Received 29 August 2015; accepted 13 December 2015; published 16 December 2015
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1. Introduction
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 [1] -[3] (Lie Bracket, Chevalley Eilenberg Cohomology...). The principal objective of this work is to introduce the notions of deformation of Lie algebra in the more general representation rather than the adjoint representation.
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
by putting the commutator structure in place.
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.
2. 2 - 3 Matrix Representation Theorem
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
by
the space of i-multilinear skew symmetric map on V
with valor in W; we also denoted by
and
respectively the basis of V and W. We also suppose
that
is a representation of the Lie algebra
where
is the associated Lie structure.
2.1. Description of Cochain Spaces
Since element of
of
skew symmetric, then for all
, we have
[4] . Let
and
, we have
and
, where
.
implies that
.
iff
is a linear map.
Then,
![]()
Lemma 1: If the
and
, then
.
Proof. Since
then,
.
Thus, we define an isomorphic map
from
to
as follows;
. □
iff
is bilinear and antisymmetric map; then
![]()
Lemma 2: If
and
then ![]()
Proof. From the expression of an element
in
from above,
can be represented as a
column matrix
of the lie constant structures. □
iff
is a tri-linear and skew symmetric map,
then
![]()
since
is a linear anti-symmetric mapping.
Lemma 3: If
and
then
.
Proof. Since for every
, we have that
from the expression of
above. □
2.2. Diagram of a Sequence of Linear Maps
According to the above results, we have the following diagram where we shall identify and define
and
in order to contruct our 2 - 3 Matrix Chevalley-Eilenberg Cohomology.
![]()
Expression of
[1] [4] :
![]()
Expression of
[1] [4] :
![]()
![]()
Expression of
[1] [4] :
![]()
![]()
since
is mapped to the zero space. A direct computation, give us [1]
![]()
Definition of
:
![]()
![]()
i.e
is the identity mappings from W to W.
Definition of
:
![]()
![]()
which is the matrix of
,
,
and
.
Definition of
:
![]()
![]()
which is the matrix of
,
,
.
2.3. Homological Differential
In this section, we are going to determine expressions of
and also prove that
for us to obtain our 2 - 3 matrix Chevalley-Eilenberg differential complex. This is possible unless by stating an important hypothesis which we call 2 - 3 matrix Chevalley-Eilenberg hypothesis.
Proposition 1: If
for all x, y in V, then
.
Proof. We assume that
for all x, y in V.
By definition, we have that
(1)
. (2)
Then by substituting equation (1) into (2),we have
![]()
by hypothesis.
Expression of
:
Let V be a two dimensional Lie-algebra with basis
and the Lie’s bracket
where
and W a three dimensional vector space with basis
. We define
by
, where
and
is a linear mapping associated to the matrix
.
Let
defined by
![]()
Therefore;
![]()
Since
![]()
Therefore,
![]()
Also, we have
,
where
![]()
, where
![]()
So,
![]()
![]()
Therefore,
![]()
Now, we compute
where
and
are basis vectors of V.
By replacing the constants
and
, we obtain
which is given as;
![]()
Thus,
.
Hence,
is defined by
with
,
.
Corollary 1: If
![]()
then
is defined by
.
2.4. Fundametal Condition of 2 - 3 Matrix Chevalley-Eilenberg Cohomology
We now state the main hypothesis for our 2 - 3 matrix Chevalley-Eilenberg Cohomology, which we suppose that ![]()
i.e ![]()
i.e
,
and
.
This is an important tool in the construction of our 2 - 3 matrix cohomology differential complex.
2.5. Expression of ![]()
From the diagram,
![]()
![]()
![]()
![]()
![]()
where
![]()
![]()
and
. Thus, using the basis vectors
and
in V, we have
![]()
Hence, the mapping
is defined as;
![]()
Corollary 2: If ![]()
then the mapping
is defined as;
![]()
The matrix
has been assigned to the matrix
to simplify the composition of
and
.
Proposition 2:
.
