There is a common way(Weinberg QFT Vol.1 P83) to introduce the central charge which I can't understand. Given a unitary projective representation $U(g)$ of Lie group $G$.
$$U(g_1)U(g_2)=e^{i \phi(g_1,g_2)}U(g_1g_2)\tag{1}$$
Using local coordinates $\{x^a \}$ near identity element, $g1.g2=g(x_1).g(x_2)=g(x_3(x_1,x_2))$ $$x^a_3(x_1,x_2)=x^a_1+x^a_2+\gamma^{abc}{x^b_1}x_2^c+\cdots\tag{2}$$ $$\phi(g_1,g_2)\equiv\phi(x_1,x_2)=\gamma^{bc}x_1^bx_2^c+\cdots\tag{3}$$ $$U(g(x))=1+ix^aT^a+\frac{1}{2}x^a x^b T^{ab}+\cdots\tag{4}$$ with $T^a$ Hermitian and $T^{ab}=T^{ba}$.
Substitude $(2,3,4)$ into $(1)$, $$-T^cT^b= i \gamma^{cb}1+i\gamma^{acb}T^a+T^{cb}\tag{5}$$
By defining, $$f^{abc}\equiv \gamma^{acb}-\gamma^{abc} \quad f^{bc}\equiv \gamma^{cb}-\gamma^{bc}$$ $$[T^b,T^c]=i f^{abc}T^a+i f^{bc}1\tag{6}$$ They call $f^{bc}$ as central charge.
My questions:
1.This derivation heavily relies on the coordinates and representation which is unfamiliar to me. From my knowledge, given a Lie group $G$, $T^a$ should be the tangent vector at identity element, that is $T^a \in T_e G = \mathfrak{g}$. The commutator of Lie algebra should still be in Lie algebra. Why can $i f^{bc}1$ occur in $(6)$ since $1$ is not an element in $\mathfrak{g}$.
2.It seems that they're talking about a specific representation $(1)$, because given an abstract Lie group $G$, $e^{i \phi(g_1,g_2)}$ can't occur in group product. Only after you find a projective representation $(1)$, i.e. $\phi(g_1,g_2)$ nonzero, you can define the central charge by this way. However textbook also says that for a simply-connected Lie group, it can have projective representation when the central charge of Lie algebra is nontrivial. Is there some circular argument here?
Or maybe the central charge is defined for specific representation. Then the question is the sufficient and necessary condition for a Lie algebra $\mathfrak{g}$ to have a representation with nontrivial central charge?
3.So what's the geometric definition of central charge? There should be some definition of central charge that doesn't depend on representation and coordinate.