in the form of a matrix  

𝔞𝔣𝔣(2) ≔ {(
𝑋 𝑌
0 0

) ;  𝑋 ∈ 𝔤𝔩2(ℝ), 𝑌 ∈ ℝ
2} ⊆ 𝔤𝔩3(ℝ)    (3) 

where 𝔤𝔩3(ℝ) is 3 × 3 real matrix. The purpose of this research is to give decompose this 
Lie algebra into Levi subalgebra and radical. 
 

METHODS  
We used literature study for the research method, especially the study of 

Frobenius Lie algebra 𝔞𝔣𝔣(2) and about Levi decomposition of Lie algebra in [18]. First, 
we given an affine Lie algebra 𝔞𝔣𝔣(2). We proved the affine Lie algebra 𝔞𝔣𝔣(2) not 
solvable. Then, it is proved that Lie algebra 𝔞𝔣𝔣(2) can be decomposed into its subalgebra 
and radical.  

Before going into the discussion, we would like to introduce the theoretical 
foundations used in this study as follows: 
Definition 1 [19] Let 𝔤 be a vector space and a bilinear form 

[. , . ]: 𝔤 × 𝔤 ∋ (𝑥, 𝑦) ↦ [𝑥, 𝑦] ∈ 𝔤. 
The bilinear form [. , . ] is called a Lie bracket for 𝔤 if the following condisitions are 
satisfied: 

1. [𝑥, 𝑦] = −[𝑦, 𝑥]; ∀ 𝑥, 𝑦 ∈ 𝔤 
2. [𝑥, [𝑦, 𝑧]] + [𝑦, [𝑧, 𝑥]] + [𝑧, [𝑥, 𝑦]] = 0; ∀ 𝑥, 𝑦, 𝑧 ∈ 𝔤. 

The vector space 𝔤 equipped by Lie brackets is called Lie algebra. 
Definition 2 [19] A linear subspace 𝔥 of 𝔤 is called a Lie sub-algebra if [𝔥, 𝔥] ⊆ 𝔥, we 
denote by 𝔥 < 𝔤. If we have [𝔤, 𝔥 ] ⊆ 𝔥, we call 𝔥 as an ideal of 𝔤 and then write 𝔥 ⊴ 𝔤. 
Definition 3 [19] Let 𝔤 be a Lie algebra. The derived series of 𝔤 is defined by  

𝐷0(𝔤) = 𝔤 and 𝐷𝑛(𝔤) = [𝐷𝑛−1(𝔤), 𝐷𝑛−1(𝔤)] ∀𝑛 ∈ ℕ   (4) 
The Lie algebra 𝔤 is said to be solvable, if there exists an 𝑛 ∈ ℕ with 𝐷𝑛(𝔤) = {0}. 
Theorem 1 Let 𝔤 be Lie algebra then,  



Levi Decomposition of Frobenius Lie Algebra of Dimension 6 

Henti 396 

i. If 𝔤 is solvable then the subalgebras and homomorphic images of 𝔤 are solvable. 
ii. If 𝔥 is a solvable ideal of 𝔤 and 𝔤/𝔥 is solvable, then 𝔤 is solvable 
iii. If 𝔥 and 𝔦 are solvable ideals of 𝔤 then 𝔥 + 𝔦 is also a solvable ideal of 𝔤. 

This theorem shows that the sum of all solvable ideals of a Lie algebra is a solvable ideal. 
So, in every finite-dimensional Lie algebra 𝔤, there exists a maximal solvable ideal. This 
ideal is called the radical of 𝔤 and denoted by  𝑅𝑎𝑑(𝔤). 
Theorem 2 [18] Let 𝑉 be vector space over a field and let 𝔤 be a subalgebra of 𝔤𝔩(𝑉), the 𝔤 
is solvable if Tr(𝑥𝑦) = 0 for all 𝑥 ∈ 𝔤 and 𝑦 ∈ [𝔤, 𝔤]. 
Theorem 3 [18] Let 𝔤 be Lie algebra over a field 𝔽, then  

