There is a common way described in Weinberg's Quantum Field Theory Vol.1 (P83) for introducing the concept of the central charge which I find difficult to grasp. This method uses a unitary projective representation $U(g)$ of a Lie group $G$.
$$U(g_1)U(g_2)=e^{i \phi(g_1,g_2)}U(g_1g_2)\tag{1}$$
Local coordinates $\{x^a \}$ near the identity element yield $$g1\cdot g2=g(x_1).g(x_2)=g(x_3(x_1,x_2))$$ with $$x^a_3(x_1,x_2)=x^a_1+x^a_2+\gamma^{abc}{x^b_1}x_2^c+\cdots\tag{2}$$ $$\phi(g_1,g_2)\equiv\phi(x_1,x_2)=\gamma^{bc}x_1^bx_2^c+\cdots\tag{3}$$ $$U(g(x))=1+ix^aT^a+\frac{1}{2}x^a x^b T^{ab}+\cdots\tag{4}$$ where $T^a$ is Hermitian and $T^{ab}=T^{ba}$.
Substituting equations $(2)$, $(3)$, and $(4)$ into equation $(1)$ yields: $$-T^cT^b= i \gamma^{cb}1+i\gamma^{acb}T^a+T^{cb}\tag{5}$$
By defining, $$f^{abc}\equiv \gamma^{acb}-\gamma^{abc} \quad f^{bc}\equiv \gamma^{cb}-\gamma^{bc}$$ $$[T^b,T^c]=i f^{abc}T^a+i f^{bc}1.\tag{6}$$ the term $f^{bc}$ is referred to as the central charge.
My questions are as follows:
This derivation heavily relies on coordinates and representations which I find unfamiliar. From what I understand, given a Lie group $G$, $T^a$ should be the tangent vector at the identity element, that is, $T^a \in T_e G = \mathfrak{g}$. The commutator of a Lie algebra should remain within the Lie algebra. Why then does the term $i f^{bc}1$ appear in equation $(6)$, given that $1$ is not an element of $\mathfrak{g}$?
The text appears to discuss a specific representation (1). For a given abstract Lie group $G$, the term $e^{i \phi(g_1,g_2)}$ cannot appear in the group product. Only after a projective representation (1) is found and $\phi(g_1,g_2)$ is non-zero, can the central charge be defined in this manner. However, the text also states that for a simply-connected Lie group, a projective representation can exist when the central charge of the Lie algebra is nontrivial. Isn't there a circular argument here?
Or perhaps the central charge is defined specifically for certain representations. If so, what are the necessary and sufficient conditions for a Lie algebra $\mathfrak{g}$ to have a representation with a nontrivial central charge?
- What would be the geometric definition of a central charge? Surely there must exist a definition of central charge that does not depend on the representation or coordinates.