A Left-Symmetric Structure on the Semi-Direct Sum Real Frobenius Lie  Algebra of Dimension 8


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 7(2) (2022), Pages 267-280 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: September 30, 2021 Reviewed: January 22, 2022 Accepted: January 26, 2021 
DOI: http://dx.doi.org/10.18860/ca.v7i1.13462       

A Left-Symmetric Structure on the Semi-Direct Sum Real 
Frobenius Lie  Algebra of Dimension 8 

 

Edi Kurniadi*, Nurul Gusriani , Betty Subartini   

Department of Mathematics of FMIPA of Universitas Padjadjaran, Indonesia  
 

*Corresponding Author 
Email: edi.kurniadi@unpad.ac.id*,  

nurul.gusriani@unpad.ac.id, betty.subartini@unpad.ac.id   

ABSTRACT  

This paper relates quasi-associative algebras or Koszul algebras of matrix Lie algebras of finite 
dimension to finite dimensional Frobenius Lie algebras which is written as a semi direct sum.  
Let M2(ℝ)⋊𝔤𝔩2(ℝ) be the Lie algebra of  the semi-direct sum of the real vector space  M2(ℝ) 
and the Lie algebra 𝔤𝔩2(ℝ) of the sets of all 2×2 real matrices. The research aims to give explicit 
formulas of left-symmetric algebra structures on M2(ℝ)⋊𝔤𝔩2(ℝ).  In this paper, a Frobenius 
functional is constructed in order to show that the Lie algebra M2(ℝ)⋊𝔤𝔩2(ℝ) is the real 
Frobenius Lie algebra of dimension 8. Moreover,  a bilinear form corresponding to this 
Frobenius functional is symplectic. Then the obtained symplectic bilinear form induces the left-
symmetric algebra structures on M2(ℝ)⋊𝔤𝔩2(ℝ). In other words, we show that  the Lie algebra  
M2(ℝ)⋊𝔤𝔩2(ℝ) is the left-symmetric algebra. Thus, we give the formulas of its left-symmetric 
algebra structure explicitely. The left-symmetric algebra structures for case of higher dimension 
of this Lie algebra type are still an open problem to be investigated.  
 

Keywords: Left-symmetric algebra; Frobenius Lie algebra;  Frobenius functional; Semi-direct  
sum;  Symplectic form. 

 

INTRODUCTION  

Let ℝ be the field of real numbers of characteristic zero and M2(ℝ) be the ℝ-vector 
space  of all 2×2 real matrices with entries contained in ℝ. The space M2(ℝ) is  
considered as the ℝ-algebra structure given by usual addition and multiplication of 
matrices. In addition, The space M2(ℝ) is the Lie algebra equipped by the Lie brackets in 
the form of the matrix commutator [𝑥,𝑦]:= 𝑥𝑦 −𝑦𝑥 for all matrices 𝑥,𝑦 ∈ M2(ℝ). We 
denote by 𝔤𝔩2(ℝ) the Lie algebra for M2(ℝ) with respect to the mentioned Lie bracket 
above. Moreover, we shall let 𝔥2 ≔ M2(ℝ)⋊𝔤𝔩2(ℝ) represent for the semi-direct sum 
Lie algebra of the vector space M2(ℝ) and the Lie algebra  𝔤𝔩2(ℝ) of dimension 8. 
Furthermore, the Lie algebra 𝔥2 has the following matrix realization: 
 

http://dx.doi.org/10.18860/ca.v7i1.13462
mailto:edi.kurniadi@unpad.ac.id*
mailto:nurul.gusriani@unpad.ac.id
mailto:betty.subartini@unpad.ac.id


A Left-Symmetric Structure on the Semi-Direct Sum Real Frobenius Lie  Algebra of Dimension 8 

Edi Kurniadi 268 

𝜎 ≔ 𝜎(𝑈,𝑋) = (
𝑋 𝑈
0 0

) ∈ 𝔥2 ⊂ M4(ℝ)                                                     (1) 

 
with 𝑈 ∈ M2(ℝ) and  𝑋 ∈ 𝔤𝔩2(ℝ). Let 𝔥2

∗ be a dual vector space of 𝔥2. We also realize in 
the following matrix  

𝜎∗ ≔ 𝜎∗(𝛼,𝛽) = (
𝛽 0
𝛼 0

) ∈ 𝔥2
∗ ⊂ M4(ℝ)                                                   (2) 

 
with 𝛼 ∈ M2(ℝ) and  𝛽 ∈ 𝔤𝔩2(ℝ). A value of a linear functional  𝜎

∗ ∈ 𝔥2
∗ on a matrix 𝜎 ∈

𝔥2 is denoted by 
 

〈𝜎∗,𝜎〉 = tr(𝑈𝛼)+tr(𝑋𝛽) = 〈𝛼,𝑈〉+〈𝛽,𝑋〉                                            (3) 
 

where  tr is denoted the matrix trace. The notion of the Lie algebra 𝔥2 comes originally 
from the Lie algebra 𝔥𝑛,𝑝 ≔ M𝑛,𝑝(𝕂)⋊𝔤𝔩𝑛(𝕂) where M𝑛,𝑝(K) is the real vector space of 

the set 𝑛 ×𝑝 matrices and 𝔤𝔩𝑛(𝕂) is the Lie algebra of the set 𝑛 ×𝑛 matrices over a field 
𝕂 of characteristic 0. This Lie algebra was introduced by [1] in the context of coadjoint 
representations of the affine Lie group G𝑛,𝑝 ≔ M𝑛,𝑝(𝕂)⋊GL𝑛(𝕂) where GL𝑛(𝕂) is the 

Lie group of all invertible matrices of the Lie algebra 𝔤𝔩𝑛(𝕂). In our case, we take 𝑛 =
𝑝 = 2 and 𝕂 = ℝ. We denote by M2(ℝ) ≔ M2,2(ℝ) and we determine the Lie algebra 𝔥2 

of the Lie group G2. 
In addition, the case 𝑝 = 1 is considered as the affine Lie algebra, denoted by 𝔞𝔣𝔣(𝑛), 

of dimension 𝑛(𝑛 +1). More precise, the realization of this affine Lie algebra 𝔞𝔣𝔣(𝑛)  is  of 
the form M𝑛,1(ℝ)⋊𝔤𝔩𝑛(ℝ) where M𝑛,1(ℝ) is isomorphic to the space ℝ

𝑛. Interestingly, 

the Lie algebra 𝔞𝔣𝔣(𝑛) is Frobenius since it has a trivial stabilizer on a certain linear 
functional. In the other words, the Lie group Aff(𝑛) of 𝔞𝔣𝔣(𝑛) has an open coadjoint orbit 
[2]. Moreover, the Lie algebra 𝔞𝔣𝔣(𝑛) is non-solvable. Regarding the notion of a Frobenius 
Lie algebra, this Lie algebra type was introduced by Ooms  in the context to answer 
Professor Jacobson’s question on properties of finite dimensional Lie algebras having an 
exact simple module over the universal enveloping algebras([3],[4]). The answer that 
the universal enveloping algebra has an exact simple module if its finite dimensional Lie 
algebra is Frobenius. Furthermore, if a Lie algebra  𝔤 is Frobenius then there exists a 
linear functional 𝜓 defined on 𝔤 such that a bilinear alternating form is non-degenerate. 
Namely, it is a symplectic linear form and such the linear functional 𝜓 is called a 
Frobenius functional [5].  

