I would like the find an embedding/faithful representation from the projective linear group $PGL_n\mathbb{C}\to GL_m\mathbb{C}$ for some $m$, and likewise for the other projective groups $PSL_n\mathbb{R}, PSO_n\mathbb{C}, PSO_n\mathbb{R}, Psp_{2n}\mathbb{C}$, and $Psp_{2n}\mathbb{R}$.

Is there an elementary way to deduce such an embedding, without using the machinery of Lie algebras or adjoint representations?

I know that we can make an argument using the adjoint representation of $PGL_n\mathbb{C}$ on its Lie algebra, but these haven't been introduced in my text yet.

  • $\begingroup$ Is the identity map for $GL_n\mathbb{C}$ considered acceptable? $\endgroup$ – user27126 Mar 31 '13 at 17:48
  • $\begingroup$ Is there a typo in this question? Based on the title, it seems you are looking for a representation $\mathrm{PGL}_n(\mathbb{C})\to\mathrm{GL}_m(\mathbb{C})$, not $\mathrm{GL}_n(\mathbb{C})\to\mathrm{GL}_m(\mathbb{C})$. $\endgroup$ – Jim Belk Mar 31 '13 at 18:30
  • $\begingroup$ Yes, for some reason there isn't a "P" in $GL_n\mathbb{C}$! Fixed it, thank you. $\endgroup$ – ff90 Mar 31 '13 at 18:55

A representation of $\text{PGL}_n(\mathbb{C})$ is, almost by definition, a representation of $\text{GL}_n(\mathbb{C})$ on which the center acts trivially. The center of a group $G$ is, almost by definition, the kernel of the homomorphism

$$G \ni g \mapsto (x \mapsto gxg^{-1}) \in \text{Aut}(G).$$

When $G = \text{GL}_n(\mathbb{C})$, this homomorphism extends to a linear representation of $\text{GL}_n(\mathbb{C})$ on $\mathcal{M}_n(\mathbb{C}) \cong \mathbb{C}^{n^2}$ on which the center acts trivially, and almost by definition, this gives a faithful representation of $\text{PGL}_n(\mathbb{C})$. (This is the adjoint representation, but you don't need to know anything about Lie algebras or even tangent spaces to recognize that $\text{GL}_n(\mathbb{C})$ embeds into $\mathcal{M}_n(\mathbb{C})$.)

  • $\begingroup$ Many thanks. As for constructing an actual representation, my understanding is that it would suffice to take any representation of $\operatorname{GL}_n(\mathbb{C})$ on $\mathcal{M}_n(\mathbb{C})$ that is invariant under multiplication by complex scalars? And the same for, say, $\operatorname{SO}_n(\mathbb{C})$ on $\mathcal{M}_n(\mathbb{C})$ for $\operatorname{PSO}_n(\mathbb{C})$ invariant under $\{ \pm I\}$? $\endgroup$ – ff90 Mar 31 '13 at 20:25
  • $\begingroup$ @ff90: yes, that would suffice. $\endgroup$ – Qiaochu Yuan Mar 31 '13 at 20:47
  • $\begingroup$ @QiaochuYuan I recently learned that a definition of a matrix Lie group is a closed subgroup of $GL_{n}(\mathbb{C})$. In this case, is the image of that faithful representation closed? $\endgroup$ – user135520 Sep 2 '15 at 3:09

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