In this very good P.E. answer, it is explained precisely what it means for a quantum system/theory to have a symmetry group $G$ (where $G$ is a Lie group): going back to first principles, it means that the Hilbert space $\mathcal H$ is equipped with a projective unitary representation of $G$, ie a morphism $G\to \operatorname{PU}(\mathcal H)$. We can then work out exactly what this means for the Hilbert space itself, in terms of linear representations, encountering central extensions of $G$ (some of which can be detected at the level of the Lie algebra, other are coverings).
My question: what is the corresponding line of reasoning for the gauge group? Working from first principles as much as possible, how does the global structure of the gauge group factor in defining the QFT?
I know that to get a positive energy from the Yang-Mills lagrangian, there needs to be a positive definite bilinear form on the Lie algebra, which implies that it is the direct product of an abelian factor and a compact semi-simple Lie algebra.