The notion of the Frobenius functional gives some important implications. Among 
them, firstly, since the alternating bilinear form corresponding to the Frobenius 
functional is the symplectic linear form, then the dimension of Frobenius Lie algebra is 
always even. Secondly, the stabilizer of Frobenius Lie algebras corresponding to the 
Frobenius functional is trivial, then we have a fact that the Frobenius Lie algebra is not 
nilpotent [6].  Appearing in many different areas of Lie algebras, for instance, in the 
study of bounded homogeneous domain, simple hypersurface singularities, and 
solutions of Yang-Baxter equation [7], Frobenius Lie algebras bring great significance in 
many areas of Lie algebras.  
In ([5],[8]), a notion of a left-symmetric algebra structure on a Lie algebra was 
introduced where the bilinear product was defined. It was proved that an 𝑛-dimensional 
Lie algebra  𝔤 has left-symmetric algebra structures if there exists a 𝔤-module 𝐾 of 
dimension 𝑛 such that the 1-cocyle space contains a bijective 1-cocyle. Indeed, not every 
type of Lie algebras has affine structure. These structures are important because it arise 



A Left-Symmetric Structure on the Semi-Direct Sum Real Frobenius Lie  Algebra of Dimension 8 

Edi Kurniadi 269 

in many areas of mathematics such as convex homogeneous cones, affine manifolds, and 
vertex algebras. Associating to the Frobenius functional, the left-symmetric algebras can 
be induced by the symplectic structure corresponding to a Frobenius functional [9]. In 
the present work, we are interested in studying the Frobenius Lie algebra and left-
symmetric algebra for the Lie algebra 𝔥2 of the Lie group G2. 

In this paper, we calculate a Frobenius functional of 𝔥2 corresponding to its Pfaffian 
and we prove that 𝔥2 is 8-dimensional Frobenius Lie algebra. The obtained Frobenius 
functional of 𝔥2 is applied to contruct a symplectic structure on 𝔥2 which induces a left-
symmetric algebra structure on 𝔥2. The research aims to give explicit formulas of left-
symmetric algebra structures on M2(ℝ)⋊𝔤𝔩2(ℝ).  Furthermore, with respect to a basis 
𝔥2, we calculte the left-symmetric algebra structures explicitely. Let 𝜀𝑖𝑗, 1 ≤ 𝑖,𝑗 ≤ 4 be 

element of 𝔥2 ⊂ M4(ℝ) of the form in the equation  (1).  
 
We organize the paper as follows. Section 1 is explained the background of this 

research, state of art, the aim of research, statement of the main results, and some basic 
notions . Section 2 is devoted to research method. Section 3 discusses our main results 
to prove that 𝔥2 ≔ M2(ℝ)⋊𝔤𝔩2(ℝ) is 8-dimensional Frobenius Lie algebra and 𝔥2 has 
left-symmectric structures. In this section, we also give explicit formulas for left-
symmetric structures of 𝔥2 and discussion for future research related to our main 
results. In the end of this paper,  the conclusion is given.  

 
 

METHODS  

Our main object was considered from the notion of the affine Lie group G𝑛,𝑝 ≔

M𝑛,𝑝(𝕂)⋊GL𝑛(𝕂) which corresponds to the notion of coadjoint representations. 

Regarding this affine Lie group, particularly, we consider the special case for 𝑛 = 𝑝 = 2 
and we obtained the Lie algebra 𝔥2 ≔ M2(ℝ)⋊𝔤𝔩2(ℝ) of the Lie group G2 ≔ M2(ℝ)⋊
GL2(ℝ).   

We studied some related papers corresponding to a Frobenius Lie algebra and we 
offered another alternative to show whether it was a Frobenius Lie algebra or not 
coresponding to its Pfaffian.  Particularly on some Oom’s work [4]. On the other hand, 
Burde introduced the notation of left-symmetric algebra structures on a Lie group and a 
Lie algebra [8]. We proved the existence of left-symmetric algebra structures on  𝔥2 and 
we listed its structures explicitely. Our constructions based on symplectic form which 
induces its left-symmetric algebra structures.  

In the next section, we shall briefly review some basic notions of Lie algebras, 
Frobenius Lie algebras and their properties, Pfaffian of a Frobenius Lie algebras, and 
left-symmetric structures on a Lie algebra. In Section Results and Discussion, we shall 
complete the proof of our main results.  

 
 
 

1.1 Frobenius Lie algebra 
 

We shall first introduce the notion of a Lie algebra as follows: 

Definition 1[10]. Let 𝔤 be a vector space. A Lie bracket on 𝔤 is a bilinear form on 𝔤×𝔤, 
usually denoted by [ , ], which satisfies: 
1. [𝑥,𝑥] = 0  for 𝑥 ∈ 𝔤, 



A Left-Symmetric Structure on the Semi-Direct Sum Real Frobenius Lie  Algebra of Dimension 8 

Edi Kurniadi 270 

2. Jacobi identity, that is 

[𝑥,[𝑦,𝑧]] = [[𝑧,𝑥],𝑦]+[[𝑥,𝑦],𝑧].                                                                 (4) 

 
A vector space 𝔤 together with a Lie bracket [ , ] is called a Lie algebra.  
 
