March 13, 2009

The Okubo Algebra

Posted by John Baez

While studying the algebra of Grand Unified Theories and the role of division algebras in supersymmetric Yang–Mills theory, I bumped into a curious entity called the ‘Okubo algebra’.

If you know anything about it, tell me!

Susumu Okubo discovered his algebra while pondering quarks. For some reason not yet known to me, this led him to seek an 8-dimensional algebra having $\mathfrak{su}(3)$ as its Lie algebra of derivations.

The octonions won’t do: their Lie algebra of derivations is the exceptional Lie algebra $\mathfrak{g}_2$, which contains $\mathfrak{su}(3)$ as a subalgebra.

The Okubo algebra can be described as follows. Let $M$ be the space of $3 \times 3$ traceless self-adjoint complex matrices. In other words, take $\mathfrak{su}(3)$ and multiply all the matrices in there by $i$. Give $M$ the product

$X \circ Y = a X Y + b Y X + \frac{1}{6} tr(X Y)$

where the product at right is just ordinary matrix multiplication, and

$a = \overline{b} = (3 + i \sqrt{3})/6$

The result is a nonunital, nonassociative algebra. That sounds bad. But, just like $\mathbb{R}, \mathbb{C}, \mathbb{H}$ and $\mathbb{O}$, it’s a composition algebra: it’s equipped with a nondegenerate quadratic form

$Q : M \to \mathbb{R}$

such that

$Q(xy) = Q(x) Q(y)$

And it’s power-associative: that is, the subalgebra generated by any one element is associative. This is equivalent to having the identities

$(A \circ A ) \circ A = A \circ (A \circ A)$

and

$((A \circ A ) \circ A) \circ A = (A \circ A ) \circ (A \circ A)$

for every element $A$.

Furthermore, the Okubo algebra is Lie-admissible: that is, the commutator

$[A, B] = A \circ B - B \circ A$

defines a Lie algebra. For the Okubo algebra, this Lie algebra is $\mathfrak{su}(3)$.

(Everyone knows that associative algebras are Lie-admissible; fewer people know the converse fails — and fewer still know that every operad gives a Lie-admissible algebra! I learned the last fact from Bill Schmitt; the construction appear in section 4.7 here.)

The Okubo algebra is not alternative, as the octonions are. Remember, an algebra is alternative if the subalgebra generated by any two elements is associative — or equivalently, if any two of these three identities hold:

$A \circ (A \circ B) = (A \circ A) \circ B$ $A \circ (B \circ B) = (A \circ B) \circ B$ $A \circ (B \circ A) = (A \circ B) \circ A$

The Okubo algebra only satisfies the third of these laws:

$A \circ (B \circ A) = (A \circ B) \circ A$

Such algebras are called flexible.

But what’s going on here, really?

I might consider the whole subject too bizarre to be worth bothering with, except that Elduque has used a $(\mathbb{Z}/3)^3$-grading on the Okubo algebra to put a similar grading on the Lie algebras $\mathfrak{f}_4$ and $\mathfrak{e}_6$. In the latter case, this relies on an amazing construction of $\mathfrak{e}_6$ as

$\mathfrak{sl}(3) \oplus \mathfrak{sl}(3) \oplus \mathfrak{sl}(3)$ $\oplus$ $\mathbb{C}^3 \otimes \mathbb{C}^3 \otimes \mathbb{C}^3$ $\oplus$ $\mathbb{C}^3 \otimes \mathbb{C}^3 \otimes \mathbb{C}^3$

Some references I should read in my copious spare time:

• S. Okubo, Pseudo-quaternion and pseudo-octonion algebras, Hadronic J. 1 (1978), 1250–1278.
• S. Okubo, Deformation of the Lie-admissible pseudo-octonion algebra into the octonion algebra, Hadronic J., 1 (1978), 1383–1431.
• S. Okubo, Octonions as traceless matrices via a flexible Lie-admissible algebra, Hadronic J. 1 (1978), 1432–1465.
• S. Okubo, A generalization of Hurwitz theorem and flexible Lie-admissible algebras, Hadronic J. 3 (1978), 1–52.
• S. Okubo, H.C. Myung, Some new classes of division algebras, J. Algebra 67 (1980), 479–490.
• S. Okubo, Introduction to Octonion and other Non-Associative Algebras in Physics, Cambridge Univ. Press, 1995.

The Hadronic Journal is a bit of a mystery to me. I’ve seen a lot of quirky papers on nonassociative algebras and physics in this journal. Are the people who write them followers of Okubo, in some sense?

Posted at March 13, 2009 1:56 AM UTC

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Re: The Okubo Algebra

Here’s something I learned about the Okubo-product from a nice talk last week by Melanie Raczek. She used it to classify degree 3 central simple algebras, but I’ll restrict to the split case, that is consider 3x3 complex matrices and look at the 8-dimensional subspace V=sl(3) of trace zero matrices.

