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- *Corresponding Author:
- Elisabeth Remm

Universite de Haute Alsace

LMIA, 4 rue des Freres Lumiere

68093 Mulhouse, France03 89 33 66 52

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**Received Date:** July 18, 2017; **Accepted Date:** July 26, 2017; **Published Date:** July 30, 2017

**Citation: **Remm E, Goze M (2017) 2-Dimensional Algebras Application to Jordan,
G-Associative and Hom-Associative Algebras. J Generalized Lie Theory Appl 11:
278. doi: 10.4172/1736-4337.1000278

**Copyright:** © 2017 Remm E, et al. This is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

We classify, up to isomorphism, the 2-dimensional algebras over a field . We focuse also on the case of characteristic 2, identifying the matrices of GL(2, 2) with the elements of the symmetric group Σ3. The classification is then given by the study of the orbits of this group on a 3-dimensional plane, viewed as a Fano plane. As applications, we establish classifications of Jordan algebras, algebras of Lie type or Hom-Associative algebras.

2-Dimensional algebras; Classification; Hom-associative algebras

An algebra over a field is -vector space equipped with a
product which corresponds to a bilinear map on with values in . For a given dimension, one of the basic problems is the determination
up to linear isomorphism of all these algebras. Sub classes of algebras
where widely studied. These subclasses where often obtained setting a
quadratic relation on *μ*. Among other examples of such classes are Lie
algebras (in this case μ is skewsymmetric and satisfies Jacodi identity),
associative algebras, Lie-admissibles algebras, Pre-Lie algebras in
particular. In all these examples, classifications where established in a
general frame work, that is, with no other hypothesis on these classes and
only in very small dimensions. For example for Lie algebras, we know
the general classifications up to the dimension 6. In bigger dimension
we impose additional algebraic properties if we hope to continue this
classification. For example simple Lie algebras are fully classified since
the work of Killing and Cartan, in any dimension. Unfortunately it is
more and less the only solved case. If we consider complexe nilpotent
Lie algebras, the classification is known only up to the dimension 7.
It is the same for the associatives algebras. If we are only interested
in general algebras, the only known cases are the dimension 2 and 3.
It is true that the problem is equivalent to the classification of tensors
of type (2,1) on a finite dimensional vector space. We are then facing
to a basic multilinear algebra problem which is subject to a lack of
informations on the tensors.

Here we reconsider this problem from the beginning, that is in
dimension 2. This work is certainly not the first one of the subject.
There is for example the work of Petersson. Our approach is not similar.
We are not fully interested by the classification up to isomorphism
but by the determination of subclasses, minimal in a certain sense,
which are invariant up to isomorphism. The motivation comes from
the constatation of what happen in greater dimensions for nilpotent
Lie algebras for example In this case, the classification is established in
dimension 7 but quasi unusable in its present forme. This means that
if we have a precise example of nilpotent Lie algebra of this dimension,
it is long and fastidious to recognize it in the given list because most
of the time it is not adapted to the invariants used to established the
classification. Moreover the length of the list can be puzzling. In
greater dimensions, the number of isomorphy classes, the need to
write invariant parametrized families seems to be an unrealistic goal.
Hence the idea to reduce the classification problem to a determination
of invariant classes. This is the aim of this work. However we will
established the link with Petersson’s work. Our approach is quite basic. In characteristic different from 2, we decompose a tensor *μ* as a skewsymmetric and symmetric one. Since the skewsymmetric
case is elementary, we classify those which are symmetric modulo
the automorphism group of the associated skeysymmetric law. In
characteristic 2, the problem is equivalent to the determination of the
orbits of the Fano plane modulo the symmetric group. Finally, we use
these results to describe or find again certain classes of algebras whose
a direct approach is rather difficult. In particular, we determine the 2-dimensional Jordan algebras and we find again the results of ref. [1],
the *G*-associative algebras and the Hom-associative algebras.

We have begun the study of the determination of general algebras in ref. [2] which was specially an introduction to a more precise work developed in this paper but with the same idea to describe "minimal" families invariant by isomorphism rather than a precise list for which the use is difficult. Recently, we were acquainted with the work of Pertersson, based on an Kaplansky result which permits to describe all the algebras from some unital algebras and to give isomorphism criteria. We try in this paper to look our description in a Petersson point of view. We note also a recent work, on the same subject of H. Ahmed, U. Bekbaev and I. Rakhimov [3].

