I was reading the second chapter of the first volume of Weinberg's books on QFT. I am quite confused by the way he derives the Lie algebra of a connected Lie group.
He starts with a connected Lie group of symmetry transformations and then consider how it would be represented on the Hilbert space. Then, he says that as it is a symmetry transformation and because it is connected to identity continuously (because it is locally path connected), the transformation U(T) corresponding to a particular symmetry transformation T must be unitary by Wigner's theorem. So, he takes a group element close to identity and expand it as a power series and considers their compositions to derive at the commutation relations for the operators representing generators of the Lie algebra.
Much of this can be followed on pages 53-55. My doubts and confusions are as follows:
Does this give a Lie algebra of the underlying group of Ts or the group induced on Hilbert space of the U(T)s?
Is it that we presuppose a Lie algebra with some generators already out there and think of all this stuff on Hilbert space as its representation and then derive the commutation relations of the representation of generators to get the abstract Lie product relations of Lie algebra? If yes, then are the Lie algebras of Ts and U(T)s same or different? Please be precise on this.
If all the above is true, then why do we need specifically representations on Hilbert space? Any representation would do, right? And where does all this talk about unitarity of representations used in the following derivation of the Lie algebra? I can't see any application of Wigner's theorem in what follows. I think that all that follows can be done on the representation on any space. But then this brings us to the question as to if a representation of the group exists on that vector space? Is it why we need Wigner's theorem and Hilbert space? Then, why don't we take the representations on 4D space-time itself where a representation exists trivially by definition? If this is true, how do we go in general about finding about what kind of representations can exist for a Lie algebra?
In general, provided with a Lie group, how does a mathematician go on to derive its Lie algebra? Should the Lie group be connected? Can Lie algebras be defined for Lie groups which are not connected?