𝑅𝑎𝑑(𝔤) = {𝑥 ∈ 𝔤 | 𝑇𝑟(ad 𝑥 ⋅ ad 𝑦) = 0}     (5) 
for all 𝑦 ∈ [𝔤, 𝔤]. 
Definition 4 [19] Let 𝔤 be a Lie algebra. If its radical is trivial i.e 𝑅𝑎𝑑(𝔤) = {0} then 𝔤 is 
called semisimple. The Lie algebra 𝔤 is said to be simple if it is not abelian and if it contains 
no ideal other than 𝔤 and {0}. 
Definition 5 [1] Let 𝑉 be a space vector. A linear map 𝜌: 𝑉 → 𝑉 is said to be 
endomorphism on 𝑉  if the following condition satisfied:  
1.  𝜌(𝑥 + 𝑦) = 𝜌(𝑥) + 𝜌(𝑦)  

2. 𝜌(𝑥𝑦) = (𝜌(𝑥))𝑦 = 𝑥(𝜌(𝑦)) 

for all 𝑥, 𝑦 ∈ 𝑉. The set of all endomorphism on 𝑉 is denoted by  𝐸𝑛𝑑(𝑉). Furthermore, the 
endomorphism 𝐸𝑛𝑑(𝑉) equipped by Lie bracket [𝑥, 𝑦] = 𝑥𝑦 − 𝑦𝑥 for all 𝑥, 𝑦 ∈ 𝐸𝑛𝑑(𝑉) is 
Lie algebra and it is called a general linear algebra, we denoted by 𝔤𝔩(V). 
Definition 6 [19] Let 𝔤 be a Lie algebra and 𝑥 ∈ 𝔤. The map 𝑎𝑑: 𝔤 → 𝔤 defined by  

ad 𝑥 ∶ 𝔤 ∈ 𝑦 ↦ ad 𝑥(𝑦) = [𝑥, 𝑦] ∈ 𝔤      (6) 
is a derivation. The map ad: 𝔤 → 𝔤𝔩(𝔤) is called an adjoint representation. 
Let a representation of Lie algebra 𝔤 in the dual vector space 𝔤∗ is denoted by ad∗ whose 
value on 𝔤 is defined by  

〈ad∗(𝑥)𝜑, 𝑦〉 = 〈𝜑, ad∗(−𝑥)𝑦〉 = 〈𝜑, [𝑦, 𝑥]〉     (7) 
for 𝜑 ∈ 𝔤∗, for all 𝑥, 𝑦 ∈ 𝔤. 
A stabilizer of Lie algebra 𝔤 at the point 𝜑 ∈ 𝔤∗ is given in the following form: 

𝔤𝜑 = {𝑥 ∈ 𝔤 | ad∗(𝑥)𝜑 = 0}       (8) 
Definition 7 [20] Let 𝔤 be a Lie algebra whose 𝔤∗ be a dual vector space of 𝔤. A Lie algebra 
𝔤 is said to be Frobenius Lie algebra if there exist linear functional 𝜑 ∈ 𝔤∗ such that the 
stabilizer of 𝔤 on 𝜑 is equal to 0. 

Furthermore, we review briefly some basic notations needed in Levi 
decomposition. We explain Levi’s theorem which states that a finite dimensional Lie 
algebra can be expressed as the semidirect sum of the Levi subalgebra and the radical. 
Theorem 4 [18] Let 𝔤 be a Lie algebra and let 𝔤 be not solvable, then 𝔤/𝑅𝑎𝑑(𝔤) is a 
semisimple Lie subalgebra. 
Theorem 5 [18] Let 𝔤 be a finite dimensional Lie algebra. If 𝔤 is not solvable, then there is 
a semisimple subalgebra 𝔰 of 𝔤 such that 

𝔤 = 𝔰 ⊕ 𝑅𝑎𝑑(𝔤).        (9) 
In this decomposition, 𝔰 ≅ 𝔤/𝑅𝑎𝑑(𝔤) and we have commutation relations as follows 

[𝔰, 𝔰] = 𝔰, [𝔰, 𝑅𝑎𝑑(𝔤)] ⊆ 𝑅𝑎𝑑(𝔤), [𝑅𝑎𝑑(𝔤), 𝑅𝑎𝑑(𝔤)] ⊆ 𝑅𝑎𝑑(𝔤).  (10) 
The example of Levi decomposition can be seen in the work of [18], one of all 

example its Levi decomposition as follows 
Example 1 [18] Let 𝔤 be a Lie algebra spanned by  



