How to find the generator of this Lie group?

The ##n## coordinates are ##X^\iota##, and we have ##n## basis vectors ##\frac{\partial}{\partial X^\iota}##. In summary, the generator of the Lie group is ##\xi^\iota_i(X^\kappa) \frac{\partial}{\partial X^\iota}##, as stated in the paper.
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Haorong Wu
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How to find the generator of this Lie group?
Hello, there. Consider a Lie group operating in a space with points ##X^\iota##. Its elements ##\gamma [ N^i]## are labeled by continuous parameters ##N^i##. Let the action of the group on the space be ##\gamma [N^i] X^\iota=\bar X^\iota (X^\kappa, N^i)##. Then the infinitesimal transformation is given by $$\gamma[\delta N^i]X^\iota=X^\iota+\left . \frac {\partial \bar X^\iota (X^\kappa, N^i)}{\partial N^i} \right |_{N^i=0}\delta N^i=X^\iota+\xi^\iota_i(X^\kappa) \delta N^i$$ in the neighborhood of the identity ##N^i=0##. According to the paper, the generators of the Lie group is ##X_i=\xi^\iota_i (X^\kappa) \frac \partial {\partial X^\iota}## I do not see how to get this conclusion. I thought the genrator should be ##\xi^\iota_i(X^\kappa) ## itself. Is it related to the fact that the genrator is a vector, and ##\frac \partial {\partial X^\iota}## is merely the basis vector as in the differential geometry?

Thanks!
 
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Haorong Wu said:
Summary:: How to find the generator of this Lie group?

Is it related to the fact that the generator is a vector, and ##\frac \partial {\partial X^\iota}## is merely the basis vector as in the differential geometry?
Yes. We have to have ##n## coordinates over all ##\iota##.
 
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Related to How to find the generator of this Lie group?

1. What is a Lie group?

A Lie group is a mathematical concept that combines the ideas of a group (a set with a binary operation) and a smooth manifold (a space that looks locally like Euclidean space).

2. How do you find the generator of a Lie group?

The generator of a Lie group is found by taking the derivative of the group's elements at the identity element. This can be done using the Lie algebra, which is a vector space that encodes the group's structure.

3. What is the significance of finding the generator of a Lie group?

The generator of a Lie group is important because it allows us to understand the group's structure and properties. It also allows us to perform calculations and make predictions about the group's behavior.

4. Are there different methods for finding the generator of a Lie group?

Yes, there are several methods for finding the generator of a Lie group, including using the Lie algebra, Lie bracket, and exponential map. The choice of method depends on the specific group and its properties.

5. Can the generator of a Lie group be used to solve problems in other fields?

Yes, the concept of a Lie group and its generator has applications in many areas of mathematics and physics, including differential geometry, quantum mechanics, and relativity. It can also be used in practical applications such as robotics and control theory.

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