# Construction of any projective representation of SU(2)

i) Can one point out any explicit construction of a projective representation of SU(2)? If yes, please give a summary and provide a Ref. If not, please state the reason and also provide a Ref.

ii) How is i) related to the fact that any group $A$ such that SU(2) is a quotient group for $A$, or SU(2) is a normal subgroup of $A$?

• For the sake of an interested topologist, what do you mean by a projective representation?
– user98602
Commented Mar 10, 2017 at 21:46
• I may be wrong, but I think it means that finding out the existence of matrix representation $\rho(g)$ where $g \in G=SU(2)$, such that $\rho(g1)\rho(g2)=\alpha(g1,g2)\rho(g1 \cdot g2)$, where $\alpha(g1,g2)\in H^2(G, \mathbb{R}/\mathbb{Z})$ is a 2-cocycle in cohomology group of $G=SU(2)$. Commented Mar 10, 2017 at 22:22
• This is correct. In less fancy language: in an ordinary (linear) representation we have that $\rho(g1)\rho(g2) = \rho(g1g2)$. (This more or less the definition of representation.) In a projection representation we loosen up this criterion and demand it only to hold 'up to multiplication by a non-zero scalar'. Equivalently $\rho$ itself is also only defined up to scalar multiplication. In other words: in a representation $\rho$ is a homomorphism to the group of linear transformations of a vector space $V$, in a projective representation $\rho$ is 'only' a map to the symmetry group of $P(V)$. Commented Sep 27, 2017 at 15:06

Every finite dimensional unitary projective representation of $SU(2)$ is linear. It's because $SU(2)$ is simply connected and it's a key fact to classification of $SO(3)$ projective representation: $1 \to \mathbf{Z}_2 \to SU(2) \to SO(3) \to 1$ is exact sequence, i.e., $SU(2)$ is a central extension of $SO(3)$ (there is a relation with spin in physics). See Bargmann's paper, section 3.
Edit: ok, it's not linear, it's equivalente to a linear representation (in the sense that you can choose another representatives to $PGL(V)$ in order to make $\rho: G \to GL(V)$ a group homomorphism). In another terms, there is a global associated 2-cocycle that is cohomologous to the trivial one. We need the topological hypothesis to globalizate the cocycles and representatives. In general, all finite dimensional unitary projective representation are (equivalente to) linear in a neighborhood of identity $e$. If $G$ is path connected and simples connected we can globalizate it.