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Let $F$ be a matrix Lie group which has as its double cover the group $G$ (both are connected but not simply connected). Suppose further that the universal cover $\mathscr{G}$ of $G$ is not a matrix group. (I have in mind the specific example $F=\text{SO}_0(2,1)$ and $G=\text{SL}(2,\mathbb{R})$ but would like to keep the situation somewhat more general if possible). I would like to obtain projective representations of the Lie group $F$ and representations of its Lie algebra $\mathfrak{F}$. To my understanding, the process one usually goes through is to find exact representations of the universal cover $\mathscr{G}$ of $F$ which are then equivalent to projective representations of $F$. How does this process change if $\mathscr{G}$ is not a matrix Lie group?

Is it possible to only work with representations of the double cover $G$ to obtain projective representations of $F$, and representations of $\mathfrak{F}$ (I suppose yes, but then I will only obtain some of the projective reps of $F$)?

How would I go about obtaining reps of $\mathfrak{F}$ given a rep of $\mathfrak{G}$? Perhaps the answer lies with the covering map: Suppose $\rho:G\rightarrow F$ is the covering map of $F$ i.e. a surjective Lie group homomorphism, then $\text{d}\rho|_e:\mathfrak{G}\rightarrow\mathfrak{F}$ is the corresponding Lie algebra homomorphism with $\exp(\text{d}\rho(\mathfrak{g}))=\rho(\exp(\mathfrak{g}))$, but these aren't representations right?

Can I obtain a representation in this way? Suppose $\text{T}:g\mapsto \text{T}(g)$ is a representation of $G$ over some vector space $V_T$, i.e. $T$ is a homomorphism. Then by the above correspondence we have $\text{dT}:\mathfrak{g}\mapsto \text{dT}(\mathfrak{G})$ with the property $\exp(\text{dT}(\mathfrak{g}))=\text{T}(\exp(\mathfrak{g}))$. In this way, the representation $\text{T}$ of $G$ induces a representation of it's Lie algebra $\mathfrak{G}$, and if it happens that $G$ and $F$ have locally isomorphic algebras, then $\text{T}$ would also induce a representation of $\mathfrak{F}$, right? Under what conditions are two Lie algebras locally isomoprhic? Is $\mathfrak{sl}(2,\mathbb{R})\cong\mathfrak{so}_0(2,1)$ locally?

Apologies if some (or all) of the above does not make sense, I have only just begun with representation theory and Lie algebras (and am more interested in its applications to physics rather than rigour, but I want to make sure what I am doing is correct).

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If $\mathscr{G}$ is not linear the process still works: it just means that you can take advantage of non-linearity to push the process further. Namely, let $Z$ be the kernel of the covering map $\mathscr{G}\to G$; let $K$ be the kernel of all (finite-dimensional) representations of $\mathscr{G}$, and $Z'=K\cap Z$. Then all projective representations of $F$ actually come from representations of $\mathscr{G}/Z'$.

In the case $G=\mathrm{SL}_2(\mathbf{R})$, you indeed have $\mathscr{G}/Z'=G$. So projective representations of $\mathrm{PSL}_2(\mathbf{R})=\mathrm{SO}_0(2,1)$ are indeed representations of $G=\mathrm{SL}_2(\mathbf{R})$.

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