1. Introduction

Throughout this paper, In 2004 Hall B. C. [1] wrote a book of Lie group for manifold theory and the relationship between Lie groups and Lie algebras. The reason of studying the representation is that a representation can be thought of as an action of group on some vector space. Such actions (representations) arise naturally in many branches of both mathematics and physics [5], [6], and it is important to understand them.

In [1], the Schur's lemma introduced the concept of action of Lie algebra on the space of linear maps from which denoted by, also introduce the concept of action on tensor product of two representation of Lie algebra.

Schur's lemma state: Suppose that and are representation of lie algebra acting.

On finite – dimensional space and, respectively. Define an action of g on for all x∊g and f . and, as equivalence of representation.

In [4], T. H. Majeed study the AAC of Lie group on and translation it to AAC _Lie algebra on.

In this paper we will present and study the concept of action on and the equivalent relation with the tensor product space. since is a vector space of all linear functional from, so is also vector space of all linear functional from into then the representation of G acting on this vector is action of G on this Hom space. Also we give an equivalent relation between AC_Lie group and AAC_Lie group on Hom and AC_Lie group with AAC_Lie group on Tensor products, and explain the actions structure by using diagram.

2. Basic Definition

Definition (2.1) [2]: A Lie group G is a finite dimensional smooth manifold G together with a group structure on G, such that the multiplication and the attaching of an inverse are smooth maps.

Definition (2.2) [3]: A matrix Lie group is any subgroup with the following property. If is any sequence of matrices in G and A_m converges to some matrix A then A ∈ G, or A is not invertible.

Definition (2.3) [7]: A finite-dimensional real (complex) representation of G is a Lie group homomorphism Generally, a Lie group homomorphism where V is a finite dimensional real (complex) vector space with dim V ≥ 1.

Definition (2.4) [7]: Let G and H are two Lie groups. A map f from G to H is called a Lie group homomorphism if f is a group homomorphism and -map on H.

Definition (2.5) [1]: If U and V are finite dimensional real or complex vector spaces, then a tensor product of U and V is a vector space W, together with a bilinear map with the following property: If ψ is any bilinear map of into a vector space W₁, then there exists a unique linear map of W into W₁, such that the following diagram commutes:

3. The Action of Lie Group on Hom Space

In this section we define an action of G on and give the equivalent relation with representation acting on the tensor space.

Note (3.1):

Diagram (1)

Put the vector space of all linear maps from, where we define an action of Lie group G on which given by:

define by:

For all a∊G, and F= F2ₒ ,

.this diagram(1) will show the structure of this action.

Proposition (3.2): Let i=1, 2; be representation of lie group G acting on, then is a representation of Lie group acting on, which called AC_Lie group on

Proof:

Since

(1)

And

(2)

Then Thus by is a representation of G. The following diagram (2) shows that is a group homomorphism:

Diagram (2)

Where

Also is a continuous map since the composition of continuous map is continuous. See diagram (3).

Diagram (3)

Hence is a representation of Lie group G, since every continuous homomorphism is smooth

Example (3.3): Suppose that be a representation from G into such that

Proposition(3.4): Let i=1,2 ;be representation of lie group G acting on then is a representation of G acting on the vector space such that: for all a∊G.

Diagram (4)

The diagram (4) use to show that is a group homomorphism of G on And also it is smooth maps. The diagram (5) is to show smoothness of this representation:

Diagram (5)

The diagonal map inclusion map and bilinear map are smooth hence the composition of this are smooth. Hence is smooth representation.

Proposition (3.5): Let be a representation of G acting on k-finite dimensional vector space respectively, then the AC _lie group of G on is equivalent to the AC_lie group on.

Proof: To show that is a bilinear map, defined by:

Where is a linear map, defined by:

For all

Other for all

So is a bilinear map, thus by using the tensor product and universal property of this tensor product, we get a unique linear map

So by universal property of tensor product there exists a unique linear map this makes the above diagram commutative.

4. AAC-Lie Group on Hom and Tensor Product

In this section we construct an action on the space of all linear functional from into.

Definition (4.1): Let i=1,2,3,4 be a representation o f Lie group G acting on finite dimensional vector space for i=1,2,3,4, then the representation of G acting on form an action, defined by:

such that:

This diagram will show the structure of this representation:

Diagram (6)

Proposition (4.2): Let be a representation of G acting on then such that is a representation of G.

Proof: Firstly we must show that is a group homomorphism. The diagram (7) below is to show proving of group homomorphism.

Diagram (7)

Secondly to prove is continuous, let and be two finite vector spaces such that and be two representation acting on W1 and W2 respectively. Diagram (8) show continuity of

Diagram (8)

Hence is a representation of G, since every continuous homomorphism is smooth.

Proposition (4.3):

Let for all i=1,2,3,4, be a representation of G acting on then such that:

for all a∈G.

Is a representation of G.

Proof: Diagram (9) below illustrates the process of proof group homomorphism.

Diagram (9)

Diagram (10) below illustrates the process of continuity. Since the composition of continuous map is continuous.

Diagram (10)

Hence is a representation of G, since every continuous homomorphism is smooth.

Example (4.4): LetAnd GL(2,R), and then the AAC-lie group on hom is: