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I know that the complex projective line $\mathbb{C}P^1$ can be embedded in the complex projective space $\mathbb{C}P^n$ (Veronese embedding). For example, $\mathbb{C}P^1\rightarrow\mathbb{C}P^3$ is given explicitly by $(z,w)\mapsto(z^3,z^2w,zw^2,w^3)$ in homogenous coordinates.

I was wondering if the same could be done with quaternionic projective spaces, i.e. is there a 'veronese type' embedding: $\mathbb{H}P^1\rightarrow\mathbb{H}P^n$ ??

(I know that the non-commutativity of quaternions make the veronese map ill defined)

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Nice question. Out of curiosity: why do you want to know? – Mariano Suárez-Alvarez Mar 5 '11 at 17:40
Could you kindly tell me the definition of $\mathbb{H}P^n$. – A.G Mar 5 '11 at 20:48
I want to build a map $S^{4n+3}\rightarrow S^4\cong\mathbb{H}P^1$ using the Hopf bundle projection $S^{4n+3}\rightarrow\mathbb{H}P^n$ and the Veronese embedding (if it exists) $\mathbb{H}P^1\rightarrow \mathbb{H}P^n$ – Dar Far Mar 5 '11 at 20:50
Anjan, it is the space of rays (one-dimensional subspaces) of the vector space $\mathbb{H}^{n+1}$ , and $\mathbb{H}$ are the quaternions. – Dar Far Mar 5 '11 at 20:53
Since you say the usual veronese map is ill defined, what properties of the veronese map are you hoping to capture with this map? – Jason DeVito Mar 14 '11 at 22:38
up vote 3 down vote accepted

Veronese embedding is the restriction of Segre embedding on the diagonal, so let's talk about quaternionic Segre embeddings instead. Now I have to admit, it's not a real answer, just 2 remarks:

  1. Segre embedding $\mathbb CP^{\infty}\times\mathbb CP^{\infty}\to\mathbb CP^{\infty}$ gives a structure of an H-space on $\mathbb CP^{\infty}$. But I don't think $\mathbb HP^{\infty}$ admits an H-space structure. (See also Hatcher. 4L.4.)
  2. Ordinary Segre embedding $\mathbb P(V)\times\mathbb P(W)\to\mathbb P(V\otimes W)$ maps a pair of 1-dimensional subspaces to their tensor product. Now, tensor product of two (say, left) quaternionic vector spaces is not a quaternionic vector space. But one can take tensor product over complex numbers — it induces a map $\mathbb P(V)\times\mathbb P(W)\to Gr_2(V\otimes_{\mathbb C}W)$.
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