The Universal Covering Group of U(n) and Projective Representations

Abstract

Using fibre bundle theory we construct the universal covering group of U(n), $\tilde{U}(n)$, and show that $\tilde{U}(n)$ is isomorphic to the semidirect product $SU(n)\bigcirc {\scriptstyle s}$ R. We give a bijection between the set of projective representations of U(n) and the set of equivalence classes of certain unitary representations of $SU(n)\bigcirc {\scriptstyle s}$ R. Applying Bargmann's theorem, we give explicit expressions for the liftings of projective representations of U(n) to unitary representations of $SU(n)\bigcirc {\scriptstyle s}$ R. For completeness, we discuss the topological and group theoretical relations between U(n), SU(n), U(1) and Z_n.