Let be a field whose characteristic will be precise later. An algebra over a field is a -vector space V with a multiplication given by a bilinear map

*μ*:*V* × *V*→*V*.

We denote by I=(*V*,*μ*) a -algebra structure on *V* with
multiplication *μ*. Throughout this paper we fix the vector space *V*.
Since we are interested by the 2-dimensional case we could assume that . Two -algebras *A*=(*V*,*μ*) and *A*′=(*V*,*μ*′) are isomorphic if there
is a linear isomorphism,

*f*:*V*→*V*

such as;

*f*(*μ*(*X*,*Y*))=*μ*′(*f*(*X*), *f*(*Y*)),

for all *X*,*Y*∈*V*. The classification of 2-dimensional -algebras is
then equivalent to the classification of bilinear maps on with
values in *V*. Let {*e*_{1},*e*_{2}} be a fixed basis of *V*. A general bilinear map *μ* has
the following expression:

and it is defined by 8 parameters. Let *f* be a linear isomorphism of *V*. In
the given basis, its matrix *M* is non degenerate. If we put:

Then,

with Δ=*ad*−*bc*≠0. The isomorphic multiplication.

Satisfies,

With,

(1)

These formulae describe an action of the linear group on parameterized by the structure constants (*α _{i}*,

We assume in this section that . We consider the bilinear
map *μ _{a}* and

for all *X*,*Y*∈*V*. The multiplication *μ _{a}* is skew-symmetric and it is a
Lie multiplication (any skew-symmetric bilinear application in is a
Lie bracket). It is isomorphic to one of the following:

In fact, if *μ _{a}* is not trivial, thus . If

We have .

If *α*=0, thus *β*≠0 and we take:

This gives . In any case, if *μ _{a}*≠0, then it is isomorphic to

**Case **

An automorphism of the Lie algebra is a linear isomorphism such that:

for every *X*,*Y*∈*A*. The set of automorphisms of this Lie algebra is
denoted by .

**Lemma 1:** We have:

Proof. In fact, assume that is the matrix of the automorphism f in the given basis . Then,

and,

Then,

*c*=0, *a*=*ad*.

But *detM*=*ad*≠0 so *a*=*ad* implies that *d*=1 This gives the lemma.

Let *μ* be a general multiplication of 2-dimensional -algebra such
that *μ _{a}* is isomorphic to . It is isomorphic to a the bilinear map
(always denoted by

The classification, up to isomorphism, of the Lie algebras (*V*,*μ*)
such that *μ _{a}* is isomorphic to is equivalent to the classification
up an isomorphism belonging to of the abelian algebras
isomorphic to:

In this case (1) is reduced to:

(2)

1. Assume that *β*_{1}≠0.

• Suppose that is algebraically closed and consider the
isomorphism . The isomorphic algebra is such that. We deduce that in this case *μ _{s}* is isomorphic to:

Then *μ* is isomorphic to:

with .

• If is not algebraically closed (for example if is a finite
field), let be the multiplicative subgroup of elements a^{2} with . In this case *μ* is isomorphic to a Lie bracket belonging to the 4
parameters family:

with and. For example, if , then λ∈{−1,1}.

2. Assume *β*_{1}=0, *β*_{2}≠0. In this case (1) is reduced to:

(3)

and taking *b*=−*β*_{4}/2*β*_{2} and , we see that *μ _{s}* is isomorphic to:

We obtain the following multiplication, being algebraically closed or not:

3. Assume now that . In this case (1) is reduced to:

(4)

and taking *b*=−*α*_{2}/*α*_{1} and , we obtain and . In this
case, *μ* is isomorphic to:

4. Assume now that . In this case,
considering , the Lie bracket *μ* is isomorphic to:

5. Assume now that . The
Lie bracket *μ* is isomorphic to:

6. If , then *μ* is isomorphic
to with *β*_{4}=2*α*_{2}

**Theorem 2:** Any 2-dimensional non commutative algebras
isomorphic to one of the following algebras:

• If is algebraically closed:

with

• If is not algebraically closed:

Let us make the link with the results of Petersson [4]. The main idea
of this work is to construct algebras from unital algebra. Recall that
an algebra *A*=(*V*,*μ*) is called unital if there exists 1∈*V* such that *μ*(1, *X*)=*μ*(*X*,1)=*X* for any *X*∈*V* for any *X*∈*V*.

