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Geometry of the Okubo algebra

Last week, Melanie Raczek gave a talk entitled ‘Cubic forms and Okubo product’ in our Artseminar, based on her paper On ternary cubic forms that determine central simple algebras of degree 3.

I had never heard of this strange non-associative product on 8-dimensional space, but I guess it is an instance of synchronicity that now the Okubo algebra seems to pop-up everywhere.

Yesterday, there was the post the Okubo algebra by John Baez at the n-cafe, telling that Susumu Okubo discovered his algebra while investigating quarks.

I don’t know a thing about the physics, but over the last days I’ve been trying to understand some of the miraculous geometry associated to the Okubo algebra. So, let’s start out by defining the ‘algebra’.

Consider the associative algebra of all 3×3 complex matrices $M_3(\mathbb{C}) $ with the usual matrix-multiplication. In this algebra there is the 8-dimensional subspace of trace zero matrices, usually called the Lie algebra $\mathfrak{sl}_3 $. However, we will not use the Lie-bracket, only matrix-multiplication. Typical elements of $\mathfrak{sl}_3 $ will be written as $X,Y,Z,… $ and their entries will be denoted as

$X = \begin{bmatrix} x_0 & x_1 & x_2 \\ x_3 & x_4 & x_5 \\ x_6 & x_7 & -x_0-x_4 \end{bmatrix} $

For any two elements $X,Y \in \mathfrak{sl}_3 $ one defines their Okubo-product to be the 3×3 matrix

$X \ast Y = \frac{1}{1-\omega}(Y.X-\omega X.Y) – \frac{1}{3}Tr(X.Y) 1_3 $

where $\omega $ is a primitive 3-rd root of unity and $1_3 $ is the identity matrix. Written out in the entries of X and Y this operation looks horribly complicated

$X \ast Y = \frac{1}{1-\omega} \begin{bmatrix} p_{11} & p_{12} & p_{13} \\ p_{21} & p_{22} & p_{23} \\ p_{31} & p_{32} & -p_{11}-p_{22} \end{bmatrix} $

with

[tex]\begin{eqalign} \\ p_{11} &= (1-\omega)x_0y_0+x_3y_1+x_6y_2-\omega(x_1y_3+x_2y_6)-\frac{1}{3}T \\ p_{12} &= x_1y_0+x_4y_1+x_7y_2-\omega(x_0y_1+x_1y_4+x_2y_7) \\ p_{13} &= x_2y_0+x_5y_1-x_0y_2-x_4y_2-\omega(x_0y_2+x_1y_5-x_2y_0-x_2y_4) \\
p_{21} &= x_0y_3+x_3y_4+x_6y_5 – \omega(x_3y_0+x_4y_3+x_5y_6) \\
p_{22} &= (1-\omega)x_4y_4+x_1y_3+x_7y_5 – \omega(x_3y_1+x_5y_7) – \frac{1}{3}T \\
p_{23} &= x_2y_3+x_5y_4-x_0y_5-x_4y_5-\omega(x_3y_2+x_4y_5-x_5y_0-x_5y_4) \\
p_{31} &= x_0y_6+x_3y_7-x_6y_0-x_6y_4-\omega(x_6y_0+x_7y_3-x_0y_6-x_4y_6) \\
p_{32} &= x_1y_6+x_4y_7-x_7y_0-x_7y_4 – \omega(x_6y_1+x_7y_4-x_0y_7-x_4y_7) \\
T &= 2x_0y_0+2x_4y_4+x_1y_3+x_2y_6+x_3y_1+x_5y_7+x_6y_2+x_7y_5+x_0y_4+x_4y_0
\end{eqalign}[/tex]

The crucial remark to make is that $X \ast Y $ is again a trace zero matrix. That is, we have defined a new operation on $\mathfrak{sl}_3 $.

$\mathfrak{sl}_3 \times \mathfrak{sl}_3 \rightarrow \mathfrak{sl}_3~\qquad~\qquad~(X,Y) \mapsto X \ast Y $

This Okubo-product is neither a Lie-bracket, nor an associative multiplication. In fact, it is a lot ‘less associative’ than that other 8-dimensional algebra, the octonions. The only noteworthy identity it has is that $X \ast (Y \ast X) = (X \ast Y) \ast X $. So, why should we be interested in this horrible algebra?

