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- 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!

Thanks!