Let $\mathbb{H}$ be a Hilbert space and $\mathbb{P}(\mathbb{H})$ the projective space of one dimensional linear subspaces of $\mathbb{H}$, that is,
$$\mathbb{P}(\mathbb{H}):=\mathbb{H}\backslash\{0\}/\sim$$
where $f\sim g$ if there is $\lambda \in \mathbb{C}^{\times}$ such that $f=\lambda g$. Let $U(\mathbb{H})$ the group of unitary operators over $\mathbb{H}$ and let $\gamma:\mathbb{H}\backslash\{0\}\to \mathbb{P}(\mathbb{H})$ be the canonical map into the quotient space. For every $A \in U(\mathbb{H})$ we define a map $\hat{\gamma}:\mathbb{P}(\mathbb{H})\to \mathbb{P}(\mathbb{H})$ by:
$$\hat{\gamma}(A)(\varphi)=\gamma (A(f))$$
for every $\varphi=\gamma(f)\in\mathbb{P}(\mathbb{H})$ and $f\in \mathbb{H}$. We define $U(\mathbb{P}):= \hat{\gamma}(U(\mathbb{H}))$. In particular $U(\mathbb{P})$ is a subgroup of ${\rm Aut}(\mathbb{P}(\mathbb{H}))$.
We then have the following theorem (c.f. for instance (1) Thm 3.10): If $G$ is a group and $T:G\to U(\mathbb{P})$ is a group homomorphism, then there is a central extension $E$ by $U(1)$ and a homomorphism $S:G \to U(\mathbb{H})$ so that the following diagram commutes:
That is, every projective representation $T:G\to U(\mathbb{P})$ can be lifted to a unitary representation of a central extension of $G$ by $U(1)$.
Is there an equivalent statement for representations of Lie algebras?
- Martin Schottenloher, A Mathematical Introduction to Conformal Field Theory