Proof. Since
and
We have:
![]()
Which gives us our 2 - 3 matrix Chevalley Eilenberg homological hypothesis
. □
Remark 1: By straightforward computation, we have
![]()
2.6. Determination of the
and ![]()
.
iff
![]()
iff
(3)
(4)
(5)
Now, we compute the
using the standard basis
![]()
If
then ![]()
If
then ![]()
If
then ![]()
If
then ![]()
If
then ![]()
If
then ![]()
Thus, we have the image matrix as follows:
![]()
Next, we calculate the rank of the matrix
which will help us to know the
and
by using the dimension rank theorem of the vector spaces [5] [6] .
We now reduce the matrix
to reduce row echelon form. We then replace the entries of the matrix
by the follows constants:
![]()
where
Let
and by dividing each of the
entries of row 1 by
and carrying out the following row operation
and
, we obtain
![]()
Let
be such that
and by carrying the following row operations
,
and
, and setting
thus we obtain the following matrix.
![]()
Let
be such that
, and by carry the following row operations
,
and
. By setting
and
. Also, if we let
,
![]()
![]()
we obtain the following matrix.
.
Hence we obtain the reduce row echelon form of
of rank 3 [5] [6] .
We wish to consider now the cases of the matrix
of rank 1 and rank 2 since the case of rank Zero is trivial.
Rank 1: By setting each of the entries on row 2 and 3 of matrix A to zero, we obtain the rank of
to be 1.
Rank 2: By setting each of the entries on row 3 of matrix B to zero, we obtain the rank of
to be 2.
Proposition 3: if
,
,
![]()
then
and the
. Further
and
.
Proposition 4: From matrix A, if
, ![]()
![]()
then
and
. Further,
and
.
Proof. Since the
, we have that
, thus
. We now show that
. By the dimension rank theorem, we have that
which is
. □
Proposition 5: From matrix B, if
,
![]()
![]()
![]()
then
![]()
and
. Further,
and
.
Proof. Since the
, we have that
, thus
. We now show that
.
By the dimension rank theorem, we have that
that is ![]()
Proposition 6: if
,
,
then
![]()
and the
. Further,
and
.
Proof. Since the
, we have that
, thus
. We now show that
. By the dimension rank theorem, we have that
that is
□
Now, we compute our quotient spaces of the 2 - 3 matrix Chevalley Eilenberg cohomology which are
,
and
.
For
, we have the following quotient space:
Case 1:
and ![]()
.
For
, we have the following quotient spaces:
Case 1:
and ![]()
.
Case 2:
and ![]()
.
Case 3:
and ![]()
.
Case 4:
and ![]()
.
Case 5:
and ![]()
.
Case 6:
and ![]()
.
Case 7:
and ![]()
.
Case 1:
and ![]()
.
Case 2:
and ![]()
.
Case 3:
and ![]()
.
Case 4:
and ![]()
.
Case 5:
and ![]()
.
Case 6:
and ![]()
.
Case 1:
and ![]()
.
Case 2:
and ![]()
.
Case 3:
and ![]()
.
Case 4:
and ![]()
.
Case 5:
and ![]()
.
Case 1:
and ![]()
.
Case 2:
and ![]()
.
Case 3:
and ![]()
.
Case 4:
and ![]()
.
Case 1:
and ![]()
.
Case 2:
and ![]()
.
Case 3:
and ![]()
.
Case 1:
and ![]()
.
Case 2:
and ![]()
.
For
, we have the following quotient spaces:
Case 1:
and ![]()
.
Case 2:
and ![]()
.
Case 3:
and ![]()
.
Case 4:
and ![]()
.
We suggest that further research in this direction is to carry out the deformation on the Cohomological spaces
,
and
which are 32 in number and apply a specific example with
. We will also carry out an extensive study on the solution of our system of linear equations on the 2 - 3 matrix Chavelley Eilenberg fundamental condition.
Acknowledgements
We thank the Editor and the referee for their comments.
NOTES
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*Corresponding author.