**Lemma 3:** If *μ _{a}* is not trivial, then

Proof. Assume that there exists 1 satisfying *μ*(1, *X*)=*μ*(*X*,1)=*X*, then:

for any *X*∈*V*. Then *μ _{a}*(1,

The algebra *A*=(*V*,*μ*) is called regular if there exists *U*,*T*∈*V* such
that the linear applications:

are linear isomorphisms. From ref. [5], for any regular algebra *A*=(*V*,*μ*)
there exist a unique, up an isomorphism, unital algebra *B*=(*V*,*μ _{u}*) and
two linear isomorphisms

for any *X*, *Y*∈*V*. The algebra *B* is called the unital heart of *A*. To
compare Theorem 2 with the Petersson results, we have to determine
the regular algebras. Let us consider the first family. The application *L _{U}* is not regular for any

Likewise *R _{T}* is not regular for any

We deduce that any algebra is regular except the algebras given by:

Let us note that is left-singular but right-regular and is right-singular and left-regular. An algebra which is left and right singular is called bi-singular. We can summarize the results in the following array:

1. regular except and .

2. is left-singular and right-regular,

3. is right-singular and left-regular,

4. is regular,

5. is regular except ,

6. is bisingular.

7. is regular except

8. is bisingular,

9. is left-singular and right-regular as soon as *β*_{4}≠0,

10. is left-regular and right-singular as soon as *β*_{4}≠0,

11. is regular except for *α*_{2}=0, 1 or −1,

12. is bisingular,

13. is left-singular and right-regular as soon as *β*_{4}≠0,

14. is left-regular and right-singular as soon as* β*_{4}≠0,

We deduce.

**Proposition 4:** We consider the following algebras,

1. with

2.

3. with

4. with *α*_{2} ≠ 0,1,−1.

For anyone of these algebras *A*, there exists an unital algebra and linear endomorphisms *f _{A}*,

This unital algebra *B _{A}* is called the unital heart of

1. Let be . If *α*_{2}≠1 or −1 then and are not
singular. In fact,

Thus,

Then the identity element of *B _{A}* is

and *B _{A}* is etale. If

2. Let be . This algebra is regular. If α1≠0, then and are not singular and *B _{A}* is etale.

**Case **

The multiplicatio *μ* n is symmetric. The group of automorphisms of *μ _{a}* is . Moreover the multiplication writes:

We assume that there exists two independent idempotent vectors.
If *e*_{1} and *e*_{2} are these vectors, then:

We obtain the following algebras:

Remark that if any element is idempotent, thus . In fact:

In the general case, if *ae*_{1}+*be*_{2} is an idempotent with *ab*≠0, then a
and *b* satisfy the system:

If 4*α*_{2}*β*_{2}=1, then the system has solutions as soon as . In
this case we obtain the multiplication and for any *a*, the vectors *ae*_{1}+(1−*a*)*e*_{2} are idempotent. If , the vector:

is an idempotent and the only idempotents are *e*_{1}, *e*_{2} and *v*. The changes
of basis {*e*_{1},*v*} or {*e*_{2}, *v*} do not simplify the number of independent
parameters.

We assume that there exists only one idempotent vector. If *e*_{1} is
this vector, thus μ(*e*_{1}, *e*_{1})=*e*_{1}. If we consider a vector *v*=*xe*_{1}+*ye*_{2} such that *μ*(*v*,*v*)=*v*, then *x* and *y* have to satisfy:

(5)

If we assume that *y*≠0, the second equation gives as soon as *β*_{2}≠0, and thus:

(6)

Let us consider a change of basis which preserves *e*_{1} that is,

(7)

with *d*≠0. Since in this new basis we have , we can find *b* such that . Then we can assume that *β*_{4}=0.

