# Central Extension of Left Alternative Algebras and the Second Cohomology Group

Authors: Said Boulmane
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

The purpose of this paper is to prove that the second cohomology groupH2A,F of a left alternative algebra A over an algebraically closed field F of characteristic 0 can be interpreted as the set of equivalent classes of one-dimensional central extensions of A.


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## Body

## 1. Introduction

In this paper, we consider finite-dimensional alternative algebras over an algebraically closed fieldF of characteristic 0. An algebra A is left alternative (resp. right alternative) if it satisfies the left alternative law (resp. the right alternative law) x2y=xxy (resp. yx2=yxx), for all x,y∈A. An alternative algebra is one which is both left and right alternative. For a general theory about alternative algebras, refer [1–6].IfA is left alternative algebra, then the algebra defined on the same vector space A with “opposite” multiplication x⋄y≔yx is a right alternative algebra and vice versa. Hence, all the statements for left alternative algebras have their corresponding statements for right alternative algebras. Thus, we will only consider the left alternative algebra case in this paper.Elhamdadi and Makhlouf [7] introduced an algebraic cohomology of left alternative algebras, and they computed the second cohomology group of the 2 by 2 matrix algebra ℳ2F (considered as left alternative algebra). In this paper, we recall the cohomology theory and birepresentations for left alternative algebra and we discuss the links between the second cohomology group H2A,F and central extensions of left alternative algebra, and then we prove that the second cohomology group H2A,F of a left alternative algebra A can be interpreted as the set of classes of one-dimensional central extensions of A.


## 2. Preliminaries

In this section, we recall some definitions and concepts of alternative algebras. This class of algebras is well studied by several authors; for more details, refer to [1, 4, 6].Definition 1.
LetA be a left alternative algebra. A birepresentation of A is a triple M,π,Π, where M is a F-vector space and π:A⟶EndM and Π:A⟶EndM are two linear maps satisfying(i)
πa2=πa2(ii)
Πaa′−Πa′Πa=Πa′,πawhere(1)Πa′,πa=Πa′πa−πaΠa′,∀a,a′∈A.
In this case,M is called bimodule of A. We denote πam (resp. Πam) by a.m (resp. m.a).Proposition 1.
LetA be a left alternative algebra and M be a bimodule of A. The direct summand A⊕M is made into a left alternative algebra, by defining multiplication as follows:(2)a+ma′+m′=aa′+a.m′+m.a′,for all a,a′∈A and m,m′∈M.Proof.
We show that the left alternative law is satisfied.
Leta,a′∈A and m,m′∈M.(3)a+m2a′+m′−a+ma+ma′+m′=a2+m.a+a.ma′+m′−a+maa′+m.a′+a.m′=a2a′+a2.m′+m.a.a′+a.m.a′−aaa′−a.m.a′−a.a.m′−m.aa′=a2.m′+m.a.a′+a.m.a′−a.m.a′−a.a.m′−m.aa′=m.a.a′+a.m.a′−a.m.a′−m.aa′=0.
Therefore,A⊕M is a left alternative algebra.


## 3. Central Extensions of Left Alternative Algebras

Definition 2.
LetA be a left alternative algebra; a one-dimensional central extension of A is an exact sequence of left alternative algebras:(4)0⟶F⟶iA˜⟶πA⟶0,such that Kerπ⊂ℨA˜.
We denote byℨA˜=a∈A˜,A˜a=0andaA˜=0 the center of A˜ and by Kerπ the Kernel of π.
Two such extensionsA˜ and A˜′ are called equivalent if there is a morphism ϕ:A˜⟶A′˜ of left alternative algebras such that the following diagram is commutative:(5)0⟶F⟶iA˜⟶πA⟶0↓id↓ϕ↓id0⟶F⟶i′A˜′⟶π′A⟶0.
Denote byCLA,F the set of equivalent classes of one-dimensional central extensions of A.Example 1.
LetA1 be a left alternative algebra, V a vector space, and φ:A1×A1⟶V be a bilinear map such that(6)φa2,b=φa,ab,for alla,b∈A1.
We define the following product on the vector space direct sumA=A1⊕V:(7)a+vb+v′≔ab+φa,b,for alla,b∈A1,v,v′∈V.
We can check easily thatA is a left alternative algebra (see Example 1 in [8]), and further we obtain an exact sequence:(8)0⟶V⟶λA1⊕V⟶μA1⟶0.
Moreover, we haveKerμ=0×V⊂ℨA1⊕V, so the extension is central and called the central extension of A1 by V via φ.