Let 𝔤 be a Lie algebra  with 𝔤∗ is dual vector space consisting of linear functionals on 𝔤. 
We denote by 𝔤𝔩(𝔤) the Lie algebra of endomorphism of 𝔤 to itself. Then a representation 
of the Lie algebra 𝔤 on himself is given by the following map 
 

ad ∶  𝔤 ∋ 𝑥 ↦ ad(𝑥) ∈ 𝔤𝔩(𝔤).                                                                            (5) 
 
Furthermore, the corresponding representation of the Lie algebra 𝔤 on the space 𝔤∗ is 
considered as the dual map to ad and is written in the following formula 
 

〈ad∗(𝑥)𝜎∗,𝑦〉 = 〈𝜎∗,−ad(𝑥)𝑦〉 = 〈𝜎∗,−[𝑥,𝑦]〉                                           (6) 
 

where 𝑥,𝑦 ∈ 𝔤and 𝜎∗ ∈ 𝔤∗.  
Let 𝜎∗ be an element of 𝔤∗ and 𝐵𝜎∗ be alternating bilinear form corresponding to 𝜎

∗ 
which is defined by 

𝐵𝜎∗ ∶ 𝔤×𝔤 ∋ (𝑥,𝑦) ↦ 〈𝜎
∗, [𝑥,𝑦]〉 ≔ 𝐵𝜎∗(𝑥,𝑦) ∈ ℝ.                                  (7) 

The kernel of 𝐵𝜎∗ is given by the following formula 
 

Ker(𝐵𝜎∗) = {(𝑥,𝑦) ∈ 𝔤×𝔤   ;  〈𝜎
∗, [𝑥,𝑦]〉 = 0 } 

= {𝑥 ∈ 𝔤   ;   〈𝜎∗, [𝑥,𝑦]〉 = 0,∀ 𝑦 ∈ 𝔤 } 
= {𝑥 ∈ 𝔤   ;   〈ad∗(𝑥)𝜎∗,𝑦〉 = 0,∀ 𝑦 ∈ 𝔤 } 
= {𝑥 ∈ 𝔤   ;   ad∗(𝑥)𝜎∗ = 0} 
 

The latter formula is nothing but a stabilizer of 𝔤 correponding to 𝜎∗ which is denoted by 

𝔤𝜎
∗
.  

 
Definition 2[11]. Let 𝔤 be a Lie algebra. The Lie algeba 𝔤 is said to be Frobenius if there 
exists a linear functional 𝜎∗  ∈ 𝔤∗ such that the alternating bilinear form  in the equation 
(7) is non-degenerate. Such 𝜎∗ satisfying the eqution (7) is called a Frobenius functional.  
 

Indeed, the non-commutative Lie algebra of dimension 2 is always Frobenius. It is 
well known that affine Lie algebra 𝔞𝔣𝔣(1) ≔ ℝ⋊ℝ is 2-dimensional Frobenius Lie 
algebras. Those Frobenius Lie algebras form a large class of Lie algebras. For example, 
some parabolic and seaweed subalgebras of semi-simple Lie algebras, 𝑗-algebras, and 
Borel subalgebras of simple Lie algebras are Frobenius Lie algebras [7].  

Now, let 𝑇 ≔ {𝑥1,𝑥2,…,𝑥𝑛} be a basis for a Frobenius Lie algebra 𝔤 of dimension 𝑛 
with 𝑛 is even. For 𝑥𝑗,𝑥𝑘 ∈ 𝑇, we denote by 𝑀𝔤(ℝ) an 𝑛 ×𝑛 matrix whose entries are 

defined by 𝑀𝔤(ℝ) ≔ ([𝑥𝑗,𝑥𝑘])𝑗,𝑘=1
𝑛 . Moreover, for 𝜎∗ ∈ 𝔤∗ we define a matrix  𝑀𝔤(ℝ)(𝜎

∗) 

stand for an 𝑛 ×𝑛 matrix whose entries are defined by   
 

𝑀𝔤(ℝ)(𝜎
∗) ≔ (〈𝜎∗, [𝑥𝑗,𝑥𝑘]〉)𝑗,𝑘=1

𝑛 .                                                           (8) 

 
We get the following theorem :   



A Left-Symmetric Structure on the Semi-Direct Sum Real Frobenius Lie  Algebra of Dimension 8 

Edi Kurniadi 271 

Theorem 1[4]. Let 𝔤 be a Lie algebra of even dimensional with a basis 𝑇 ≔ {𝑥1,𝑥2,…,𝑥𝑛}. 
The Lie algebra 𝔤 is Frobenius if one of the following equivalent  conditions is satisfied: 
1. The stabilizer 𝔤𝜎

∗
= {0} for some 𝜎∗ ∈ 𝔤∗.  

2. The determinant of the matrix 𝑀𝔤(ℝ) is not equal to zero. 

3. The determinant of the matrix  𝑀𝔤(ℝ)(𝜎
∗) is not equal to zero  for some Frobenius 

functionals  𝜎∗ ∈ 𝔤∗.  

The relation of alternating bilinear form and a linear functional on 𝔤 in the term of a 
Frobenius Lie algebra can be explained as follows:  
In general, let 𝜓 be alternating bilinear form in a Lie algebra 𝔤. If 𝜓 is non-degenerate 
closed 2-form then 𝔤 is said to be a quasi-Frobenius Lie algebra. Furthermore, if there 
exists a linear functional 𝜎∗ ∈ 𝔤∗ such that 

 
𝜓(𝛼,𝛽) = d𝜎∗(𝛼,𝛽) = −〈𝜎∗, [𝛼,𝛽]〉   (𝛼,𝛽 ∈ 𝔤)                                  (9) 

 
with d𝜎∗ denotes the differential of 𝜎∗, then 𝔤 is also called a Frobenius Lie algebra [5].  
Indeed, this statement is equivalent to definition of the Frobenius Lie algebra. 

 
Example 1. The nice examples of low dimensional Frobenus Lie algebras are 4-

dimensional Frobenius Lie algebras over a field with characteristic ≠ 2 and 6-

dimensional Frobenius Lie algebras over an algebraically closed field classified in [6]. In 

the other hand, the (2𝑛 +1)-dimensional heisenberg Lie algebra is not Frobenius Lie 

algebra.  

 

Since we shall relate a Frobenius functional to Pfaffian, we need to recall the notion 

of Pfaffian for the square matrix 𝐴 ≔ (𝐴𝑗𝑘)𝑗,𝑘=1
2𝑛  where 𝐴 is an alternating matrix [12]. 

The formula of  Pfaffian is given by  

 

Pf(𝐴) ≔
1

2𝑛𝑛!
∑ sgn(𝜏)∏ 𝐴𝜏(2𝑖−1)𝜏(2𝑖)

𝑛
𝑘=1𝜏∈𝑆(2𝑛) .                              (10) 

 

The Pfaffian for the Lie algebra 𝔤 is defined as the Pfaffian of matrix 𝑀𝔤(ℝ).  

 

Remark 1 [12]. Let 𝐴 be an even dimensional  square-alternating matrix. Then det(𝐴) =

Pf(𝐴)2.   

