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The main objective of this paper is to provide the tool rather than the classical adjoint representation of Lie algebra; which is essential in the conception of the Chevalley Eilenberg Cohomology. We introduce the notion of representation induced by a 2 - 3 matrix. We construct the corresponding Chevalley Eilenberg differential and we compute all its cohomological groups.

This work is included in the domain of differential geometry which is the continuation of infinitesimal calculation. It is possible to study it due to the new techniques of differential calculus and the new family of topological spaces applicable as manifold. The study of Lie algebra with classical example puts in place with so many homological materials [

This work is base on 2 - 3 matrix Chevally Eilenberg Chohomology representation, in which our objective is to fixed a matrix representation and comes out with a representation which is different from the adjoint repre- sentation. Further, given a Lie algebra V, W respectively of dimension 2 and 3, we construct a linear map that will define a Lie algebra structure from a Lie algebra V into

This does lead us to a fundamental condition of our 2 - 3 matrix Chevalley Eilenberg Cohomology. We com- pute explicitly all the associated cohomological groups.

We begin by choosing V to be a 2-dimensional vector space and W a 3-dimensional vector space, then we called our cohomology on a domain vector space V and codomain W a 2 - 3 matrix Chevalley Eilenberg Cohomology. In what follow, we denoted for all

with valor in W; we also denoted by

that

Since element of

Then,

Lemma 1: If the

Proof. Since

Thus, we define an isomorphic map

Lemma 2: If

Proof. From the expression of an element

column matrix

then

since

Lemma 3: If

Proof. Since for every

According to the above results, we have the following diagram where we shall identify and define

Expression of

Expression of

Expression of

since

Definition of

i.e

Definition of

which is the matrix of

Definition of

which is the matrix of

In this section, we are going to determine expressions of

Proposition 1: If

Proof. We assume that

By definition, we have that

Then by substituting equation (1) into (2),we have

by hypothesis.

Expression of

Let V be a two dimensional Lie-algebra with basis

Let

Therefore;

Since

Therefore,

Also, we have

So,

Therefore,

Now, we compute

By replacing the constants

Thus,

Hence,

Corollary 1: If

then

We now state the main hypothesis for our 2 - 3 matrix Chevalley-Eilenberg Cohomology, which we suppose that

i.e

i.e

This is an important tool in the construction of our 2 - 3 matrix cohomology differential complex.

From the diagram,

where

Hence, the mapping

Corollary 2: If

then the mapping

The matrix

Proposition 2:

Proof. Since

Which gives us our 2 - 3 matrix Chevalley Eilenberg homological hypothesis

Remark 1: By straightforward computation, we have

iff

Now, we compute the

If

If

If

If

If

If

Thus, we have the image matrix as follows:

Next, we calculate the rank of the matrix

We now reduce the matrix

where

entries of row 1 by

Let

Let

we obtain the following matrix.

Hence we obtain the reduce row echelon form of

We wish to consider now the cases of the matrix

Rank 1: By setting each of the entries on row 2 and 3 of matrix A to zero, we obtain the rank of

Rank 2: By setting each of the entries on row 3 of matrix B to zero, we obtain the rank of

Proposition 3: if

then

Proposition 4: From matrix A, if

Proof. Since the

Proposition 5: From matrix B, if

and

Proof. Since the

By the dimension rank theorem, we have that

Proposition 6: if

and the

Proof. Since the

Now, we compute our quotient spaces of the 2 - 3 matrix Chevalley Eilenberg cohomology which are

For

Case 1:

For

Case 1:

Case 2:

Case 3:

Case 4:

Case 5:

Case 6:

Case 7:

Case 1:

Case 2:

Case 3:

Case 4:

Case 5:

Case 6:

Case 1:

Case 2:

Case 3:

Case 4:

Case 5:

Case 1:

Case 2:

Case 3:

Case 4:

Case 1:

Case 2:

Case 3:

Case 1:

Case 2:

For

Case 1:

Case 2:

Case 3:

Case 4:

We suggest that further research in this direction is to carry out the deformation on the Cohomological spaces

We thank the Editor and the referee for their comments.

JosephDongho,EpizitoneDuebe-Abi,Shuntah RolandYotcha, (2015) On 2 - 3 Matrix Chevalley Eilenberg Cohomology. Advances in Pure Mathematics,05,835-849. doi: 10.4236/apm.2015.514078