Operations and Actions of Lie Groups on Manifolds

As recounted in this paper, the idea of groups is one that has evolved from some very intuitive concepts. We can do binary operations like adding or multiplying two elements and also binary operations like taking the square root of an element (in this case the result is not always in the set). In this paper, we aim to find the operations and actions of Lie groups on manifolds. These actions can be applied to the matrix group and Bi-invariant forms of Lie groups and to generalize the eigenvalues and eigenfunctions of differential operators on n  . A Lie group is a group as well as differentiable manifold, with the property that the group operations are compatible with the smooth structure on which group manipulations, product and inverse, are distinct. It plays an extremely important role in the theory of fiber bundles and also finds vast applications in physics. It represents the best-developed theory of conti-nuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. Here we did work flat out to represent the mathematical aspects of Lie groups on manifolds.


Introduction
In the present era, the study of the group related to the Lie group is essential for the sake of its comprehensive applications in several fields. Through the mathematical analysis, representations of groups on the manifold are vital due to it al-lows many group-theoretic problems in form of linear algebra problems, which is well perceived. However, Lie groups were significantly studied by Marius Sophus Lie (1842-1899), who manipulated to solve ordinary differential equations. Lie groups and Lie algebras, together with acquainted Lie theory which plays an effective role in the branch of pure and applied mathematics that is utilized in modern physics as well as an active area of research.
By the literature view, we can observe that many researchers have endeavored to analyze and discuss the importance of operations and actions of Lie groups on manifolds in various fields. In [1], Nahm et al. defined strictly semi-simple graded Lie algebras and depicted to have properties similar to semi-simple Lie algebras, presented their classification. Cahen et al. [2], proved the existence of a * product on the cotangent bundle of a parallelizable manifold M. Besides, Gelfand et al. [3], presented some important nonlocal representations of the group Gx consisting of C-mappings of a Riemannian Manifold X to a compact semisimple Lie Group G are constructed. The irreducibility, as well as non-equivalence of the introduced representations corresponding to different Riemannian metrics, is also proved in their paper. Moreover, in [4], Albeverio et al. constructed a unitary representation of the Sobolev-Lie group of C, mappings from an orientable Riemann manifold M to a Lie Group G with compact Lie algebra. The representation is given in terms of the energy function on Cj(M, G) and provides a new type of representations of current algebras. Celledoni et al. [5] showed how the methods look when applied to the rigid body equations in particular and indicated how the methods work in general. In their paper, they also discussed how the computation of these maps can be optimized for the rigid body case, and provided numerical experiments which give an idea of the performance of Lie group methods compared to other known integration schemes. In [6], Gepner et al. addressed a number of issues concerning affine Lie algebras and string propagation on group manifolds. They also showed that a 1 + 1 dimensional quantum field theory which gives a realization of current algebra. Besides, Iserles et al. [7] represented the solution as an infinite series whose terms are indexed by binary trees, by building upon an earlier work of Wilhelm Magnus. In [8], Goldman et al. studied one large class of groups: fundamental groups of compact Kahler manifolds; and deduced a general local property of their varieties of representations near representations satisfying fairly general conditions. What is more, Iserles et al. [9] surveyed the novel theory of numerical integrators that respect Lie-group structure, highlighting theory, algorithmic issues and various applications. In [10], Jean Gallier et al. introduced a framework based on Riemannian Geometry for defining some basic statistical notions on curved spaces and to make a relation between Lie groups and manifolds. Also, Neeb et al. [11] reported on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modelled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. In [12], Ferreira et al. presented lecture notes based on elements of group theory, lie groups and Lie alge-bras, representation theory of Lie algebras.
In the present tasks, we discussed the Group and Abelian groups also Subgroups with the Lie groups based on the smooth manifolds that assist us to understand the properties and actions of the Lie groups on Manifolds. We also reveal some example which shows a clear view of operations and actions of Lie groups on Manifolds.
The paper is organized as follows: in Sections 2 and 3, we introduce an overview of the group, Abelian group and the Lie group that are used in the next Section. Section 4 shows smooth mapping on Lie group with the manifold. In Section 5 includes a conclusion and recommendations of our works.

Introduction
Group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces.
Agroup is a set G equipped with a binary operation: G G G × → that associates an element a b G ⋅ ∈ to every pair of elements , a b G ∈ , and having the following properties:· is associative, has an identity element e G ∈ , and every element in G is invertible (w.r.t.). More explicitly, this means that the following equations hold for all , , a b c G ∈ : (inverse). Otherwise, we say two elements, 1 g and 2 g of a group commute with each other if their product is independent of the order, i.e., if 1 2 .
If all elements of a given group commute with one another then we say that this group is Abelian. The real numbers under addition or multiplication (without zero) form an Abelian group. The cyclic groups Z n are Abelian for any n. The symmetric group S n is not Abelian for 2 n > , but it is Abelian for n = 2. Simplexes are building blocks of a polyhedron. As shown in Figure 1, a 0-simplex 0 P is a point or a vertex, and a 1-simplex 1 0 P P is a line or an edge. A 2-simplex 0 1 2 P P P is defined to be a triangle with its interior included and a 3-simplex 0 1 2 3 P P P P is a solid tetrahedron.

Example of Group
Let us consider some groups of order two, i.e., with two elements. The elements 0 and 1 form a group under addition modulo 2. We have 0 0 0; 0 1 1; 1 0 1; 1 1 0 The elements 1 and −1 also form a group but under multiplication. We have ( ) ( ) ( ) The symmetric group of degree 2, S 2 has two which is shown in Figure 2.

