# Relation between projective representations, connectivity of a group manifold and number of equivalence classes of paths

The projective unitary representations of a multiply-connected group $G$ is defined as $$U(g_1)U(g_2)=c(g_1,g_2)U(g_1g_2)$$ where $c(g_1,g_2)$ is phase. Reading various articles, and this old post of mine, it appears to me that the topological connectivity of a multiply connected group $G$ determines the number of inequivalent projective representations in the Hilbert space i.e., the number of values $c(g_1,g_2)$ can take. And different inequivalent projective representations will be charecterised by different values of $c(g_1,g_2)$ for the same elements $g_1$ and $g_2$. For $\textrm{SO}(3)$, as I guess, $c$ takes two different values $\pm 1$.

Question From the fact that a “$n$-”ply connected group has $n$ different equivalence classes of paths, I would like to understand how does it lead to $n$ different values of $c$.

• Why do you say that it is difficult to find out the projective reps of $SO(3)$? Or rather, what do you mean by difficult here? – Ruben Verresen Jul 2 '16 at 19:58
• I have edited the question which now I think addresses your point. – SRS Jul 2 '16 at 20:11
• I explain the classification of projective representations in this Q&A of mine. Non-simply connected (but still e.g. semi-simple with $H^2(\mathfrak{g},\mathbb{R})=0$) groups are case two - projective representations of (the identity component of) a non-simply connected group are in bijection to linear representations of its simply connected universal cover - or also just to linear representations of its algebra. I'm actually not sure what exactly you're asking in this question, especially with the "difficult". – ACuriousMind Jul 2 '16 at 20:29
• @Acuriousmind: I think I will edit question 2 later. But what do you have to say about 1 and 3? – SRS Jul 2 '16 at 20:42
• For 1, read my post - projective representations are linear representations of central extensions, and being non-simply-connected means that there is a central extension - that to the simply-connected universal cover.. 3. doesn't make much sense to me - of course SU(2) has projective representations - it's just that every such representation already comes from a linear representation of it(because SU(2) doesn't have central extensions, as it is simply connected). – ACuriousMind Jul 2 '16 at 20:51