Levi Decomposition of Frobenius Lie Algebra of Dimension 6 

Henti 397 

{𝑥1 = (
0 1 0
1 0 0
0 0 1

) , 𝑥2 = (
1 0 0
0 −1 0
0 0 1

) , 𝑥3 = (
0 1 0
0 0 0
0 0 1

) , 𝑥4 = (
0 1 0
1 0 0
0 0 0

)} (11) 

where Lie bracket non-zero is 
[𝑥1, 𝑥2] = 4𝑥1 − 4𝑥3 − 2𝑥4, [𝑥1, 𝑥3] = 𝑥1 − 𝑥2 − 𝑥4,  
[𝑥2, 𝑥3] = −2𝑥1 + 2𝑥3 + 2𝑥4, [𝑥2, 𝑥4] = −4𝑥1 + 4𝑥3 + 2𝑥4,  
[𝑥3, 𝑥4] = −𝑥1 + 𝑥2 + 𝑥4.  

The Lie algebra 𝔤 can be express in 𝔤 = 𝑅𝑎𝑑(𝔤) ⋊ 𝔥 where radical 𝑅𝑎𝑑(𝔤) =

𝑠𝑝𝑎𝑛 {(
0 0 0
0 0 0
0 0 1

)} and Levi subalgebra 𝔥 is spanned by {𝑧1 = (
0 1 0
1 0 0
0 0 0

) , 𝑧2 =

(
1 0 0
0 −1 0
0 0 0

) , 𝑧3 = (
0 1 0
0 0 0
0 0 0

)}. 

RESULTS AND DISCUSSION  

In this section, let 𝔞𝔣𝔣(2) be the affine Lie algebra and let 𝔞𝔣𝔣(2) be realized in the 
following matrix form  

𝔞𝔣𝔣(2) = {(
𝑎 𝑏 𝑥
𝑐 𝑑 𝑦
0 0 0

) | 𝑎, 𝑏, 𝑐, 𝑑, 𝑥, 𝑦 ∈ ℝ} ⊆ 𝔤𝔩3(ℝ),  (12) 

with the standard basis for 𝔞𝔣𝔣(2), we have 

𝑆 = {𝑥1 = (
1 0 0
0 0 0
0 0 0

) , 𝑥2 = (
0 1 0
0 0 0
0 0 0

) , 𝑥3 = (
0 0 0
1 0 0
0 0 0

) , 𝑥4 = (
0 0 0
0 1 0
0 0 0

) ,

𝑥5 = (
0 0 1
0 0 0
0 0 0

) , 𝑥6 = (
0 0 0
0 0 1
0 0 0

)}. 

          (13) 
The Lie brackets for the affine Lie algebra 𝔞𝔣𝔣(2) is defined by [𝑎, 𝑏] = 𝑎𝑏 − 𝑏𝑎, ∀ 𝑎, 𝑏 ∈
𝔞𝔣𝔣(2) such that the non-zero Lie brackets for the affine Lie algebra 𝔞𝔣𝔣(2) as follows 
 

  
   

(14) 
Theorem 5 [8] Let 𝔞𝔣𝔣(2) be a Lie algebra of dimension 6 with basis in the equation (13). 
Let 𝔞𝔣𝔣(2)∗ be its dual vector space of 𝔞𝔣𝔣(2). Then there exist a linear functional 𝜑 = 𝑥2

∗ +
𝑥6

∗ ∈ 𝔞𝔣𝔣(2)∗ such that 𝔤𝜑 = {0}. Therefore, the affine Lie algebra 𝔞𝔣𝔣(2) is Frobenius. 
In this section of the discussion is our main result, we will prove the Proposition 

1 and the Proposition 2 as follows. 
Proposition 1. The affine Lie algebra 𝔞𝔣𝔣(2) is not solvable.  
Proof.  We have that 

𝐷1(𝔞𝔣𝔣(2)) = [𝐷(𝔞𝔣𝔣(2)), 𝐷(𝔞𝔣𝔣(2))] = 𝑠𝑝𝑎𝑛{𝑥1 − 𝑥4, 𝑥2, 𝑥3, 𝑥5, 𝑥6}.  
Next, we compute 𝐷2(𝔞𝔣𝔣(2)) also obtained 

𝐷2(𝔞𝔣𝔣(2)) = [𝐷1(𝔞𝔣𝔣(2)), 𝐷1(𝔞𝔣𝔣(2))] = 𝑠𝑝𝑎𝑛{𝑥1 − 𝑥4, 𝑥2, 𝑥3, 𝑥5, 𝑥6}.  

