Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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.