Well, let us consider the subset of $\mathfrak{sl}_3 $ consisting of those matrices X satusfying $Tr(X^2)=0 $. That is, with the above notation, all matrices X such that

$x_0^2+x_4^2+x_1x_3+x_2x_6+x_5x_7=0 $

In the 8-dimensional affine space $\mathfrak{sl}_3 $ these matrices form a singular quadric with top the zero-matrix. So, it is better to go projective. That is, any non-zero matrix $X \in \mathfrak{sl}_3 $ determines a point in 7-dimensional projective space $\mathbb{P}^7 $ with homogeneous coordinates

$\overline{X} = [x_0:x_1:x_2:x_3:x_4:x_5:x_6:x_7] \in \mathbb{P}^7 $

and the points $\overline{X} $ corresponding to solutions of $Tr(X^2)=0 $ form a smooth 6-dimensional quadric $Q \subset \mathbb{P}^7 $ with homogeneous equation

$Q = \mathbb{V}(x_0^2+x_4^2+x_1x_3+x_2x_6+x_5x_7) $

6-dimensional quadrics may be quite hard to visualize, so it may help to recall the classic situation of lines on a 2-dimensional quadric (animated gif taken from surfex).

A 2-dimensional quadric contains two families of lines, often called the ‘blue lines’ and the ‘red lines’, each of these lines isomorphic to $\mathbb{P}^1 $. The rules-of-intersection of these are :

  • different red lines are disjoint as are different blue lines
  • any red and any blue line intersect in exactly one point
  • every point of the quadric lies on exactly one red and one blue line

The lines in either family are in one-to-one correspondence with the points on the projective line. We therefore say that there is a $\mathbb{P}^1 $-family of red lines and a $\mathbb{P}^1 $-family of blue lines on a 2-dimensional quadric.

A 6-dimensional quadric $Q \subset \mathbb{P}^7 $ contains two families of ‘3-planes’. That is, there is a family of red $\mathbb{P}^3 $’s contained in Q and a family of blue $\mathbb{P}^3 $’s. Can we determine these red and blue 3-planes explicitly?

Yes we can, using the Okubo algebra-product on $\mathfrak{sl}_3 $. Take $X \in \mathfrak{sl}_3 $ defining the point $\overline{X} \in Q $ (that is, $Tr(X^2)=0 $). then all 3×3 matrices one obtains by taking the Okubo-product with left X-factor form a 4-dimensional linear subspace in $\mathfrak{sl}_3 $

$L_X = { X \ast Y~|~Y \in \mathfrak{sl}_3 } \simeq \mathbb{C}^4 \subset \mathfrak{sl}_3 $

so its non-zero matrices determine a 3-plane in $\mathbb{P}^7 $ (consisting of all points with homogeneous coordinates $[p_{11}:p_{12}:p_{13}:p_{21}:p_{22}:p_{23}:p_{31}:p_{32}] $, using the above formulas) which actually lies entirely in the quadric Q. These are precisely the bLue 3-planes in Q. That is, the family of all bLue 3-planes consists precisely of the 3-planes

$\mathbb{P}(L_X) $ with $X \in \mathfrak{sl}_3 $ satisfying $Tr(X^2)=0 $

Phrased differently, any point $\overline{X} \in Q $ determines a blue 3-plane $\mathbb{P}(L_X) $.

Similarly, any point $\overline{X} \in Q $ determines a Red 3-plane by taking Okubo-products with Right X-factor, that is, $\mathbb{P}(R_X) $ is a 3-plane for Q where

$R_X = { Y \ast X~|~Y \in \mathfrak{sl}_3 } \simeq \mathbb{C}^4 \subset \mathfrak{sl}_3 $

and all Red 3-planes for Q are of this form. But, this is not all… these correspondences are unique! That is, any point on the quadric defines a unique red and a unique blue 3-plane, or, phrased differently, there is a Q-family of red 3-planes and a Q-family of blue 3-planes in Q. This is a consequence of triality.

To see this, note that the automorphism group of a 6-dimensional smooth quadric is the rotation group $SO_8(\mathbb{C}) $ and this group has Dynkin diagram $D_4 $, the most symmetrical of them all!

In general, every node in a Dynkin diagram has an interesting projective variety associated to it, a so called homogeneous space. I’ll just mention what these spaces are corresponding to the 4 nodes of $D_4 $. Full details can be found in chapter 23 of Fulton and Harris’ Representation theory, a first course.

The left-most node corresponds to the orthogonal Grassmannian of isotropic 1-planes in $\mathbb{C}^8 $ which is just a fancy way of viewing our quadric Q. The two right-most nodes correspond to the two connected components of the Grassmannians of isotropic 4-planes in $\mathbb{C}^8 $, which are our red resp. blue families of 3-planes on the quadric. Now, as the corresponding dotted Dynkin diagrams are isomorphic



there corresponding homogeneous spaces are also isomorphic. Thus indeed, there is a one-to-one correspondence between points of the quadric Q and red 3-planes on Q (and similarly with blue 3-planes on Q).

Okay, so the Okubo-product allows us to associate to a point on the 6-dimensional quadric Q a unique red 3-plane and a unique blue 3-plane (much as any point on a 2-dimensional quadric determines a unique red and blue line). Do these families of red and blue 3-planes also satisfy ‘rules-of-intersection’?

