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I learned from Baez's notes on octonions that the classical simple Lie algebras can be identified with the Lie algebras of isometry groups of projective spaces over $\mathbb{R}, \mathbb{C}$ and $\mathbb{H}$, and that there is a way to generalize this for the exceptional algebras $\mathfrak{f}_4, \mathfrak{e}_6, \mathfrak{e}_7$ and $\mathfrak{e}_8$ as isometry groups of "projective planes" over the octonions $\mathbb{O}$ (the Cayley plane), bioctonions $\mathbb{C} \otimes \mathbb{O}$, quateroctonions $\mathbb{H} \otimes \mathbb{O}$ and octooctonions $\mathbb{O} \otimes \mathbb{O}$ respectively, which leads to the Magic Square construction.

The exceptional algebra $\mathfrak{g}_2$, on the other hand, is characterized as the algebra of derivations of the octonions. Why isn't there a projective space corresponding to it, like for the other simple Lie algebras? Or is there?


Edit: here's some new information that may help in answering the question.

  • I've found that there exists an extension of the magic square construction, called the magic triangle (Deligne, 1996), which does include $\mathfrak{g}_2$. However, the column in which it appears corresponds to a "dimension" $2\nu - 2$ of $-2/3$, so if we were to extend the projective plane interpretation we would have

    $$\mathfrak{g}_2 = \mathfrak{isom}((\mathbb{A} \otimes \mathbb{O})\mathbb{P}^2)$$

    where $\mathbb{A}$ would be a division algebra (superalgebra?) over $\mathbb{R}$ of dimension $-2/3$. I haven't heard of vector spaces with negative fractional dimension, but perhaps there is a way to make sense out of this.

  • Someone has pointed me to the classification of symmetric spaces https://en.wikipedia.org/wiki/Symmetric_space#Classification_result of compact type, which include all the "classical" projective spaces as special cases of real (BDI), complex (AIII) and quaternionic (CII) Grassmanians, and also the exceptional Cayley and Rosenfeld projective planes (FII, EIII, EVI and EVIII) mentioned above.

    The last space G, related to the group $G_2$, is the "space of subalgebras of the octonion algebra $\mathbb{O}$ which are isomorphic to the quaternion algebra $\mathbb{H}$". But there's no mention of it being a projective space. Could this be what I'm looking for?

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    $\begingroup$ @DietrichBurde But the Fano plane only contains the unit imaginary octonions, no? And the automorphism group of the Fano plane itself is the discrete group PGL(3,2). $\endgroup$ – pregunton Aug 19 '15 at 19:48
  • $\begingroup$ I mean the other automorphism group, i.e., $Aut(\mathbb{O})=G_2$, and $\mathbb{O}$ is "represented" in $\mathbb{F}_2\mathbb{P}^2$. $\endgroup$ – Dietrich Burde Aug 19 '15 at 20:15
  • $\begingroup$ For the construction of $\mathbb{O}$ using the Fano plane, see section $2$ here. $\endgroup$ – Dietrich Burde Aug 19 '15 at 20:23
  • $\begingroup$ @DietrichBurde Still, I think that doesn't answer my question. The construction I'm referring to in my post is precisely in section $4$ of the paper you linked. It identifies, for example, the algebra of isometries $\mathfrak{isom}(\mathbb{RP}^n)$ with $\mathfrak{so}(n+1)$ and $\mathfrak{isom}(\mathbb{OP}^2)$ with $\mathfrak{f}_4$. My question is whether there is a projective space $X$ such that $\mathfrak{isom}(X) \cong \mathfrak{g}_2$. $\endgroup$ – pregunton Aug 19 '15 at 20:54
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    $\begingroup$ Baez says that only $4$ exceptional Lie algebras come from the isometry groups of the projective planes related to octonions, but $G_2$ does not (page $3$ of your link). He continues, that on the other hand, $G_2$ arises directly as automorphism group of the octonions. I do not know why there is no projective space corresponding to it. $\endgroup$ – Dietrich Burde Aug 20 '15 at 17:58

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