If moreover *α*_{2}≠0, taking , we obtain and we have
the algebra:

Equation (6) simplifies as:

(8)

If we assume that is algebraically closed, then this equation has
in general two roots. It has no root if *β*_{2}=0 which is excluded. Then to
have only one idempotent, 0 must be the only root which is equivalent
to *α*_{4}=0 and *β*_{2}=1/2. We obtain the following algebra:

If is not algebraically closed, then we have no idempotent other
than 0 if *α*_{4}=0 and *β*_{2}=1/2 and we obtain the previous algebra *μ*^{7} or if is irreducible in . We obtain:

with irreducible in (so *α*_{4}≠0).

If *α*_{2}=0 and if is algebraically closed, we consider in the change of
basis (7) defined above, *b*=0 and if *α*_{4}≠0:

There exits only one idempotent if and only if *β*_{2}=1/2. We obtain
the following algebra:

If *α*_{2}=*α*_{4}=0, we have only one idempotent if and only if 2*β*_{2}≠1. We
obtain:

Assume not algebraically closed and *α*_{2}=0. If the equation d^{2}*α*_{4} has a root in , we find *μ*^{8}. If not, let such that In this case we have only one idempotent if and only if (2*β*_{2}=1) or . We obtain:

and,

Assume now that *β*_{2}=0. Then (5) implies *y*^{2}*β*_{4}=*y*. If *β*_{4}=0, then *y*=0
and we have:

The change of basis . We obtain:

if *α*_{2}≠0. Assume now that *α*_{2}=0 and *α*_{4}≠0. If is algebraically close, we
obtain:

No vector is idempotent. If there exists *v* with *μ*(*v*,*v*)≠0, thus we can
consider that *μ*(*e*_{1},*e*_{1})=*e*_{2} that is,

1. If *α*_{4}=0, that is , then the vector is idempotent as soon as *β*_{4}≠0. Then the hypothesis implies *β*_{4}=0. Let be *v*=*xe*_{1}+*ye*_{2}. The equation *μ*(*v*,*v*)=*v* is equivalent to:

that is,

If *α*_{2}=0, then *x*=*y*=0, and no elements are idempotent. We obtain
the algebras, corresponding to *β*_{2}≠0 or *β*_{2}=0

If *α*_{2}≠0 and then x satisfies the equation:

(9)

If is algebraically closed, such equation admits a non trivial
solution. This is not compatible with our hypothesis. Assume that is not algebraically closed. If *β*_{2}≠0, the change of basis and permits to consider *β*_{2}=1 and the (9) becomes,

This equation has a non solution if where. We obtain the algebras:

and,

2. If *α*_{4}≠0 the vector *v*=*xe*_{1}+*ye*_{2} is idempotent if and only if:

Then . Let us note that 1−2*yα*_{2}≠0 because 1−2*yα*_{2}=0
implies *y*^{2}*α*_{4}=0 that is *y*=0 and in this case *x*=0 and *v*=0 . We deduce
that *y* is a root of the equation:

that is:

If is algebraically closed, this equation admits always a solution except if:

Then . We note that *β*_{2}=0 implies,
if the characteristic of is not 3, *α*_{2}=*α*_{4}=0. From hypothesis, we can
assume that *β*_{2}≠0 and the change of basis which preserves the condition *e*_{1}*e*_{1}=*e*_{2} changes *β*_{2} in *kβ*_{2} and we can take *β*_{2}=3. Then , then *α*_{2}=−2 and *α*_{4}=4, *β*_{4}=8 and we
obtain the algebra:

Let us note that if the characteristic of is 3, then *α*_{4}*β*_{2}=0 and *β*_{2}=0. This gives *α*_{2}(*α*_{2}+*β*_{4})=0 and . Since *α*_{2}=0 implies *α*_{4}=0 and 4_{2}+β_{4}=*α*_{2}+*β*_{4}=0 we obtain *β*_{4}=2*α*_{2} and . By a change of basis we can take *α*_{2}=1 and we obtain the algebra:

which correspond to *μ*_{15} in characteristic 3.