## 4. Interpretation ofH2A,F

Now, we recall the cohomology theory for left alternative algebra introduced by M. Elhamdadi and A. Makhlouf [7].LetA be a left alternative algebra and M an A-bimodule. If p≥1, a p-cochain of A with values in M are a p-linear mapping of Ap in M. We denote by CpA,M the space of the p-cochains of A. For p<0, we put CpA,M=0 and C0A,M=M.Forφ∈C1A,M, we define the first differential δ1φ∈C2A,M by(9)δ1φa,b=φa.b+a.φb−φab,for alla,b∈A.Letφ∈C2A,M, we define the second differential δ2φ∈C3A,M by(10)δ2φ=δH2φ°id3+σ1,where σ1 is the trilinear map on A3 defined by σ1a,b,c=b,a,c and δH2 denotes the Hochschild differential of degree 2 defined by(11)δH2φa,b,c=a.φb,c−φab,c+φa,bc−φa,b.c.Call anyp-form φ∈CpA,M a p-cocycle if and only if δpφ=0 and denote the subspace of p-cocycles by ZpA,M.Thep-th cohomology group HpA,M is defined to be the factor space:(12)ZpA,Mδp−1Cp−1A,M,forp≥1.Remark 1.
In Example1, if we consider V as an A1-trivial bimodule, then(13)δH2φa,b,c=φa,bc−φab,c.
It follows that(14)δ2φa,b,c=δH2φa+b,a+b,2c,=φa+b,a+b2c−φa+ba+b,2c=0.
Consequently,A1⊕V is a left alternative algebra if and only if φ∈Z2A1,V, (i.e., φ is a 2-cocycle).
Now, from our previous discussion, we obtain the main result of this paper:Proposition 2.
LetA be a left alternative algebra and consider F as a trivial A-bimodule, then the cohomology group H2A,F can be interpreted as the set of equivalent classes of one-dimensional central extensions of A (i.e., there is a bijection between H2A,F and CLA,F).Proof.
Letφ∈Z2A,F, we can associate to φ the extension(15)0⟶F⟶iA⊕F⟶πA⟶0,where the product of A⊕F is given by a,αb,β=ab,φa,b, for all a,b∈A and α,β∈F. If φ¯=φ′¯, then φ′=φ−δ1θ, with θ∈C1A,F. Moreover, φ′ defines a one-dimensional central extension:(16)0⟶F⟶iA⊕F⟶πA⟶0,where the product of A⊕F is given by(17)a,αb,β=ab,φ′a,b=ab,φa,b−δ1θa,b.
The mappingϕ:A⊕φF⟶A⊕φ′F defined by a,α⟶a,α+θa is a morphism of left alternative algebras. Indeed, let a,b∈A and α,β∈F.(18)ϕa,αb,β=ϕab,φa,b,=ab,φa,b+θab=ab,φa,b−δ1θa,b,Sinceδ1θa,b=−θab=ab,φ′a,b=a,α+θab,β+θb=ϕa,αϕb,β,which gives the equivalent between these extensions. So, cohomologous cocycles correspond to equivalent one-dimensional central extensions. We also prove that equivalent one-dimensional central extensions induce cohomologous cocycles. In order to show the subjectivity, we have to construct a 2-cocycle from a given one-dimensional central extension of A:(19)0⟶F⟶iA˜⟶πA⟶0.
This is given as follows:
There is a linear mapβ:A⟶A˜ with π°β=idA. Let φa,b=βaβb−βab, for a,b∈A. As π is a morphism of left alternative algebras, we have πφa,b=0; thus, φa,b has values in kerπ=imi=F. Then, one can check that the map φ:A×A⟶F is a 2-cocycle. We thus obtain(20)H2A,F≅CLA,F.



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