What's the geometric (or representation independent) definition of central charge of Lie algebra $\mathfrak{g}$?

Question

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.

Answer 1

The very well written summary of the topic by ACuriousMind which he linked to in the comments section should normally answer your question(s). You need to just put it all together, that is do some thinking by yourself. So don't jump to any of the texts below.

You asked me for some concrete literature on the subject which is probably also familiar to ACuriousMind - other than the source you quoted, i.e. the 2nd chapter of Weinberg I.

Let me see.

• I would start with the famous article by Valja Bargmann On Unitary Ray Representations of Continuous Groups officially at http://www.jstor.org/stable/1969831

• Then I should be prepared to do some extra reading from a diff. geom. / Lie group text such as Helgason, S. - Differential geometry, Lie groups and symmetric spaces (GSM 34, AMS, 1978) to be able to tackle the short generic monograph Parthasarathy, K. - Multipliers on Locally Compact Groups (LNM 93, Springer, 1969) and the applied monograph to the restricted Poincaré group which is Simms, D.J. - Lie Groups and Quantum Mechanics (LNM 52, Springer, 1968, ---+).

• Moving to newer literature, we have the monograph by Azcarraga, J., Izquierdo, J. - Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics (CUP, 1995) which could be the sum of all literature quoted in the other two paragraphs above and two below and the famous Geometry of Quantum Theory by Varadarajan (Chapter 7, Springer, 1968).

• A less technical approach to the Poincaré group compared to Simms is offered by B. Thaller in his book The Dirac Equation (Springer, 1992), section 2.4 which starts at page 62.

• The lightest version to the whole topic of projective representations and Lie group/Lie algebra extensions is offered by Morton Hamermesh in his book Group Theory and Its Applications to Physical Problems (Dover, 1962, Chapter 12).

• Last, but not least, it all started with the work by Chevalley and Eilenberg freely available at http://www.ams.org/journals/tran/1948-063-01/S0002-9947-1948-0024908-8/S0002-9947-1948-0024908-8.pdf