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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeOct 28th 2013
   • (edited Oct 28th 2013)
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexHere is another stub: [[Albert algebra]]. It would be nice to get a reference to clear up the number of (real) Albert algebras. John Baez\'s [octonion paper](http://math.ucr.edu/home/baez/octonions/), among other literature (including our [[Jordan algebra]]), takes it for granted that there is only one (which is true, over the complex numbers, but people are usually working over the real numbers). But John himself points out on [a Wikipedia talk page](https://en.wikipedia.org/wiki/Talk:_Albert_algebra) that there are two (and that\'s what I followed).

   Here is another stub: Albert algebra.

   It would be nice to get a reference to clear up the number of (real) Albert algebras. John Baez's octonion paper, among other literature (including our Jordan algebra), takes it for granted that there is only one (which is true, over the complex numbers, but people are usually working over the real numbers). But John himself points out on a Wikipedia talk page that there are two (and that's what I followed).

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 26th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added a brief remark on the [relation to E6](https://ncatlab.org/nlab/show/Albert+algebra#Automorphisms)

   I have added a brief remark on the relation to E6

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeJan 26th 2016
   • (edited Jan 26th 2016)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI added the reference to Albert's 1934 paper. Wikipedia says there are _three_ nonisomorphic Albert algebras over the reals... Admittedly, the nLab page clarifies that the two it gives are the ones that aren't "special", but doesn't say what this means, and only links to [[Jordan algebra]] on the key word 'special'.

   I added the reference to Albert’s 1934 paper.

   Wikipedia says there are three nonisomorphic Albert algebras over the reals…

   Admittedly, the nLab page clarifies that the two it gives are the ones that aren’t “special”, but doesn’t say what this means, and only links to Jordan algebra on the key word ’special’.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 27th 2016
   • (edited Jan 27th 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI don't have time right now to check, but it would be nice to have this sorted out. What I am really after is an characterization of $E_7$ analogous to the one of $E_6$ as the $\mathbb{O}$-determinant-preserving linear isos on (one of) the Albert algebras. The characterizations of $E_7$ that are commonly stated say that $E_7$ is linear isos on more or less two copies of the (vector space underlying) the Albert algebra that preserve a certian symplectic form and a certain trilinear form. There must be a way to say this that makes very explicit use of the octonionic structure, instead of just writing out the symplectic form in components? What I would really like to understand is however this: in a note _[[schreiber:From higher to exceptional geometry]]_ a while back I had recorded the observation that 1. the old "hidden super-Lie algebra" of D'Auri-Fré '82 may be interpreted as providing a tangent-space-wise moduli space for choices of 3-form connections on those 2-gerbes whose curvature 4-form is the M2-brane cocycle; 1. its bosonic body happens to arise in the direct sum form that enters the definition of the vector space underlying the $\mathbf{56}$ rep of $E_7$, which makes it very suggestive that we want to be thinking of these moduli as transforming in that rep, instead of in all of $GL(56)$. But beyond it being suggestive, I don't understand yet, at a deeper level, what it is about $E_7$ at this point that makes us want to prefer it over $GL(56)$. I am hoping that maybe if I understand its action on two copies of the Albert algebra better, that might shed light on what's really going on.

   I don’t have time right now to check, but it would be nice to have this sorted out.

   What I am really after is an characterization of $E_7$ analogous to the one of $E_6$ as the $\mathbb{O}$-determinant-preserving linear isos on (one of) the Albert algebras.

   The characterizations of $E_7$ that are commonly stated say that $E_7$ is linear isos on more or less two copies of the (vector space underlying) the Albert algebra that preserve a certian symplectic form and a certain trilinear form.

   There must be a way to say this that makes very explicit use of the octonionic structure, instead of just writing out the symplectic form in components?

   What I would really like to understand is however this: in a note From higher to exceptional geometry (schreiber) a while back I had recorded the observation that

   1. the old “hidden super-Lie algebra” of D’Auri-Fré ’82 may be interpreted as providing a tangent-space-wise moduli space for choices of 3-form connections on those 2-gerbes whose curvature 4-form is the M2-brane cocycle;

   2. its bosonic body happens to arise in the direct sum form that enters the definition of the vector space underlying the $\mathbf{56}$ rep of $E_7$, which makes it very suggestive that we want to be thinking of these moduli as transforming in that rep, instead of in all of $GL(56)$.

   But beyond it being suggestive, I don’t understand yet, at a deeper level, what it is about $E_7$ at this point that makes us want to prefer it over $GL(56)$. I am hoping that maybe if I understand its action on two copies of the Albert algebra better, that might shed light on what’s really going on.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeFeb 21st 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added a remark on the relation of the exceptional Jordan algebra to $\mathbb{R}^{10,1\vert \mathbf{16}}$, [here](https://ncatlab.org/nlab/show/Albert+algebra#RelationTo10dSuperSpacetime)

   I have added a remark on the relation of the exceptional Jordan algebra to $\mathbb{R}^{10,1\vert \mathbf{16}}$, here

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeFeb 21st 2018
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAnd we still want to know whether something here emerges from the brane bouquet associated to the superpoint $R^{0|3}$. In case some enthusiastic young person is tuning in, this idea was to do for other superpoints what Urs and John Huerta did for $R^{0|2}$ in [M-theory from the Superpoint](https://arxiv.org/abs/1702.01774). Urs had pointed out that this might be of interest: J. Ambjorn, Y. Watabiki, [Creating 3, 4, 6 and 10-dimensional spacetime from W3 symmetry](https://arxiv.org/abs/1703.04402).

   And we still want to know whether something here emerges from the brane bouquet associated to the superpoint $R^{0|3}$.

   In case some enthusiastic young person is tuning in, this idea was to do for other superpoints what Urs and John Huerta did for $R^{0|2}$ in M-theory from the Superpoint.

   Urs had pointed out that this might be of interest: J. Ambjorn, Y. Watabiki, Creating 3, 4, 6 and 10-dimensional spacetime from W3 symmetry.