 

 

Example 2. Let 𝔞𝔣𝔣(1) be 2-dimensional affine Lie algebra with non-zero bracket is 

[𝑎,𝑏] = 𝑏. We obtain Pf(𝔞𝔣𝔣(1)) = 𝑏. Moreover, the 4-dimensional Frobenius Lie algebra 

Σ with non-zero brackets in the following formulas [6] 

 

[𝑑,𝑎] = 𝑎,            [𝑐,𝑏] = 𝑎 

[𝑑,𝑏] =
1

2
𝑏, [𝑑,𝑐] =

1

2
𝑐 

 

has Pfaffian of the form Pf(Σ) = 𝑎2 [13].  



A Left-Symmetric Structure on the Semi-Direct Sum Real Frobenius Lie  Algebra of Dimension 8 

Edi Kurniadi 272 

The notion of Pfaffian is very important in representation theory of Lie groups, for 

instance, in notion of square-integrable representation, the Duflo-Moore operator can be 

associated to Pfaffian [13]. In this paper, we shall relate Pfaffian  to a Frobenius 

functional contruction.  

 

 1.2 Left-Symmetric Algebra Structure 
 
The notion of left-symmetric algebra first have been arisen in the theory of Lie groups 𝐺 
endowed a left-invariant affine structure [14]. Let 𝐴,𝐵, and 𝐶 left-invariant vector fields 
of 𝐺 that are the section of the map 𝜓 ∶ 𝑇𝐺 → 𝐺 with 𝑇𝐺 is a tangent bundle of 𝐺. Let ∇ 
be a connection in 𝑇𝐺 with both curvature and torsion are zero, namely : 
 

[𝐴,𝐵] = ∇A𝐵 −∇𝐵𝐴, 

∇[A,B]𝐶 = ∇𝑋∇𝑌𝑍 −∇𝑌∇𝑋𝑍.                                                                     (11) 

Let 𝑎,𝑏, and 𝑐 be elements of a Lie algebra 𝔤 of 𝐺. Firstly, we define 𝑎𝑏 ≔ 𝑎 ∗𝑏 = ∇A𝐵. 
Let ∆(𝑎,𝑏,𝑐) be associator for  𝑎,𝑏, and 𝑐 given by 
 

∆(𝑎,𝑏,𝑐) ≔ (𝑎𝑏)𝑐 −𝑎(𝑏𝑐).                                                                   (12) 
 
In addition, a left-symmectric or a non-associative algebra structure on the Lie algebra  𝔤 
is given by the following formula 
 

∆(𝑎,𝑏,𝑐) = ∆(𝑏,𝑎,𝑐).                                                                                (13) 
 

We start with some basic definitions of left-symmetric algebra structure which shall 
be needed for the next section.  

 
Definition 3[8]. A non-associative algebra 𝐿 is called a left-symmetric algebra (LSA) if a 
bilinear product defined by 𝐿 ×𝐿 ∋ (𝑎,𝑏) ↦ 𝑎𝑏 ≔ 𝑎 ∗𝑏 ∈ 𝐿 satisfies the equation (15).  
 
A given Lie bracket by the following commutator 
 

[𝑎,𝑏] = 𝑎𝑏 −𝑏𝑎                                                                                     (14) 
 

for all 𝑎,𝑏,𝑐 ∈ 𝐿  defines a Lie algebra 𝔤 of 𝐿.  In other words, the algebra 𝐿 is a Lie 
algebra denoted by  𝔤 = 𝔤(𝐿) with respect to the Lie bracket in the equation (14). We 
call a left-symmetric algebra by a Lie admissible Lie algebra or Vinberg algebra [8].  
 
Example 3[5]. Let 𝔞𝔣𝔣(1) be a 2-dimensional affine lie algebra with basis 𝐵 ≔ {𝑎,𝑏} and 
non-zero brackets [𝑎,𝑏] = 𝑏. The left-symmetric algebra structures on 𝔞𝔣𝔣(1) are given 
by  
 

𝑎2 = −𝑎, 𝑎𝑏 = 0, 𝑏𝑎 = −𝑏,  𝑏2 = 0.                                                  (15) 
 

Indeed, 𝔞𝔣𝔣(1)  is LSA.  Furthermore, we can see that the equation (15) satisfies the 
equation (14). 
  



A Left-Symmetric Structure on the Semi-Direct Sum Real Frobenius Lie  Algebra of Dimension 8 

Edi Kurniadi 273 

Remark 2[8]. A Lie algebra 𝔤 has structure affines if it satisfies the equations (13) and 
(14).  
 
We mention here that not every Lie algebra has left-symmetric algebra structure. For 
examples, Filiform Lie algebras 𝔤 of dimension 10 ≤ dim𝔤 ≤ 13 do not have left-
symmetric algebra structure ( [8], [14] ).  

Let 𝔤 be a Frobenius Lie algebra whose Frobenius functional is 𝜎∗. We define an 
alternating bilinear form 𝐵𝜎∗ as in the equation (7) whose a representation matrix is in 
the equation (8). For all 𝑎,𝑏,𝑐 ∈ 𝔤, using a product  𝑎𝑏 ≔ 𝑎 ∗𝑏, then we have that the 
symplectic form   𝐵𝜎∗ induces a left-symmetric algebra structure( [5], [9] ) defined  by  

 
𝐵𝜎∗(𝑎𝑏,𝑐):= −𝐵𝜎∗(𝑏,[𝑎,𝑐]) = −〈𝜎

∗, [𝑏, [𝑎,𝑐]]〉.                                                            (16) 

 
The non-degeneracy of 𝐵𝜎∗ and the Jacobi identity in the equation (4) guarantee that the 
equation (16) satisfies the equations (13) and (14). In other words, we find that the 
symplectic form 𝐵𝜎∗ induces a left-symplectic algebra structure. We also observe that 
the determinant of a representation matrix of 𝐵𝜎∗ is not equal to zero since 𝔤 is the  
Frobenius Lie algebra. Furthermore, we shall apply the equation (16) to find left-
symmetric algebra structure explicitely.  
 