Yes they do and, once again, the Okubo-product clarifies them. Here they are :

  • two different red 3-planes intersect in a unique line (as do different blue 3-planes)
  • the bLue 3-plane $\mathbb{P}(L_X) $ intersects the Red 3-plane $\mathbb{P}(R_Y) $ in a unique point if and only if the Okubo-product $X \ast Y \not= 0 $
  • the bLue 3-plane $\mathbb{P}(L_X) $ intersects the Red 3-plane $\mathbb{P}(R_Y) $ in a unique 2-plane if and only if the Okubo-product $X \ast Y = 0 $

That is, Right and Left Okubo-products determine the Red and bLue families of 3-planes on the 6-dimensional quadric as well as their intersections!

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Connes & Consani go categorical

Today, Alain Connes and Caterina Consani arXived their new paper Schemes over $ \mathbb{F}_1$ and zeta functions. It is a follow-up to their paper On the notion of geometry over $ \mathbb{F}_1$, which I’ve tried to explain in a series of posts starting here.

As Javier noted already last week when they updated their first paper, the main point of the first 25 pages of the new paper is to repace abelian groups by abelian monoids in the definition, making it more in tune with other approaches, most notably that of Anton Deitmar. The novelty, if you want, is that they package the two functors $\mathbf{rings} \rightarrow \mathbf{sets} $ and $\mathbf{ab-monoid} \rightarrow \mathbf{sets} $ into one functor $\mathbf{ring-monoid} \rightarrow \mathbf{sets} $ by using the ‘glued category’ $\mathbf{ring-monoid} $ (an idea they attribute to Pierre Cartier).

In general, if you have two categories $\mathbf{cat} $ and $\mathbf{cat’} $ and a pair of adjoint functors between them, then one can form the glued-category $\mathbf{cat-cat’} $ by taking as its collection of objects the disjoint union of the objects of the two categories and by defining the hom-sets between two objects the hom-sets in either category (if both objects belong to the same category) or use the adjoint functors to define the new hom-set when they do not (the very definition of adjoint functors makes that this doesn’t depend on the choice).

Here, one uses the functor $\mathbf{ab-monoid} \rightarrow \mathbf{rings} $ assigning to a monoid $M $ its integral monoid-algebra $\mathbb{Z}[M] $, having as its adjoint the functor $\mathbf{rings} \rightarrow \mathbf{ab-monoid} $ forgetting the additive structure of the commutative ring.

In the second part of the paper, they first prove some nice results on zeta-functions of Noetherian $\mathbb{F}_1 $-schemes and extend them, somewhat surprisingly, to settings which do not (yet) fit into the $\mathbb{F}_1 $-framework, namely elliptic curves and the hypothetical $\mathbb{F}_1 $-curve $\overline{\mathbf{spec}(\mathbb{Z})} $.

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math2.0-setup : final comments

Last time I promised to come back explaining how to set-up LaTeX-support, figuring I had to tell you about a few modifications I had to make in order to get Latexrender run on my mac…

A few google searches made it plain how out of touch I am on these matters (details below). But first, there was this comment to this series by Link Starbureiy :

“I took part in Gowers’ blog discussion. My input was to move things over to Google collaboration tools, like Google Knol, and perhaps Google Sites. However, those tools for large-scale collaboration may not be the best solution anymore. I like the NSN idea, but worry about it’s very long-term stability. Would you consider porting the project over to the Google App Engine so that it can be played with in the orkut sandbox (http://sandbox.orkut.com)?”

I thought I made it clear from the outset that I didn’t want to spend the rest of my life web-mastering a site such as NSN. All I wanted to show is that the technology is there free for the taking, and show that you do not have to be a wizard to get it running even on a mac…

I would really love it when some groups, or universities, on institutes, would set up something resembling this dedicated to a single arXiv-topic. Given our history, Antwerp University might be convinced to do this for math.RA but (a) I’m not going to maintain this on my own and (b) there may very well be a bandwidth problem if such a thing would become successful… (although, from past experiences and attempts I’ve made over the years, this is extremely unlikely for this target-group).

So please, if your group has some energy to spare, set-up your own math2.0-network, port it to Google Apps, Knol, Orkut or whatever, and I’d love to join and contribute to it.

As to LaTeX-support : this is trivial these days. First you need a working LaTeX-system on your virgin macbook. The best way is to download The MacTeX-2008 Distribution at work (it is a huge 1.19Gb download…). Next, install the fauxml-wordpress plugin (that is, download it to YourHome/Downloads and then drag the file faux-ml.php to the Library/WebServer/Documents/wp-content/plugins/ directory. Next, install likewise the WP-LateX plugin following the instructions, go to the configuring page and set the directory for latex and dvipng (if you follow my instructions they should be located at /usr/texbin/latex and /usr/texbin/dvipng), fill in the text color and background color you desire and clip your default latex-documentstyle/includepackages/newcommands section from your latest paper into the LaTeX Preamble window and believe me, you’re done!!!



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