Lie groups of bundle automorphisms and their extensions

We describe natural abelian extensions of the Lie algebra $\text{aut}(P)$ of infinitesimal automorphisms of a principal bundle over a compact manifold $M$ and discuss their integrability to corresponding Lie group extensions. Already the case of a trivial bundle $P=M\times K$ is quite interesting. In this case, we show that essentially all central extensions of the gauge algebra $C^{\mathrm{\infty }}(M,\text{fk})$ can be obtained from three fundamental types of cocycles with values in one of the spaces $\text{fz}:=C^{\mathrm{\infty }}(M,V)$, ${\mathrm{\Omega }}^1(M,V)$ and ${\mathrm{\Omega }}^1(M,V)\mathrm{/}\text{dd}{C}^{\mathrm{\infty }}(M,V)$. These cocycles extend to $\text{aut}(P)$, and, under the assumption that $TM$ is trivial, we also describe the space $H^2(\mathcal{V}(M),\text{fz})$ classifying the twists of these extensions. We then show that all fundamental types have natural generalizations to non-trivial bundles and explain under which conditions they extend to $\text{aut}(P)$ and integrate to global Lie group extensions.