If is not algebraically closed, we have to consider all the algebras for which the polynomial:

(10)

has no root this is equivalent to say that *P _{A}* is irreducible. If we
consider the coefficient of

**Proposition 5:** The algebra *A* is regular if and only if *P _{A}*(

It remains to examine the case *μ*(*v*,*v*)=0 for any *v*. That is:

If *α*_{2}*β*_{2}≠0 we can find some idempotents. In all the others cases, we
have no idempotent. We obtain:

**Theorem 6:** Any commutative 2-dimensional algebra over an
algebraically closed field is isomorphic to one of the following:

If is not algebraically closed, we have also the following algebras where

Let us examine the property of regularity for these algebras. Since they are commutative, the left and right regularity are equivalent notions. Computing directly the determinant of the operator we deduce in the case algebraically closed:

1. The algebras are regular,

2. is regular if *β*_{2}≠0,

3. The algebras and are bisingular.

Let be a field of characteristic 2. Assume that . If *A* is a
2-dimensional -algebra and if {*e*_{1}, *e*_{2}} is a basis of *A*, then the values of
the different products belong to {*e*_{1}, *e*_{2}, *e*_{1}+*e*_{2}}. If f is an isomorphism of
A, it is represented in the basis {*e*_{1}, *e*_{2}} by one of the following matrices:

Each of these matrices corresponds to a permutation of the finite
set {*e*_{1}, *e*_{2}, *e*_{3}=*e*_{1}+*e*_{2}}. If fact we have the correspondance:

where _{ij} is the transposition between *i* and *j* and *c* the cycle {231}. In
fact, the matrix *M*_{2} corresponds to the linear transformation *f*_{2}(*e*_{1})=*e*_{2}, *f*(*e*_{2})=*e*_{1} and in the set (*e*_{1}, *e*_{2}, *e*_{3}) we have the transformation whose image
is (*e*_{1}, *e*_{2}, *e*_{3}) that is the transposition τ_{12}. The matrix *M*_{3} corresponds to
the linear transformation *f*_{2}(*e*_{1})=*e*_{1}+*e*_{2}, *f*(*e*_{2})=*e*_{2} which corresponds to the
permutation (*e*_{3}, *e*_{2}, *e*_{1}) that is τ_{13}. For all other matrices we have similar
results. We deduce:

**Theorem 7:** There is a one-to-one correspondance between the
change of -basis in Aand the group Σ_{3}.

If we want to classify all these products of *A*, we have to consider
all the possible results of these products and to determine the orbits of
the action of Σ_{3}. More precisely the product *μ*(*e _{i},e_{j}*) is in values in the set (

Let us consider the following sequence:

As , if and then with the relations:

for *i*, *j*, *k* all different and non zero. Thus the four first terms of this
sequence determine all the other terms. More precisely, such a sequence
writes:

**Consequence:** We have 4^{4}=256 sequences, each of these sequences
corresponds to a 2-dimensional -algebra.

Let us denote by S the set of these sequences. We have an action of
Σ_{3} on *S*: if *σ*∈Σ_{3} and *s*∈*S*, thus *s*′=*σs* is the sequence:

with when and *R _{k}*≠0.
If

1. The isotropy subgroup is Σ_{3}. In this case we have the following
sequence (we write only the 4 first terms which determine the algebras:

Recall that means means and so on.

2. The isotropy subgroup is We have only one orbit:

3. The isotropy subgroup is of order 2.

4. The isotropy subgroup is trivial. In this case any orbit contains 6
elements. As there are 256−46=46=210 elements having Σ_{3} as isotropy
group, we deduce that we have 35 distinguished non isomorphic classes.

We have 52 classes of non isomorphic algebras of dimension 2 on
the field *F*_{2}.

**Applications : 2-dimensional G-associative and Jordan
algebras**

**G****-associative commutative algebras**

The notion of *G*-associativity has been defined in ref. [4]. Let *G* be a
subgroup of the symmetric group Σ_{3}. An algebra whose multiplication
is denoted by *μ* is *G*-associative if we have:

where *ε*(*σ*) is the signum of the permutation . Since we assume that *μ* is commutative, all these notions are trivial or coincide with the
simple associativity. Now, if the algebra is of dimension 2, then the
associativity is completely determined by the identities:

We deduce that the only associative commutative 2-dimensional algebras are:

• *μ*^{6} for (*α*_{2}, *β*_{2})∈{(0,1),(1,0),(0,0)},

• *μ*^{9} for *β*_{2}=0 or 1,

•*μ*^{12}, *μ*^{16}, *μ*^{17} .

• if for *β*_{2}=1 and λ=−1.

We find again the classical list [6].

