The Action of Hesienberg Group on Finite Dimention Manifold

Our goal in this paper is introduce the action of hesienberg group on finite dimention manifold M. the interrelation ship between hesienberg group and the finite dimention manifold M is an old and vast subject. To simplify this treatment we work with liealgebra defined on a finite dimention manifold M, the heisenberg group forned an action over these liealgebra on finite manifold M. No doubt, anotion on the heisnberg group can constitute avery important situation in the a differential finite manifold M, therefore, our work presents a key role mainly in some properties and characteriztions of the action of heinberg group on finite dimention manifold M, and also we study characterization on the relation between heisnberg group and finite dimention manifold M, then we introduce an an action of heisnberg group by the tensor product of the two repersentation which are (Acolyte groups) on hom (V2, V1’) be the tensor product of two representations of heisberg group and construct the definition of AC-heisnberg group, also study the properties of this action.


Introduction
The Heis (R) group is the group of 3*3 upper triangular matrices of the form [2], under the operation of matrix multiplication elements x,x′ and x″ can be taken from any commutative ring with identity [3], [4]. In the three-dimensional case, the product of two heisenberg matrices is given by: Since the multication is not commutative, the group is non-abelian [5], [6].
The neutral element of heis (R) group is the identity matrix and inverse given by: The author in [7] the pursue examples -Higher dimensions.
-Heis (R) group modulo an odd prime P.
Adifferential manifold M can be described by a collection of charts, named an atlas [10].
In differential geometry, the heis(R) group action on a finite dimention manifold M is a group action by lie algebra group G on M that is a differentiable map action with liealgebra group action G [11].
We will define an action of heis(R) group on hom (V 2 , V 1 ′) by:
From the following diagram we show that γ is a group homomrph is m:

Example (2.3):
Suppose that γ1 is a representation from Lie algebra in to Heis(R) group, such that:  From the diagram γ is agroup homomorphism of Heis(R) group on GL (V 2 ′ * V 1 ′) and it is smooth map.

Proposition (2.5):
Let γ 1 and γ 2 bbe two representation of Heis(R) group acting on k-finite dimensional vector space V 1` , V 2 respectively then the AC-Heis(R) group of G on Hom (V 2 ,V 1`) are equivalent to the AC-Heis(R) group on V 2` * V 1`