RESULTS AND DISCUSSION  

In this section, we shall prove our main result as  we summarize as follows :  
 

Proposition 1. Let 𝑆 = {𝜀11,𝜀12,𝜀21,𝜀22,𝜀13,𝜀14, 𝜀23, 𝜀24}  be a basis for the Lie algebra 𝔥2. 
Then 𝔥2 is 8-dimensional Frobenius Lie algebra whose the Frobenius functional 𝜎0

∗ ≔ 𝜀14
∗ +

𝜀23
∗ ∈ 𝔥2

∗ is considered  with respect to the  Pfaffian of 𝔥2, denoted by Pf(𝔥2). Its Pfaffian 
Pf(𝔥2) is written in the following form : 
 

Pf(𝔥2) = (𝜀13𝜀24 −𝜀14𝜀23)
2 ∈ S(𝔥2),                                                     (17) 

 

where S(𝔥2) is a symmetric algebra of degree 4.  Furthermore, the Lie algebra 𝔥2 is left-

symmetric algebra whose formulas are in the following products  

  

𝜀11
2       = −𝜀11 𝜀11𝜀12 = 0 𝜀11𝜀21 = −𝜀21 

𝜀11𝜀22 = 0 𝜀11𝜀13 = 𝜀13 𝜀11𝜀14 = 0 

𝜀11𝜀23 = 0 𝜀11𝜀24 = −𝜀24 𝜀12𝜀21 = −𝜀22 

𝜀12𝜀11 = −𝜀12 𝜀12
2       = 0 𝜀12𝜀14 = −𝜀13 

𝜀12𝜀22 = 0 𝜀12𝜀13 = 0 𝜀21
2       = 0 

𝜀12𝜀23 = 𝜀13 𝜀12𝜀24 = 𝜀14 −𝜀23 𝜀21𝜀24 = 0 

𝜀21𝜀11 = 0 𝜀21𝜀12 = −𝜀11 𝜀22𝜀21 = 0 



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Edi Kurniadi 274 

𝜀21𝜀22 = −𝜀21 𝜀21𝜀13 = 𝜀23 −𝜀14 𝜀22𝜀14 = 0 

𝜀21𝜀14 = 𝜀24 𝜀21𝜀23 = −𝜀24 𝜀21𝜀24 = 0 

𝜀22𝜀11 = 0 𝜀22𝜀12 = −𝜀12 𝜀22𝜀21 = 0 

𝜀22
2       = −𝜀22 𝜀22𝜀13 = −𝜀13 𝜀13𝜀21 = −𝜀14 

𝜀22𝜀23 = 0 𝜀22𝜀24 = 𝜀24 𝜀13𝜀14 = 0 

𝜀13𝜀11 = 0 𝜀13𝜀12 = 0 𝜀14𝜀21 = 0 

𝜀13𝜀22 = −𝜀13 𝜀13
2       = 0 𝜀14

2       = 0 

𝜀13𝜀23 = 0 𝜀13𝜀24 = 0 𝜀23𝜀21 = −𝜀24 

𝜀14𝜀11 = −𝜀14 𝜀14𝜀12 = 0 𝜀23𝜀14 = 0 

𝜀14𝜀22 = 0 𝜀14𝜀13 = 0 𝜀24𝜀21 = 0 

𝜀14𝜀23 = 0 𝜀14𝜀24 = 0 𝜀24𝜀14 = 0 

𝜀23𝜀11 = 0 𝜀23𝜀12 = 0  

𝜀23𝜀22 = −𝜀23 𝜀23𝜀13 = 0  

𝜀23
2       = 0 𝜀23𝜀24 = 0  

𝜀24𝜀11 = −𝜀24 𝜀24𝜀12 = −𝜀23  

𝜀24𝜀22 = 0  𝜀24𝜀13 = 0  

𝜀24𝜀23 = 0 𝜀24
2      = 0                

where the products on 𝔥2  is defined by 
 

𝔥2  ×𝔥2  ∋ (𝜀𝑖𝑗,𝜀𝑘𝑙) ↦ 𝜀𝑖𝑗𝜀𝑘𝑙 ≔ 𝜀𝑖𝑗 ∗𝜀𝑘𝑙 ∈ 𝔥2 .                     (18) 

 
Proof.  Let 𝜀𝑗𝑘  be elements of 𝔥2 ⊂ M4(ℝ)  with 1 ≤ 𝑗,𝑘 ≤ 4 corresponding to the form 

in the equation (1). We let the set  𝑆 = {𝜀11,𝜀12,𝜀21,𝜀22, 𝜀13, 𝜀14,𝜀23,𝜀24} be a basis for 𝔥2.  
The Lie algebra 𝔥2 is endowed with the Lie brackets written as the matrix commutator 
with respect to the basis  𝑆 as follows: 
 

[𝜀𝑗𝑘,𝜀𝑖𝑙] = 𝜀𝑗𝑘𝜀𝑖𝑙 −𝜀𝑖𝑙𝜀𝑗𝑘                                         (19) 

 
where 1 ≤ 𝑗,𝑘, 𝑖, 𝑙 ≤ 4. In addition, we give the non-zero brackets for 𝔥2 in the following 
formulas :  
 
 
[𝜀12,𝜀21] = 𝜀11 −𝜀22 [𝜀11,𝜀21] = −𝜀21, 



A Left-Symmetric Structure on the Semi-Direct Sum Real Frobenius Lie  Algebra of Dimension 8 

Edi Kurniadi 275 

[𝜀11,𝜀12] = 𝜀12 [𝜀21,𝜀22] = −𝜀21,     

[𝜀12,𝜀22] = 𝜀12 [𝜀11,𝜀13] = 𝜀13, 

[𝜀11,𝜀14] = 𝜀14 [𝜀12,𝜀23] = 𝜀13, 

[𝜀12,𝜀24] = 𝜀14 [𝜀21,𝜀13] = 𝜀23,  

[𝜀21,𝜀14] = 𝜀24 [𝜀22,𝜀23] = 𝜀23, 

[𝜀22,𝜀24] = 𝜀24.              (20) 

 
Firstly, in order to show that 𝔥2 is the Frobenius Lie algebra, we just compute the 

Pfaffian of the matrix 𝑀𝔥2(ℝ) ≔ ([𝜀𝑘𝑗,𝜀𝑖𝑙])𝑗,𝑘,𝑖,𝑙=1
4  defined before with respect to the 

basis 𝑆. Since we have that  the determinant for 𝑀𝔥2(ℝ) in the following form 

 
det𝑀𝔥2 (ℝ) = (𝜀13𝜀24 −𝜀14𝜀23)

4,                                                        (21) 

 
is not equal to zero, then 𝔥2 is the Frobenius Lie algebra as desired. In the next step, 
using the Theorem 5, we can also see that  𝔥2 is the Frobenius Lie algebra by 
constructing a Frobenius functional such that a stabilizer of  𝔥2 ata that point is trivial. 
Secondly, to prove that 𝔥2 is left-symmetric algebra, we should find a Frobenius 
functional corresponding the Pfaffian for 𝔥2. We observe, using Remark 7, we have that 
the Pfaffian of 𝔥2  can be written of the form  
 

Pf(𝔥2 ) = (𝜀13𝜀24 −𝜀14𝜀23)
2,                                                                   (22) 

 
which is contained in the symmetric algebra 𝑆(𝔥2) of degree 4. Since 𝔥2 is Frobenius Lie 
algebra, the existence of some Frobenius functionals are guaranteed. From the equation 
(22), we first claim that a Frobenius functional is 𝜎𝑚𝑛

∗ ≔ 𝜀14
∗ +𝜀23

∗  which is contained in 
the dual space 𝔥2

∗  of vector space 𝔥2. We recalll the value of a linear functional 𝜎𝑚𝑛
∗  on 𝔥2 

is defined by 〈𝜎𝑚𝑛
∗ , 𝜀𝑗𝑘〉 = 1 for 𝑚 = 𝑗,𝑛 = 𝑘 and 0 otherwise. Let 𝑀𝔥2(ℝ)(𝜎

∗) be the 

matrix defined in the equation (10). Namely,  we have   the 8×8 matrix 𝑀𝔥2(ℝ)(𝜎
∗) ≔

(〈𝜎𝑚𝑛
∗ , [𝜀𝑘𝑗,𝜀𝑖𝑙]〉)𝑗,𝑘,𝑖,𝑙=1

4  which can be seen in the following form 

 

𝑀𝔥2(ℝ)(𝜎𝑚𝑛
∗ ) =

(

 
 
 
 
 

0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0
0 0 −1 0 0 0 0 0
−1 0 0 0 0 0 0 0
0 0 0 −1 0 0 0 0
0 −1 0 0 0 0 0 0)

 
 
 
 
 

.                                      (23)       

 

Since the determinant of matrix 𝑀𝔥2(ℝ)(𝜎𝑚𝑛
∗ ) is equal to 1, then 𝔥2 is again the 

Frobenius Lie algebra. In other words, we have shown that our claim is true. Thus, 𝜎𝑚𝑛
∗  is 

the Frobenius functional. Indeed, one can also check that the stabilizer of 𝔥2 at this 
Frobenius functional 𝜎𝑚𝑛

∗   is trivial and this again  show that 𝜎𝑚𝑛
∗  is the Frobenius 

functional.  



A Left-Symmetric Structure on the Semi-Direct Sum Real Frobenius Lie  Algebra of Dimension 8 

Edi Kurniadi 276 

Corresponding to the Frobenius functional 𝜎𝑚𝑛
∗ , we construct the symplectic linear 

form 𝐵𝜎𝑚𝑛∗  as defined in the equation (7). We shall show that 𝐵𝜎𝑚𝑛∗  induces left-

symmetric algebra structure in the equations (13) and (14) with respect to the the 
product 𝔥2 ×𝔥2 ∋ (𝑎,𝑏) ↦ 𝑎𝑏 ≔ 𝑎 ∗𝑏 ∈ 𝔥2. Let 𝑎,𝑏,𝑐, and 𝛼 be elements of 𝔥2. Firstly, 
we shall show that the equation (13). Namely, we shall show that  

 
(𝑎𝑏)𝑐 −(𝑏𝑎)𝑐 = 𝑎(𝑏𝑐)−𝑏(𝑎𝑐).                                                   (24) 

Let us observe that  

𝐵𝜎𝑚𝑛∗ ((𝑏𝑎)𝑐 −(𝑎𝑏)𝑐,𝛼) = 𝐵𝜎𝑚𝑛∗ ((𝑏𝑎)𝑐,𝛼)−𝐵𝜎𝑚𝑛∗ ((𝑎𝑏)𝑐,𝛼),   

= 𝐵𝜎𝑚𝑛∗  (𝑐, [𝑎𝑏,𝛼])−𝐵𝜎𝑚𝑛∗ (𝑐,[𝑏𝑎,𝛼]), 

= 〈𝜎𝑚𝑛
∗ , [𝑐, [𝑎𝑏,𝛼]]〉−〈𝜎𝑚𝑛

∗ , [𝑐, [𝑏𝑎,𝛼]]〉, 

= 〈𝜎𝑚𝑛
∗ , [𝑐, [𝑎𝑏,𝛼]]−[𝑐,[𝑏𝑎,𝛼]]〉, 

= 〈𝜎𝑚𝑛
∗ , [𝑐, [𝑎𝑏,𝛼] −[𝑏𝑎,𝛼]]〉, 

= 〈𝜎𝑚𝑛
∗ , [𝑐, [𝑎𝑏 −𝑏𝑎,𝛼]]〉, 

= 〈𝜎𝑚𝑛
∗ , [𝑐, [[𝑎,𝑏],𝛼]]〉, 

= 𝐵𝜎𝑚𝑛∗ (𝑐,[[𝑎,𝑏],𝛼]).                               

Therefore, we get the nice formula as follows : 

𝐵𝜎𝑚𝑛∗ ((𝑏𝑎)𝑐 −(𝑎𝑏)𝑐,𝛼) = 𝐵𝜎𝑚𝑛∗ (𝑐,[[𝑎,𝑏],𝛼]).                       (25) 

  In the similar way, we have 

𝐵𝜎𝑚𝑛∗ (𝑏(𝑎𝑐)−𝑎(𝑏𝑐),𝛼) = 𝐵𝜎𝑚𝑛∗ (𝑐, [[𝑎,𝑏],𝛼]).                        (26) 

By the equality of equations (25) and (26), then we have 

𝐵𝜎𝑚𝑛∗ ((𝑏𝑎)𝑐 −(𝑎𝑏)𝑐 +𝑎(𝑏𝑐)−𝑏(𝑎𝑐),𝛼) = 0.                          (27) 

But the non-degeneracy of the bilinear form 𝐵𝜎𝑚𝑛∗  guarantees that (𝑏𝑎)𝑐 −(𝑎𝑏)𝑐 +

𝑎(𝑏𝑐)−𝑏(𝑎𝑐) = 0. In other words, the equation (13) holds for all 𝑎,𝑏,𝑐 ∈ 𝔥2. Therefore, 
the symplectic bilinear form 𝐵𝜎𝑚𝑛∗  corresponding to 𝜎𝑚𝑛

∗  induces the left-symmetric on 

the Frobenius Lie algebra  𝔥2.   

