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