Abstract
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 Rn. 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 continuous 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.
Highlights
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
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
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
Summary
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.). This means that the following equations hold for all a,b, c ∈ G :. If all elements of a given group commute with one another we say that this group is Abelian. A 2-simplex P0 P1P2 is defined to be a triangle with its interior included and a 3-simplex P0 P1P2 P3 is a solid tetrahedron
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