Secondly, we shall show that the equation (14) holds. Let us observe that 

𝐵𝜎𝑚𝑛∗ (𝑎𝑏 −𝑏𝑎 −[𝑎,𝑏],𝛼) 

= 𝐵𝜎𝑚𝑛∗ (𝑎𝑏,𝛼)−𝐵𝜎𝑚𝑛∗ (𝑏𝑎,𝛼)−𝐵𝜎𝑚𝑛∗ ([𝑎,𝑏],𝛼), 

= 𝐵𝜎𝑚𝑛∗ (𝑎, [𝑏,𝛼])−𝐵𝜎𝑚𝑛∗ (𝑏,[𝑎,𝛼])−𝐵𝜎𝑚𝑛∗ ([𝑎,𝑏],𝛼), 

= 𝐵𝜎𝑚𝑛∗ (𝑎, [𝑏,𝛼])+𝐵𝜎𝑚𝑛∗ (𝑏,[𝛼,𝑎])+𝐵𝜎𝑚𝑛∗ (𝛼,[𝑎,𝑏]), 

= 〈𝜎𝑚𝑛
∗ , [𝑎,[𝑏,𝛼]]〉+〈𝜎𝑚𝑛

∗ , [𝑏, [𝛼,𝑎]]〉+〈𝜎𝑚𝑛
∗ , [𝛼,[𝑎,𝑏]]〉, 

= 〈𝜎𝑚𝑛
∗ , [𝑎,[𝑏,𝛼]]+[𝑏,[𝛼,𝑎]]+[𝛼,[𝑎,𝑏]]〉, 



A Left-Symmetric Structure on the Semi-Direct Sum Real Frobenius Lie  Algebra of Dimension 8 

Edi Kurniadi 277 

= 〈𝜎𝑚𝑛
∗ ,0〉 = 0. 

Thus, we obtain  

𝐵𝜎𝑚𝑛∗ (𝑎𝑏 −𝑏𝑎 −[𝑎,𝑏],𝛼) = 0.                                                     (28) 

By the same argument of non-degeneracy of 𝐵𝜎𝑚𝑛∗ , we get 𝑎𝑏 −𝑏𝑎 −[𝑎,𝑏] = 0. Thus, the 

equation (14) holds. We proved that, the alternating bilinear form 𝐵𝜎𝑚𝑛∗  induces the left-

symmetric structure on 𝔥2. 
More specific, for 𝑎,𝑏 ∈ 𝔥2, there exist scalars 𝛼𝑘𝑗 and 𝛽𝑖𝑙 where 1 ≤ 𝑗,𝑘, 𝑖, 𝑙 ≤ 4 

such that  
 

𝑎 ≔ ∑ 𝛼𝑘𝑗𝜀𝑘𝑗1≤𝑗,𝑘≤4 , 𝑏 ≔ ∑ 𝛽𝑖𝑙𝜀𝑖𝑙1≤𝑖,𝑙≤4 .                                   (29) 

Moreover, since 𝑎𝑏 ≔ 𝑎 ∗𝑏 ∈ 𝔥2, then there also exists scalars  𝜑𝑠𝑡 where 1 ≤ 𝑠,𝑡 ≤ 4 
such that  

𝑎𝑏 ≔ ∑ φ𝑠𝑡𝜀𝑠𝑡1≤𝑠,𝑡≤4 .                                                                       (30) 

We shall calculate scalar 𝜑𝑠𝑡 written as products of scalars 𝛼𝑘𝑗 and 𝛽𝑖𝑙 where 1 ≤

𝑗,𝑘, 𝑖, 𝑙,𝑠,𝑡 ≤ 4 satisfying the equationss (13) and (14). Applying induced left-symmetric 
structure from the symplectic form 𝐵𝜎𝑚𝑛∗  corresponding to the Frobenius fuctional 

𝜎𝑚𝑛
∗ ≔ 𝜀14

∗ +𝜀23
∗  then we get the following products 

𝜑𝑠𝑡 = ∑ 𝛼𝑘𝑗𝛽𝑖𝑙.1≤𝑗,𝑘,𝑖,𝑙≤4                                                                         (31) 

Therefore, by choosing suitable scalars 𝛼𝑘𝑗 and 𝛽𝑖𝑙, we have the products 𝜀𝑗𝑘𝜀𝑖𝑙. Indeed, 

these product satisfies the equationss (13) and (14) because they are induced by 
simplectic form for  𝔥2. Thus, 𝔥2 is left-symmetric algebra. 

We apply formulas in the  equations (16), (29), (30), and (31) to find the explicit 
left-symmetric structures on 𝔥2. We compute these left-symmetric structures with 
respect to the basis 𝑆 as mentioned above.  Then we obtain :  

 
𝐵𝜎𝑚𝑛∗  (𝑎𝑏,𝜀𝑘𝑗) = −𝐵𝜎𝑚𝑛∗  (𝑏,[𝑎,𝜀𝑘𝑗]), 

                  = −〈𝜎𝑚𝑛
∗ , [𝑏,[𝑎,𝜀𝑘𝑗]]〉, 

           = −〈𝜎𝑚𝑛
∗ , [∑ 𝛽𝑖𝑙𝜀𝑖𝑙1≤𝑖,𝑙≤4 , [∑ 𝛼𝑘𝑗𝜀𝑘𝑗1≤𝑗,𝑘≤4 , 𝜀𝑘𝑗]]〉,                                                  (32) 

where   𝜀𝑘𝑗 ∈ 𝑆 ⊂ 𝔥2 for  1 ≤ 𝑗,𝑘 ≤ 4.  On the other hand, we have the following formulas 

:  

𝐵𝜎𝑚𝑛∗  (𝑎𝑏,𝜀𝑘𝑗) = 𝐵𝜎𝑚𝑛∗  ( ∑ φ𝑠𝑡𝜀𝑠𝑡
1≤𝑠,𝑡≤4

,𝜀𝑘𝑗), 

= 〈𝜎𝑚𝑛
∗ , [ ∑ φ𝑠𝑡𝜀𝑠𝑡

1≤𝑠,𝑡≤4

, 𝜀𝑘𝑗 ]〉,                        (33) 

where   𝜀𝑘𝑗 ∈ 𝑆 ⊂ 𝔥2 for  1 ≤ 𝑗,𝑘 ≤ 4. Let 𝜀𝑘𝑗 = 𝜀11, then the equation (32) can be 

computed with respect the Lie brackets in the equation (20) as follows : 



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Edi Kurniadi 278 

𝐵𝜎𝑚𝑛∗  (𝑎𝑏,𝜀11) = −〈𝜎𝑚𝑛
∗ , [𝑏,[𝑎,𝜀𝑘𝑗]]〉,      

= −𝛽8𝛼2 +𝛽5𝛼3 +𝛽3𝛼5 +𝛽1𝛼6.                                                                         (34) 

