TITLE: Lie group structure on the complex projective space
QUESTION [5 upvotes]: There is a famous theorem about when $S^n$ has the structure of a Lie group. What about the complex projective space $\mathbb CP^n$? For example, why $\mathbb CP^2$ is not a Lie group (without using classification for low dimension compact Lie groups)?

REPLY [3 votes]: You can actually generalise everything in the question to show that $\mathbb{C}P^n$ ($1\leq n<\infty$) cannot even admit a Hopf structure (https://en.wikipedia.org/wiki/H-space). And one way to see this is to demonstrate the existence of a non-trivial Whitehead product in $\pi_*\mathbb{C}P^n$. I'll point out that you already have fantastic answers, and most of them can be generalised directly to cover this case. This answer is only supposed to add another perspective.
Recall the quotient map $\gamma_n:S^{2n+1}\rightarrow \mathbb{C}P^n$ and the fact that it induces isomorphisms on $\pi_*$ for $*>2$. In particular $\gamma_{n*}:\pi_{4n+1}S^{2n+1}\xrightarrow{\cong}\pi_{4n+1}\mathbb{C}P^n$ is an isomorphism that takes the Whitehead square $\omega_{2n+1}=[\iota_{2n+1},\iota_{2n+1}]\in\pi_{4n+1}S^{2n+1}$ to the Whitehead product
$$\gamma_{n*}\omega_{2n+1}=[\gamma_n,\gamma_n]\in\pi_{4n+1}\mathbb{C}P^n.$$
It is classical fact related to the Hopf invariant one problem that for odd $k$, $\omega_k$ vanishes exactly when $k=1,3$ or $7$. Therefore, since $\gamma_{n*}$ is an isomorphism, $[\gamma_n,\gamma_n]$ is non-zero in $\pi_{4n+1}\mathbb{C}P^n$ as long as $n\neq 1,3$. Now $\mathbb{C}P^1\cong S^2$ is not an $H$-space exactly because of Adam's solution to the Hopf invariant one problem (and more in line with the current trail of thought, $[\iota_2,\iota_2]=-2\eta\in\pi_3S^2$).
Therefore we single out $\mathbb{C}P^3$ as the interesting case for which this line of reasoning does not apply. In fact all Whitehead products vanish in $\mathbb{C}P^3$ (Stasheff: "On homotopy Abelian H-spaces") and $\Omega \mathbb{C}P^3$ is homotopy commutative. I assure you still that $\mathbb{C}P^3$ is not an $H$-space, since either Qiaochu's or Eric's answers for compact Lie groups apply more or less verbatim to finite H-spaces.