Central Extension of Left Alternative Algebras and the Second Cohomology Group

Abstract

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

1. Introduction

In this paper, we consider finite-dimensional alternative algebras over an algebraically closed field of characteristic 0. An algebra is left alternative (resp. right alternative) if it satisfies the left alternative law (resp. the right alternative law) (resp. ), for all . An alternative algebra is one which is both left and right alternative. For a general theory about alternative algebras, refer [1–6].

If is left alternative algebra, then the algebra defined on the same vector space with “opposite” multiplication 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 (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 and central extensions of left alternative algebra, and then we prove that the second cohomology group of a left alternative algebra can be interpreted as the set of classes of one-dimensional central extensions of .

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. Let be a left alternative algebra. A birepresentation of is a triple , where is a -vector space and and are two linear maps satisfying(i)(ii)whereIn this case, is called bimodule of . We denote (resp. ) by (resp. ).

Proposition 1. Let be a left alternative algebra and be a bimodule of . The direct summand is made into a left alternative algebra, by defining multiplication as follows:for all and .

Proof. We show that the left alternative law is satisfied.
Let and .Therefore, is a left alternative algebra.

3. Central Extensions of Left Alternative Algebras

Definition 2. Let be a left alternative algebra; a one-dimensional central extension of is an exact sequence of left alternative algebras:such that .
We denote by the center of and by the Kernel of .
Two such extensions and are called equivalent if there is a morphism of left alternative algebras such that the following diagram is commutative:Denote by the set of equivalent classes of one-dimensional central extensions of .

Example 1. Let be a left alternative algebra, a vector space, and be a bilinear map such thatWe define the following product on the vector space direct sum :We can check easily that is a left alternative algebra (see Example 1 in [8]), and further we obtain an exact sequence:Moreover, we have , so the extension is central and called the central extension of by via .

4. Interpretation of

Now, we recall the cohomology theory for left alternative algebra introduced by M. Elhamdadi and A. Makhlouf [7].

Let be a left alternative algebra and an -bimodule. If , a -cochain of with values in are a -linear mapping of in . We denote by the space of the -cochains of . For , we put and .

For , we define the first differential by

Let , we define the second differential bywhere is the trilinear map on defined by and denotes the Hochschild differential of degree 2 defined by

Call any -form a -cocycle if and only if and denote the subspace of -cocycles by .

The -th cohomology group is defined to be the factor space:

Remark 1. In Example 1, if we consider as an -trivial bimodule, thenIt follows thatConsequently, is a left alternative algebra if and only if , (i.e., is a 2-cocycle).
Now, from our previous discussion, we obtain the main result of this paper:

Proposition 2. Let be a left alternative algebra and consider as a trivial -bimodule, then the cohomology group can be interpreted as the set of equivalent classes of one-dimensional central extensions of (i.e., there is a bijection between and ).

Proof. Let , we can associate to the extensionwhere the product of is given by , for all and . If , then , with . Moreover, defines a one-dimensional central extension:where the product of is given byThe mapping defined by is a morphism of left alternative algebras. Indeed, let and .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 :This is given as follows:
There is a linear map with . Let , for . As is a morphism of left alternative algebras, we have ; thus, has values in . Then, one can check that the map is a 2-cocycle. We thus obtain

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.