Therefore, there not exist 𝑛 > 0 that causes 𝐷𝑛(𝔞𝔣𝔣(2)) = {0}. Thus, the affine Lie 

algebra 𝔞𝔣𝔣(2) is not solvable.  
∎  

Proposition 2. Let 𝔞𝔣𝔣(2) be Frobenius affine Lie algebra whose basis 𝑆 = {𝑥𝑖}𝑖=1
6  where 

the non-zero brackets for 𝔞𝔣𝔣(2) in the equation (14). The affine Lie algebra 𝔞𝔣𝔣(2) is not 

[𝑥1, 𝑥2]  =  𝑥2,   [𝑥1, 𝑥3]  =  −𝑥3,  [𝑥1, 𝑥5]  =  𝑥5,  
[𝑥2, 𝑥3]  =  𝑥1  − 𝑥4,  [𝑥2, 𝑥4]  =  𝑥2,  [𝑥2, 𝑥6]  =  𝑥5,  
[𝑥3, 𝑥4]  =  −𝑥3,  [𝑥3, 𝑥5]  =  𝑥6,  [𝑥4, 𝑥6]  =  𝑥6.  



Levi Decomposition of Frobenius Lie Algebra of Dimension 6 

Henti 398 

solvable then there exist 𝔥 = 𝑠𝑝𝑎𝑛{𝑥1, 𝑥2 + 𝑥5 + 𝑥6, 𝑥3 + 𝑥5 + 𝑥6, 𝑥4} is the semisimple Lie 
subalgebra of 𝔞𝔣𝔣(2) and 𝑅𝑎𝑑(𝔞𝔣𝔣(2)) = 𝑠𝑝𝑎𝑛{𝑥5, 𝑥6} is the radical of 𝔞𝔣𝔣(2)  such that  

 𝔞𝔣𝔣(2) = 𝑠𝑝𝑎𝑛{𝑥5, 𝑥6} ⋊ 𝑠𝑝𝑎𝑛{𝑥1, 𝑥2 + 𝑥5 + 𝑥6, 𝑥3 + 𝑥5 + 𝑥6, 𝑥4}.  (15)  
Proof. Firstly, we have the structure matrix of 𝔞𝔣𝔣(2) is 

𝐶(𝔞𝔣𝔣(2)) =

(

 
 
 

0 𝑥2 −𝑥3 0 𝑥5 0
−𝑥2 0 𝑥1 − 𝑥4 𝑥2 0 𝑥5
𝑥3 𝑥4 − 𝑥1 0 −𝑥3 𝑥6 0
0 −𝑥2 𝑥3 0 0 𝑥6

−𝑥5 0 −𝑥6 0 0 0
0 −𝑥5 0 −𝑥6 0 0 )

 
 
 

.     (16) 

Then, we find the maximal linearly independent set in the structure matrix such that 
basis  

𝐵 = {𝑦1 = 𝑥1 − 𝑥4, 𝑦2 = 𝑥2, 𝑦3 = 𝑥3, 𝑦4 = 𝑥5, 𝑦5 = 𝑥6} 
of the product space [𝔞𝔣𝔣(2), 𝔞𝔣𝔣(2)]. Next, calculate ad 𝑥𝑖  and ad 𝑦𝑗 for 1 ≤ 𝑖 ≤ 6, 1 ≤ 𝑗 ≤

5, we get 

ad 𝑥1 =

[
 
 
 
 
 
0 0 0 0 0 0
0 1 0 0 0 0
0 0 −1 0 0 0
0 0 0 0 0 0
0 0 0 0 1 0
0 0 0 0 0 0]

 
 
 
 
 

, ad 𝑥2 =

[
 
 
 
 
 

0 0 1 0 0 0
−1 0 0 1 0 0
0 0 0 0 0 0
0 0 −1 0 0 0
0 0 0 0 0 1
0 0 0 0 0 0]

 
 
 
 
 

,  

ad 𝑥3 =

[
 
 
 
 
 
0 −1 0 0 0 0
0 0 0 0 0 0
1 0 0 −1 0 0
0 1 0 0 0 0
0 0 0 0 0 0
0 0 0 0 1 0]

 
 
 
 
 

, ad 𝑥4 =

[
 
 
 
 
 
0 0 0 0 0 0
0 −1 0 0 0 0
0 0 1 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 1]

 
 
 
 
 

, 

ad 𝑥5 =

[
 
 
 
 
 

0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0

−1 0 0 0 0 0
0 0 −1 0 0 0]

 
 
 
 
 

, ad 𝑥6 =

[
 
 
 
 
 
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 −1 0 0 0 0
0 0 0 −1 0 0]

 
 
 
 
 

, 

ad 𝑦1 =  ad 𝑥1 − 𝑥4 =

[
 
 
 
 
 
0 0 0 0 0 0
0 2 0 0 0 0
0 0 −2 0 0 0
0 0 0 0 0 0
0 0 0 0 1 0
0 0 0 0 0 −1]

 
 
 
 
 

.  

