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davi2686
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I learned a lie group is a group which satisfied all the conditions of a diferentiable manifold. that is the real rigour definition or just a simplified one?
thanks
thanks
zinq said:It is possible to have a DS only on a topological space that is a topological manifold
Mathematicians have a strange kind of humor: "Convenient Category". Considering axiom 3 of Chen-spaces and the following examples I'm curious to see what will be left under so much generalization.micromass said:Uh, that is not necessarily true. But it is true that things are well-behaved only on topological manifolds. A differentiable structure can be put on very general spaces however. https://arxiv.org/abs/0807.1704
fresh_42 said:Mathematicians have a strange kind of humor: "Convenient Category". Considering axiom 3 of Chen-spaces and the following examples I'm curious to see what will be left under so much generalization.
A lie group is a type of mathematical group that is also a differentiable manifold. It combines the concepts of a group, which is a set of elements with a binary operation, and a manifold, which is a topological space that locally resembles Euclidean space.
A lie group is defined as a group that is also a differentiable manifold, meaning that the group operations are continuous and differentiable. It can be described using a set of generators and defining equations, or through its presentation as a matrix group.
Lie groups are important in mathematics and theoretical physics. They are used to study symmetries and transformations in various fields of science, including geometry, mechanics, and quantum mechanics. They also have applications in engineering, computer graphics, and cryptography.
A group is a lie group if it satisfies the four defining properties: closure, associativity, identity element, and inverse element. In addition, it must also be a differentiable manifold, meaning that it is locally similar to Euclidean space and its group operations are differentiable.
One example of a lie group is the special orthogonal group, \(SO(n)\), which is the group of all real orthogonal matrices with determinant 1. It has applications in geometry, physics, and engineering, and is an important example in the study of lie groups.