*G*-associative noncommutative algebras

Let us consider now the noncommutative case. From Theorem 2,
the multiplication μ is isomorphic to some *μ ^{i}*,i=1,…,5 (we consider
here that is algabraically closed). Let

Now, for any nonassociative algebra, we examine the *G _{i}*-associativity.
Note that all these algebras are Lie-admissible, that is Σ

**Proposition 8:** Any 2-dimensional noncommutative *G*_{2}-associative
algebra is isomorphic to one of the following:

1. or , that is *μ* is associative,

2.

3.

4.

5.

6.

**Jordan algebras**

In a Jordan algebra, the multiplication *μ* satisfies:

for all *v*,*w*. We assume in this section that is algebraically closed and
that the Jordan algebra are of dimension 2. Thus the multiplication μ is
isomorphic to *μ _{i}* for

*v*(*vw*)=*v*(*vw*)

for any *w*, that is, this identity is always satisfied.

**Lemma 9:** If *v*_{1} and *v*_{2} are idempotent vectors, thus:

for any *w*.

Proof. In the Jordan identity, we replace *v* by *v*_{1}+*v*_{2}. We obtain:

Since *v*_{1} and (*v*_{2}) are idempotent, this equation reduces:

**Proposition 10:** If *v*_{1} and *v*_{2} are idempotent vectors such that *v*_{1}*v*_{2} and *v*_{1}+*v*_{2} are independent, thus the Jordan algebra is associative.

Proof. Let *x* and *y* be two vectors of the algebra. Thus, by hypothesis, . Thus:

and,

*x*( *yw*) = *y* (*xw*)

By commutativity we obtain:

*x*( *yw*) = *x*(*wy*) = *y* (*xw*) = (*xw*) *y*

this proves that the algebra is associative.

If *μ* is given by,

the Jordan algebra admits two idempotents *e*_{1} and *e*_{2}. Since , the vectors *e*_{1}*e*_{2} and *e*_{1}+*e*_{2} are independent if and only
if *α*_{2}≠*β*_{2}. In this case the algebra can be associative and we obtain the
following associative Jordan algebra corresponding to:

1. *α*_{2}=1, *β*_{2}=0

2. *α*_{2}=0, *β*_{2}=1

These Jordan algebras are isomorphic. This gives the following Jordan algebra:

If *e*_{1}*e*_{2} and *e*_{1}+*e*_{2} are dependent, that is *e*_{1}*e*_{2}=λ(*e*_{1}+*e*_{2}), then λ=−1
or ½ or 0. If *e*_{1}*e*_{2}=0, the product is not a Jordan product. If λ=−1 the
product is never a Jordan product. If ë = ½, we obtain the following
Jordan algebra,

*μ* is given by:

This product is a Jordan product if *β*_{2}=1 or 0. We obtain:

If *μ*=*μ*_{11} we have also a Jordan structure,

*μ*=0, we have the trivial Jordan algebra.

• If is not algebraically closed, we consider,

We obtain a Jordan structure:

We find the list established in ref. [1].

**2-dimensional Hom-algebra**

The notion of Hom-algebra was introduced to generalized form of
Hom-Lie algebra which appeared naturally when we are interested by
the notion of *q*-derivation on the Witt algebra. In dimension 2, this
notion is equivalent to the classical notion of Lie algebra. In dimension
3, we have shown that any skew-symmetric algebra is a Hom-Lie
algebra. Then our interest concerns Hom-associative algebra [7,8], that
is algebra *A*=(*V*,*μ*) such that there exists *f*∈End(*V*) satisfying the Hom-
Ass identity:

for any *X*, *Y*, *Z*∈*V*. Using previous notations, we consider the
algebras *A*^{(Id, f)} and its opposite *A*^{(f, Id)}. Their multiplication law are
respectively defined by:

and the Hom-Ass identity can be written:

Assume now that the algebra A is regular. In this case, assuming
that the field is algebraically closed, there exists an unital algebra whose
product is denoted *X*⋅*Y* and two endomorphisms *u* and *v* of *V* such
that:

*μ* ( *X* ,*Y* ) = *u*( *X* ) ⋅ *v*(*Y* )

Then,

Then the Hom-Ass identity becomes:

Maybe, it is better to look the Hom-Ass identity from the previous list. Assume that A is non commutative.