That satisfies
, , e e e e a a e a a a e ⋅ = ⋅ = ⋅ = ⋅ = these three examples of groups are different realizations of the same abstract group. If we make the identifications as shown above. We see that the structure of these groups is the same.
In Figure 3, we say that these groups are isomorphic.

Isomorphism and Homomorphism of Group
Two groups G and G' are isomorphic if their elements can be put into a one-to-one correspondence which is preserved under the composition laws of the groups. The mapping between these two groups is called an isomorphism. Homomorphism 2.3.2.
A group G being mapped into another group G' but not in a one-to-one manner, i.e. two or more elements of G are mapped into just one element of G'.
If such mapping respects the product law of the groups we say they are homomorphic. The mapping is then called a homomorphism between G and G'.
From Figure 4, let G and G' be Abelian groups. A map : f G G′ → is said to be a homomorphism if   is a homomorphism between Z 6 and Z 2 .

Subgroup
Given a group G, a subset H of G is a subgroup of G iff 1) The identity element e of G also belongs to H ( e H ∈ ); 2) For all 1  A subset H of a group G which satisfies the group postulates under the same composition law used for G, is said to be a subgroup of G. The identity element and the whole group G itself are subgroups of G. They are called improper subgroups. All other subgroups of a group G are called proper subgroups. If H is a subgroup of G, and K a subgroup of H, then K is a subgroup of G.

Cosets
If H is a subgroup of G and g G ∈ is any element, the sets of the form gH are called left cosets of H in G and the sets of the form Hg are called right cosets of H in G.
Left cosets (resp. right cosets) of H induce an equivalence relation defined as follows: For all 1 2 , g g G ∈ , 1 2 g g iff 1 2 g H g H = (resp. 1 2 g g iff 1 2 Hg Hg = Obviously, is an equivalence relation.
Given a group G and a subgroup H of G we can divide the group G into disjoint sets such that any two elements of a given set differ by an element of H multiplied from the right. That is, we construct the sets gH ≡ {All elements gh of G such that h is any element of H and g is a fixed element of G}.

Theorem
The order of a subgroup of a finite group is a divisor of the order of the group. In addition, the coset space G/H, for the case in which H is invariant, has the structure of a group and it is called the factor group or the quotient group. To show this, we consider the product of two elements of two different cosets. We get The inverse element is This completes the proof of the theorem.

Lie Groups
Let G is a differentiable manifold which is endowed with a group structure such that the group operations.
It can be shown that G has a unique analytic structure with which the product and the inverse operations are written as a convergent power series.
The unit element of a Lie group is written as e. The dimension of a Lie Group G is defined to be the dimension of G of a manifold. The product symbol may be omitted and 1 2 g g ⋅ is usually written as 1 2 g g . If the product is commutative, namely 1 2 1 2 g g g g ⋅ = ⋅ , we often use the additive symbol + instead of the product symbol·.  A real Lie group is called a compact Lie group (or connected, simply connected Lie group, etc) if its underlying topological space is compact (or connected, simply connected, etc.).

Some Basic Examples of Lie Groups
1) The real line  with its standard smooth structure and the group operation being addition is a Lie group. So is every Cartesian space n  with the component wise addition of real numbers.
2) The quotient of  by the subgroup of integers →   is the circle 3) The automorphism group of any Lie group is canonically itself a Lie group.
4) The general linear group GL(n). 5) The orthogonal group O(n) and special orthogonal group SO(n).

Theorem
The Special Linear group SL(n) is a Lie Group.
Proof: Let  ∈ . We can take the matrix Consequently, SL(n) is a sub manifold of GL(n). Therefore, the group multiplication and inversion are differentiable, so SL(n) is a Lie Group.

Definition
A Lie group is a smooth manifold whose underlying set of elements is equipped with the structure of a group such that the group multiplication and inverse-assigning functions are smooth functions. The first order infinitesimal approximation to a Lie group is its Lie algebra.
By Figure 6, it is obvious that a Lie group is a smooth manifold such that every connected finite-dimensional real Lie group is homeomorphic to a product of a compact Lie group (its maximal compact subgroup) and a Euclidean space.

Operations of Lie Group on the Manifold
We suppose that this Lie group operation which may be considered as a mapping is smooth and also that the map With each element x in G there is associated a transformation L x of G called left translation: Let denote the unit element of G. the left translation

Example
We shall determine the local structure of all one-dimensional groups.
Solution: Let t be a parameter on G, chosen so that 0 t = is the identity e. Let σ be a nontrivial left invariant one form; locally, We integrate σ to get a new parameter for G,  Hence, G is Abelian (commutative).

The Action of Lie Groups on the Manifold
Let G denote a Lie group as the unit element of which is 1 and the Lie algebra of American Journal of Computational Mathematics

Conclusion
Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. From this paper, we have seen that Lie group embodies three different forms of mathematical structure. Firstly, it has a group structure. Secondly, the elements of this group also form a "topological space" so that it may be described as being a special case of "topological group". Finally, the elements also constitute an "analytic manifold". More generally Lie groups are differentiable manifolds which have very important consequences on Manifolds which are locally Euclidean spaces. We have summarized that by using differentiable structure, we can approximate the neighbourhood of any point of a Lie group by a Euclidean space which is the tangent space to the Lie group at that particular point. This approximation is some sort of local linearization of the Lie group.