(17) 

Furthermore, compute the radical of 𝔞𝔣𝔣(2) where 𝑥 = ∑ 𝛼𝑖𝑥𝑖
6
𝑖=1 ∈ 𝑅𝑎𝑑(𝔞𝔣𝔣(2)), then find 

value 𝛼𝑖 using equations (17), we have 

∑ 𝛼𝑖

6

𝑖=1

𝑇𝑟 (𝑎𝑑 𝑥𝑖 ⋅ 𝑎𝑑 𝑦𝑗) = 0 ; 1 < 𝑗 ≤ 5 

𝛼1

[
 
 
 
 
5
0
0
0
0]
 
 
 
 

+ 𝛼2

[
 
 
 
 
0
5
5
0
0]
 
 
 
 

+ 𝛼3

[
 
 
 
 
0
5
0
0
0]
 
 
 
 

+ 𝛼4

[
 
 
 
 
−5
0
0
0
0 ]

 
 
 
 

+ 𝛼5

[
 
 
 
 
0
0
0
0
0]
 
 
 
 

+ 𝛼6

[
 
 
 
 
0
0
0
0
0]
 
 
 
 

=

[
 
 
 
 
0
0
0
0
0]
 
 
 
 

. (18) 



Levi Decomposition of Frobenius Lie Algebra of Dimension 6 

Henti 399 

Next, we solve linear equations (18) such we find that 
𝛼1 − 𝛼4 = 0, 𝛼2 = 0, 𝛼3 = 0, 𝛼5 = 𝑠, 𝛼6 = 𝑡,  

and with 𝛼1 = 0, then we get 

𝑥 = ∑ 𝛼𝑖𝑥𝑖

6

𝑖=1

= 0𝑥1 + 0𝑥2 + 0𝑥3 + 0𝑥4 + 𝑠𝑥5 + 𝑡𝑥6 = 𝑠𝑥5 + 𝑡𝑥6. 

Therefore, we obtain the radical of 𝔞𝔣𝔣(2) is 
𝑅𝑎𝑑(𝔞𝔣𝔣(2)) = 𝑠𝑝𝑎𝑛{𝑥5, 𝑥6} = 𝑠𝑝𝑎𝑛{𝑟1, 𝑟2}.   (19) 

Next, we find basis Levi subalgebra of 𝔞𝔣𝔣(2). In this cases, 𝑅𝑎𝑑(𝔞𝔣𝔣(2)) is abelian 

because [𝑟𝑖, 𝑟𝑗] = 0 for all 1 ≤ 𝑖, 𝑗 ≤ 2. Complement on 𝔞𝔣𝔣(2) respect to 𝑅𝑎𝑑(𝔞𝔣𝔣(2)) 

spanned by {𝑥1, 𝑥2, 𝑥3, 𝑥4}. The quotient algebra 𝔞𝔣𝔣(2)/𝑅𝑎𝑑(𝔞𝔣𝔣(2)) is spanned by 

�̅�1, �̅�2, �̅�3, �̅�4 and we have its brackets as follows 
[�̅�1, �̅�2] = �̅�2, [�̅�1, �̅�3] = −�̅�3, [�̅�2, �̅�3] = �̅�1 − �̅�4, [�̅�2, �̅�4] = �̅�2, [�̅�3, �̅�4] = −�̅�3. (20) 

We set Levi subalgebra spanned by  
𝑧1 = 𝑥1 + 𝛼1𝑟1 + 𝛼2𝑟2, 
𝑧2 = 𝑥2 + 𝛽1𝑟1 + 𝛽2𝑟2, 
𝑧3 = 𝑥3 + 𝛾1𝑟1 + 𝛾2𝑟2, 
𝑧4 = 𝑥4 + 𝛿1𝑟1 + 𝛿2𝑟2.     (21) 