1. , let *f* be an endomorphism of *V* satisfying the Hom-Ass identity. To simplify notations we write *XY* for *μ*(*X*,*Y*) and [*X*,*Y*] for *μ _{a}*(

We deduce *f*(*e*_{1})=*ae*_{2}. Likewise we have [*e*_{2}*e*_{2},*f*(*e*_{2})]=0 and *f*(*e*_{2})=*k*(*α*_{4}*e*_{1}+*β*_{4}*e*_{1}). Other identities give :

(a) implies *a*=0 or *e*_{2}*e*_{2}=0.

(b) If *a*=0, then (*e*_{1},*e*_{2})f(*e*_{2})(*e*_{1}*e*_{1})=0 implies f(*e*_{2})*e*_{2}=0 and (*e*_{1},*e*_{1}) *f*(*e*_{2})−*f*(*e*_{1})(*e*_{1}*e*_{2})=0 implies *e*_{2}*f*(*e*_{2})=0. Then [*e*_{2}, *f*(*e*_{2})]=0 and *f*(*e*_{2})=*ke*_{2}. This gives 0=*f*(*e*_{2})*e*_{2}=*be*_{2}*e*_{2} that is *f*=0 or *e*_{2}*e*_{2}=0. But we have seen that , then in all the cases, *f*=0.

(c) If *a*≠0, then *e*_{2}*e*_{2}=0 and *f*(*e*_{2}_{})=0. We deduce that (*e*_{1}*e*_{2})*f*(*e*_{1})−*f*(*e*_{1})
(*e*_{2}*e*_{1})=0 implies *α*_{2}=*β*_{2}=0. Thus (*e*_{2}*e*_{1})*f*(*e*_{1})−*f*(*e*_{2})(*e*_{1}*e*_{1})=−*a*(*e*_{1}*e*_{2})=−*ae*_{1}=0
and *a*=0.

We deduce that the algebra is not a Hom-associative algebra.

2. . With similar simple computation we can look that also this algebra is not a Hom-Ass algebra.

3. . In this case also, if we compute , we obtain *f*(*e*_{1})=*k*_{1}*e*_{1}. Also
we have and *f*(*e*_{1})=0. We deduce *e*_{1}*f*(*e*_{2})=0 and *f*(*e*_{2})*e*_{1}=0 and *f*(*e*_{2})=0. Thus *f*=0 and *A*^{3} is not a Homassociative
algebra.

4. . If *β*_{4}≠0, then the Hom-Ass condition implies *α*_{2}=1 or
−1. We obtain the following Hom-Ass algebras:

In each of these two cases, *f* is a diagonal endomorphism. These
algebras are for *β*_{4}≠2 or −2, not associative.

5.. If *α*_{2}=0, any linear endomorphism with values in satisfies the Hom-Ass identity. Then the following algebra is Homassociative:

Assume now that *α*_{2}≠0. If *á*_{2} ≠ ±1 , then any endomorphism
satisfying the Hom-Ass identity is trivial. If *α*_{2}=1 or −1, we have non
trivial solution and the following algebras are Hom-associative algebras:

with in the first case and in the second case.

Then we have the list of noncommutative Hom-associative algebras. The commutative case can be established in the same way. In this case the Hom-Ass identity is reduced to:

Then *f* is in the kernel of the linear system whose matrix is:

Then *A* is a Hom-associative algebra if and only if *H*(*A*)=det(*HA _{A}*)=0.
We deduce that the set of 2-dimensional commutative Hom-associative
algebra can be provided with an algebraic hypersurface embedded in
the affine variety . From Theorem 6, when is algebraically closed,
we obtain:

1. . It is equal to 0 for *α*_{2}=0 or *β*_{2}=0 or *α*_{2}=1−*β*_{2} or or

2. and *A*^{7} is not a Hom-associative algebra.

3. and *A*^{8} is not a Hom-associative algebra.

4. *H*(*A ^{i}*)=0 for

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- Goze M, Remm E (2011) 2-dimension algebras. African Journal of Mathematical Physics 110: 81-91.
- Ahmed H, Bekbaev U, Rakhimov I (2017) Complete classification of two-dimensional algebras. AIP Conference Proceedings 1830 (1): 10.1063/1.4980965.
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- Petersson HP (2000) The classification of two-dimensional nonassociative algebras. Result Math 3: 120-154.
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