In the other hand, we obtain from the equation (33) that 

𝐵𝜎𝑚𝑛∗  (𝑎𝑏,𝜀11) = 〈𝜎𝑚𝑛
∗ , [ ∑ φ𝑠𝑡𝜀𝑠𝑡

1≤𝑠,𝑡≤4

, 𝜀𝑘𝑗 ]〉, 

= −φ14.                                                                                   (35) 

Therefore, we get 

φ14 = 𝛽8𝛼2 −𝛽5𝛼3 −𝛽3𝛼5 −𝛽1𝛼6.                          (36) 

 

In the similar way, we obtain the following formulas using the equations (20), (32), and 
(33) simultaneously : 

φ11 = −(𝛽1𝛼1 +𝛽2𝛼3), 

φ12 = −(𝛽1𝛼2 +𝛽2𝛼4), 

φ21 = −(𝛽3𝛼1 +𝛽4𝛼3), 

φ22 = −(𝛽3𝛼2 +𝛽4𝛼4), 

φ13 = 𝛽5𝛼1 −𝛽6𝛼2 +𝛽7𝛼2 −𝛽5𝛼4 −𝛽4𝛼5 −𝛽2𝛼6, 

φ23 = 𝛽5𝛼3 −𝛽8𝛼2 −𝛽4𝛼7 −𝛽2𝛼8, 

φ24 = 𝛽6𝛼3 −𝛽8𝛼1 −𝛽7𝛼3 +𝛽8𝛼4 −𝛽3𝛼7 −𝛽1𝛼8.         (34) 

Therefore, the following formula  

𝑎𝑏 = ( ∑ 𝛼𝑘𝑗𝜀𝑘𝑗
1≤𝑗,𝑘≤4

)( ∑ 𝛽𝑖𝑙𝜀𝑖𝑙
1≤𝑖,𝑙≤4

) = ∑ φ𝑠𝑡𝜀𝑠𝑡
1≤𝑠,𝑡≤4

 

is determined by the equation (34). By choosing suitable 𝛼𝑘𝑗 and 𝛽𝑖𝑙 then we have the 

left-symmetric structure as stated in Proposition 1. For example if we fix 𝛼1 = 1 and 
𝛽1 = 1, then we have the product 𝜀11 ∗𝜀11 = 𝜀11

2 = −𝜀11. 
∎ 

 
As discussion, there is a still open problem for a generalization of  left-symmetric 

structure of  𝔥2 to left-symmetric structure of   𝔥𝑛,𝑝:= M𝑛,𝑝(ℝ)⋊𝔤𝔩𝑛(ℝ). The Lie algebra 

𝔥𝑛,𝑝 is Frobenius Lie algebra whenever 𝑝 is factor of 𝑛 [1]. We offered an alternative 

proof to show 𝔥𝑛,𝑝:= M𝑛,𝑝(ℝ)⋊𝔤𝔩𝑛(ℝ) is Frobenius for 𝑛 = 𝑝 = 2 as explained before. 

Since 𝔥𝑛,𝑝 is Frobenius Lie algebra, then there exists a Frobenius Functional. Therefore, 

we can construct a symplectic linear form which induces left-symmetric structure for 
𝔥𝑛,𝑝. Thus, we consider the following conjecture: 

 
 



A Left-Symmetric Structure on the Semi-Direct Sum Real Frobenius Lie  Algebra of Dimension 8 

Edi Kurniadi 279 

Conjecture 1. The Lie algebra 𝔥𝑛,𝑝:= M𝑛,𝑝(ℝ)⋊𝔤𝔩𝑛(ℝ) where 𝑝 devides 𝑛 has left-

symmetric structures. These structures are induced by a symplectic form of 𝔥𝑛,𝑝.  

 
One of the interesting problems is how to find a Frobenius functional for 𝔥𝑛,𝑝 

corresponding to the Pfaffian of 𝔥𝑛,𝑝  in order to give explicit formulas for left-symmetric 

structure for 𝔥𝑛,𝑝.  

Furthermore, if the Conjecture 12 is true, then the next question is how about left-
symmetric structures for general Frobenius Lie algebras. We also proved that that all 4-
dimensional Frobenius Lie algebra are left-symmetric algebras. As mentioned before 
that not all Lie algebras have left-symmetric structure. But we guess that Lie algebras of 
Frobenius types in general have left-symmetric structures. In other words, in more 
general case we consider the following conjecture 
 
Conjecture 2. Let 𝔤 be a finite dimensional Frobenius Lie algebra. Then 𝔤 is left-symmetric 
algebra whose left-symmetric structures are induced by a symplectic form corresponding 
to Frobenius functional of 𝔤. 

 
It would be interesting to study the completeness of left-symmetric algebra of 𝔥𝑛,𝑝 

whenever a Frobenius Lie algebra is equal to its radical and in general, the completeness 
of any finite dimensional Frobenius Lie algebra.  

 
 

CONCLUSIONS 

We showed that 𝔥2 ≔ M2(ℝ)⋊𝔤𝔩2(ℝ) is the 8-dimensional Frobenius Lie algebra. 
Furthermore, we proved the existence of left-symmetric structures on the Frobenius Lie 
algebra 𝔥2 and we listed the explicit formulas of left-symmetric structures. Therefore, 𝔥2 
is left-symmetric algebra.  Our construction based on the symplectic form corresponding 
to a Frobenius functional of 𝔥2 which induced the left-symmetric structures on 𝔥2. Our 
result can motivate the left-symmetric structure for 𝔥𝑛,𝑝:= M𝑛,𝑝(ℝ)⋊𝔤𝔩𝑛(ℝ) and  for 

general case of Frobenius Lie algebras. 
For future research, we stated Conjecture 12 and Conjecture 13 which are still open 

problem to be investigated. This is very interesting if Conjecture 12 and Conjecture 13 
are true because we can study the radical of 𝔥𝑛,𝑝, denoted by Rad( 𝔥𝑛,𝑝), and if we have 

𝔥𝑛,𝑝 = Rad( 𝔥𝑛,𝑝), then we come to the notion of a completeness of left-symmetric 

algebra of 𝔥𝑛,𝑝. Moreover, left-symmetric algebras relate affine structures and it is 

interesting to investigate for the Lie groups case[15] .  
 
 

ACKNOWLEDGMENTS 

We thank Universitas Padjadjaran who has funded the work through Riset 

Percepatan Lektor Kepala (RPLK) in the year 2021 with the contract number 

1959/UN6.3.1/PT.00/2021.   

  



A Left-Symmetric Structure on the Semi-Direct Sum Real Frobenius Lie  Algebra of Dimension 8 

Edi Kurniadi 280 

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