What is the connection between the generator of a Lie group, the element of the group, and the exponential?

Question

(I'm not a mathematician, unfortunately, and I don't really master my own question. I have looked at all questions close to my question, but I could not understand anything because the level is much too high in mathematics.).

*Let's consider some generators g of a Lie group G. What are its elements ?

*If the group would not be a a Lie group, would the elements be different ones ?

*What is the connection between the generator of a Lie group, the element of the group, and the exponential function ?

*If the group is not a Lie group, is the expential disappearing ?

Answer 1

Your questions 1 and 3 have a reasonable answer, but 2 and 4 are utterly meaningless and I will not discuss these. I will assume that you are familiar with some elementary linear algebra, more precisely, the notion of a basis of a vector space.

1. It's an old-fashioned terminology that mathematicians prefer to avoid but which you frequently find in physics literature:

If $G$ is a Lie group with the Lie algebra ${\mathfrak g}$ and $\{X_1,...,X_n\}$ is a basis of ${\mathfrak g}$, then the elements $X_1,...,X_n$ are called generators of $G$. See for instance this answer.

However, one should avoid saying that an element of a Lie algebra is a generator of the Lie group. Instead, talk about a generating set.

3. Suppose now that $\exp: {\mathfrak g}\to G$ is the exponential map. Then for every element $X\in {\mathfrak g}$, $\exp(X)$ is an element of the group $G$. The same, of course, applies to generating sets: $$ \exp(X_1),..., \exp(X_n) $$ are elements of $G$. Conversely, if $g\in G$ is sufficiently close to the identity element, then there exist $X\in {\mathfrak g}$ such that $\exp(X)=g$. In terms of the basis $\{X_1,...,X_n\}$, there are real numbers $t_1,...,t_n$ (if your Lie group is real) such that $$ \exp(t_1X_1+...+t_nX_n)=g. $$ Lastly, if $G$ is compact and connected, then the above claim holds for all elements of $G$ regardless of how far they are from the identity element. This answers Question 3 about the relation.