How to find the generator of this Lie group?

[Haorong Wu]

TL;DR Summary

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!

 

[fresh_42]

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 $\iota$.