It's not at all obvious to me why a connected and simply connected Lie group has only single valued linear irreducible representations. This would come as a particular case to a more general statement: a connected n-fold connected Lie group has at most n-valued linear irreducible representations.
Can one prove this general statement mixing somehow the homotopy group and representation morphism? I haven't seen a proof for this, but the result is taken for granted in the physicists' community.
Thanks, BR, Daniel
I want to present Elie Cartan's argument about representations of simply connected Lie groups:
[...] it is seen that the space of the unimodular unitary group is a manifold in which each point is defined by four real numbers a1, a2, b1, b2 for which the sum of squares equals 1; i.e., it is a spherical space of three dimensions (the hyper- sphere of unit radius in Euclidean space of four dimensions). This space is simply connected in the sense that all closed contours can be reduced to a point by continuous deformation. This can easily be seen by considering the inverse of the hypersphere in four dimensions with respect to a point of itself (stereographic projection); this inverse is a three-dimensional Euclidean space (including the point at infinity). Then it can be shown that if the unimodular group had a multi-valued representation, on following the continuous variation of the representing matrix as the point in group space describes a suitable closed contour starting and finishing at some origin, the matrix would start as the unit matrix and finish as a different matrix. On continuously deforming the contour the final matrix will remain the same. But the contour can be deformed so as to reduce to one point—the origin. This gives a
contradiction.[...]