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Does the projective linear group $PSL_2(\mathbb{R})$ admit faithful linear representations? In other words, does there there exist a homomorphism $SL_2(\mathbb{R}) \to GL_n(\mathbb{R}),$ for some $n$, having kernel $\{I, -I\}$?

More generally I would be curious to hear any comments about linear representations of projective linear groups. Are such things vacuous/trivial?

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Let $V$ be the defining $2$-dimensional representation of $SL_2(\mathbb{R})$. It has symmetric powers $S^n (V)$ for all positive integers $n$, and you can verify that when $n$ is even, the kernel of $S^n(V)$ is precisely $\pm 1$, so all of these are faithful linear representations of $PSL_2(\mathbb{R})$.

More generally, if $G$ is any connected Lie group, the adjoint representation of $G$ on its Lie algebra $\mathfrak{g}$ has kernel precisely the center of $G$, and so furnishes a faithful linear representation of $G/Z(G)$. All of the projective linear groups arise in this way. In the above example, $S^2(V)$ is the adjoint representation.

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