Lifting Projective Representations

We investigate lifting projective representations in matrix Lie groups to unitary representations. After establishing preliminary notions, we discuss finite- and infinite-dimensional representations of matrix Lie groups, emphasizing the relationship between SU(2) and SO(3) and thus providing applications in quantum physics. The application of cohomological methods in studying the lifting of projective representations is then highlighted, along with the concept of one-dimensional central extensions. We then outline a proof of Bargmann’s theorem, which addresses the existence of lifts for possibly infinite-dimensional projective representations of Lie groups whose second group cohomology vanishes.