Next, we calculate to determine the four unknown 𝛼, 𝛽, 𝛾, 𝛿 such that 𝑧1, 𝑧2, 𝑧3, 𝑧4 span a 

semisimple Lie algebra that is isomorphic to 𝔞𝔣𝔣(2)/𝑅𝑎𝑑(𝔞𝔣𝔣(2)). Since 𝑧1, 𝑧2, 𝑧3, 𝑧4 have 

the same commutation relations as �̅�𝑖, 1 ≤ 𝑖 ≤ 4 written in equations (20), we then get  
[𝑧1, 𝑧2] = 𝑧2, [𝑧1, 𝑧3] = −𝑧3, [𝑧2, 𝑧3] = 𝑧1 − 𝑧4, [𝑧2, 𝑧4] = 𝑧2, [𝑧3, 𝑧4] = −𝑧3.  (22) 

We substitution the equation (22) onto (23) such that equation can be written as  
[𝑥1 + 𝛼1𝑟1 + 𝛼2𝑟2, 𝑥2 + 𝛽1𝑟1 + 𝛽2𝑟2] = 𝑥2 + 𝛽1𝑟1 + 𝛽2𝑟2,           (23) 
[𝑥1 + 𝛼1𝑟1 + 𝛼2𝑟2, 𝑥3 + 𝛾1𝑟1 + 𝛾2𝑟2] = −(𝑥3 + 𝛾1𝑟1 + 𝛾2𝑟2),          (24) 
[𝑥2 + ∑ 𝛽𝑗𝑟𝑗

2
𝑗=1 , 𝑥3 + ∑ 𝛾𝑗𝑟𝑗

2
𝑗=1 ] = (𝑥1 + ∑ 𝛼𝑗𝑟𝑗

2
𝑗=1 ) − (𝑥4 + ∑ 𝛿𝑗𝑟𝑗

2
𝑗=1 )(25) 

[𝑥2 + 𝛽1𝑟1 + 𝛽2𝑟2, 𝑥4 + 𝛿1𝑟1 + 𝛿2𝑟2] = 𝑥2 + 𝛽1𝑟1 + 𝛽2𝑟2,           (26) 
[𝑥3 + 𝛾1𝑟1 + 𝛾2𝑟2, 𝑥4 + 𝛿1𝑟1 + 𝛿2𝑟2] = −(𝑥3 + 𝛾1𝑟1 + 𝛾2𝑟2).          (27)  

Then, we apply the equations (23), (24), (25), (26), and (27) to compute 
𝛼𝑖, 𝛽𝑖, 𝛾𝑖, 𝛿𝑖, 1 ≤ 𝑖 ≤ 2. From equations (23) and (26) obtained that 𝛽1 = 𝛽2 = 1. From 
equations (24) and (27) obtained that 𝛾1 = 𝛾2 = 1. For equations (25), we obtained that 
𝛼1 − 𝛿1 = 𝛼2 − 𝛿2 = 0 and let 𝛼𝑖 = 0, such that we have 

𝑧1 = 𝑥1, 𝑧2 = 𝑥2 + 𝑟1 + 𝑟2, 𝑧3 = 𝑥3 + 𝑟1 + 𝑟2, 𝑧4 = 𝑥4. 
Thus, the Levi subalgebra spanned by 

{𝑧1 = (
1 0 0
0 0 0
0 0 0

) , 𝑧2 = (
0 1 1
0 0 1
0 0 0

) , 𝑧3 = (
0 0 1
1 0 1
0 0 0

) , 𝑧4 = (
0 0 0
0 1 0
0 0 0

)}.   (28) 

∎  

CONCLUSIONS 

It has been proven in Proposition 2 that the affine Lie algebra 𝔞𝔣𝔣(2) can be 
decomposed into its subalgebra and radicals which written in the equations (15). From 
our result of this paper, other research can study about decomposition of the general 
formula affine Lie algebra 𝔞𝔣𝔣(n) of dimension 𝑛(𝑛 + 1). For future research, the 
decomposition process can be expanded from the decomposition result of 𝔞𝔣𝔣(𝑛) in its 
radical and Levi subalgebra form such that we can find structure Frobenius Lie algebra 
𝔞𝔣𝔣(𝑛) of its decomposition. 



Levi Decomposition of Frobenius Lie Algebra of Dimension 6 

Henti 400 

REFERENCES 

[1] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory. Springer, 
1972. 

