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The following paragraph is from page 150 in the book Representation Theory by Fulton and Harris.

Observe that when we exponentiate the image of $sl_2 \mathbb{C}$ under the embedding $sl_2\mathbb{C} \rightarrow sl_{n+1}\mathbb{C}$ corresponding to the representation $\mathrm{Sym}^n V$, we arrive at the group $SL_2 \mathbb{C}$ when $n$ is odd and $PGL_2 \mathbb{C}$ when $n$ is even. Thus, the representation of the group $PGL_2 \mathbb{C}$ are exactly the even powers $\mathrm{Sym}^{2n}V$.

Here $V$ is the standard representation of $sl_2\mathbb{C}$ and $\mathrm{Sym}^n$ denotes the symmetric $n$th power.

I want to understand the embedding $sl_2\mathbb{C} \rightarrow sl_{n+1}\mathbb{C}$. This cannot be the obvious one because then we would get $SL_2\mathbb{C}$ for every $n$. There should be an embedding corresponding to $\mathrm{Sym}^n V$. How can I find this embedding?

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  • $\begingroup$ As you say, it's the map corresponding to $\text{Sym}^n(V)$. The action of $\mathfrak{sl}_2(\mathbb{C})$ on this rep gives a map $\mathfrak{sl}_2(\mathbb{C}) \to \mathfrak{gl}_{n+1}(\mathbb{C})$ that you can show lands in $\mathfrak{sl}$. $\endgroup$ – Qiaochu Yuan Apr 9 '17 at 7:04
  • $\begingroup$ @Qiaochu Yuan, thanks, I was stupid. $\endgroup$ – Hwang Apr 9 '17 at 7:49

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