[2] M. A. Alvarez, M. C. Rodríguez-Vallarte, and G. Salgado, “Contact and Frobenius 
solvable Lie algebras with abelian nilradical,” Commun. Algebr., vol. 46, no. 10, pp. 
4344–4354, 2018, doi: 10.1080/00927872.2018.1439048. 

[3] P. Turkowski, “Low-dimensional real Lie algebras,” J. Math. Phys., vol. 29, no. 10, 
pp. 2139–2144, 1988, doi: 10.1063/1.528140. 

[4] M. Rais, “La représentation coadjointe du groupe affine,” vol. 4, pp. 913–937, 1978. 
[5] M. A. Alvarez, M. C. Rodríguez-Vallarte, and G. Salgado, “Contact nilpotent Lie 

algebras,” Proc. Am. Math. Soc., vol. 145, no. 4, pp. 1467–1474, 2017, doi: 
10.1090/proc/13341. 

[6] I. V. Mykytyuk, “Structure of the coadjoint orbits of Lie algebras,” J. Lie Theory, vol. 
22, no. 1, pp. 251–268, 2012. 

[7] A. Diatta and B. Manga, “On properties of principal elements of frobenius lie 
algebras,” J. Lie Theory, vol. 24, no. 3, pp. 849–864, 2018. 

[8] B. Csikós and L. Verhóczki, “Classification of frobenius Lie algebras of dimension ≤ 
6,” Publ. Math., vol. 70, no. 3–4, pp. 427–451, 2007. 

[9] E. Kurniadi, “DEKOMPOSISI DAN SIFAT MATRIKS STRUKTUR PADA ALJABAR LIE 
FROBENIUS BERDIMENSI 4,” no. 1, pp. 392–399, 2021. 

[10] Henti, E. Kurniadi, and E. Carnia, “On Frobenius functionals of the Lie algebra 
M3(ℝ) ⊕ gl3(ℝ),” J. Phys. Conf. Ser., vol. 1872, no. 1, 2021, doi: 10.1088/1742-
6596/1872/1/012015. 

[11] Henti, E. Kurniadi, and E. Carnia, “Quasi Associative ALgebras on the Frobenius Lie 
Algebra M3(ℝ) ⊕ gl3(ℝ),” Al-Jabar  J. Pendidik. Mat., vol. 12, no. 1, pp. 1–16, 2021, 
[Online]. Available: http://ejournal.radenintan.ac.id/index.php/al-
jabar/article/view/2014/1564. 

[12] E. Kurniadi, N. Gusriani, and B. Subartini, “A Left-Symmetric Structure on The 
Semi-Direct Sum Real Frobenius Lie Algebra of Dimension 8,” Cauchy, vol. 7, no. 2, 
pp. 267–280, 2022, doi: 10.18860/ca.v7i2.13462. 

[13] E. Kurniadi and H. Ishi, “Harmonic analysis for 4-dimensional real frobenius lie 
algebras,” Springer Proc. Math. Stat., vol. 290, no. March, pp. 95–109, 2019, doi: 
10.1007/978-3-030-26562-5_4. 

[14] Diatta, A., B. Manga, and A. Mbaye, “On Systems of Commuting Matrices, Frobenius 
Lie Algebras and Gerstenhaber’s Theorem,” arXiv:2002.08737., 2020. 

[15] D. N. Pham, “$\frak{g}$-quasi-Frobenius Lie algebras,” Arch. Math., vol. 52, pp. 
233–262, 2016, [Online]. Available: http://arxiv.org/abs/1701.01680. 

[16] M. Goze and E. Remm, “Contact and Frobenius forms on Lie groups,” Differ. Geom. 
its Appl., vol. 35, pp. 74–94, 2014, doi: 10.1016/j.difgeo.2014.05.008. 

[17] T. Barajas, E. Roque, and G. Salgado, “Principal derivations and codimension one 
ideals in contact and Frobenius Lie algebras,” Commun. Algebr., vol. 47, no. 12, pp. 
5380–5391, 2019, doi: 10.1080/00927872.2019.1623238. 

[18] M. Dagli, “Levi Decomposition of Lie Algebras; Algorithms for its Computation,” 
Iowa State University, 2004. 

[19] J. Hilgert and K.-H. Neeb, Structure and Geometry of Lie Groups. New York: Springer 
Monographs in Mathematics, 2012. 

[20] A. I. Ooms, On frobenius lie algebras, vol. 8, no. 1. 1980.