On this space there is a nondegenerate quadratic form Tr(A2) (here A is any trace zero matrix). Going projective, 3x3 matrices satisfying Tr(A)=Tr(A2)=0, form a 6-dimensional quadric in P7. A nice fact about 6-dimensional quadrics is that any point determines two 3-planes through the point and contained in the quadric (compare this to the two lines through a point on a 2-dimensional quadric). The Okubo product on V describes these.

For any two trace-zero 3x3 matrices A and B define the Okubo product to be

A*B =(1/(1-w))(AB-wBA) - Tr(AB)

where w is a third root of unity.

If a matrix A determines a point on the quadric, then one of the 3-planes through it is the projective space of the 4-dimensional subspace of all Okubo products A*B with B ranging over V (the other coming from the 4-dimensional subspace of all products B*A).

Posted by: lievenlb on March 13, 2009 8:46 PM | Permalink | Reply to this

Re: The Okubo Algebra

Oops, I made a few mistakes. There should be a 1/3 in front of the Tr and, more importantly, these are NOT the 3-planes passing through the point…

Anyway, the situation is even more beautiful than I thought bringing in triality. A bit much to put in a comment, so I made a post on what I hope to understand about the relation between the Okubo-algebra and the geometry of a 6-dimensional smooth quadric.

Posted by: lievenlb on March 14, 2009 5:16 PM | Permalink | Reply to this

Re: The Okubo Algebra

Thanks for all the information! What a nice coincidence, that you went to a talk on the Okubo algebra just last week.

An email correspondent, who may prefer to remain anonymous, suggested that I look at the account of Okubo and para-Cayley algebras in Chapter VIII of The Book of Involutions by Knus, Merkurjev, Rost and Tignol. He said “it might lead you to consider them as less bizarre; maybe even as inevitable.”

Posted by: John Baez on March 15, 2009 2:40 AM | Permalink | Reply to this

Re: The Okubo Algebra

Talking of coincidences, in the post Lieven mentioned above, he has a nice animation of the two families of lines on a 2-dimensional quadric being discussed here.

Posted by: David Corfield on March 17, 2009 5:26 PM | Permalink | Reply to this

Re: The Okubo Algebra

Yeah! So, if I ever put the puzzle pieces together, they should all fit quite prettily.

That animation on Lieven’s page is not anything unusual, by the way: it’s just a hyperboloid, and you can see models like this in math departments worldwide:

Posted by: John Baez on March 17, 2009 6:14 PM | Permalink | Reply to this

Re: The Okubo Algebra

My anonymous correspondent says that The Book of Involutions gives a nice conceptual explnation of the Okubo algebra. You can see the table of contents, bibliography and index of this book on Markus Rost’s webpage.

The idea, apparently, is to take the “triality” outer automorphism of a suitable group of type $D_4$ and twisting it by a cube root of unity when one is available. (The Book of Involutions works over an arbitrary field; if you only care about the complex numbers then you’ve got your cube root of unity, and this explains the funny numbers in some of the explicit formulas above.)

Fans of exceptional algebraic structures will know that the untwisted version of this construction gives a $G_2$ type group as the fixed points of the outer automorphism, and an octonion algebra as the associated composition algebra. The twisted version gives a non-split $A_2$ type group (for example $\mathfrak{su}(3)$) as the fixed point group and an Okubo algebra as the associated composition algebra!

I need to get ahold of this book. In my last round of studying the octonions, I found its concern with arbitrary fields quite frustrating. Now I’m more fond of Galois theory.

By the way: if ‘$\mathfrak{su}$’ looks like meaningless junk to you, instead of a Gothic font version of ‘su’, post a comment to let me know. I can avoid these characters if most people don’t have that font installed.

Posted by: John Baez on March 15, 2009 5:36 PM | Permalink | Reply to this

Re: The Okubo Algebra

Actually, fonts usually don’t show up under my installation of Safari. It make the readings amusing, I guess. I have been too lazy to look up font installations.

Posted by: Scott Carter on March 15, 2009 8:08 PM | Permalink | Reply to this

Re: The Okubo Algebra

I run firefox on OSX 10.5, and as far as I am aware, have most of the fonts required for MathML (I haven’t updated recently, and I’m no computing guru anyway), and your fraktur su’s come out as boxes filled with gunk.

Posted by: David Roberts on March 16, 2009 4:26 AM | Permalink | Reply to this

Re: The Okubo Algebra

I would suggest looking at these instructions. With the advent of Firefox 3.x and the STIX fonts, there’s a certain amount of cruft that may need to be cleared out before things (once again) work as expected.

Posted by: Jacques Distler on March 16, 2009 6:02 AM | Permalink | PGP Sig | Reply to this

Re: The Okubo Algebra

Ooh, fancy! I almost want to be using these fraktur letters rather than the LaTeX ones ;)

Posted by: David Roberts on March 17, 2009 4:14 AM | Permalink | Reply to this

Re: The Okubo Algebra

It looks like meaningless junk to me. Fortunately the occurrence of meaningless junk letters at the $n$-Café coincides closely with the occurrence of Lie theory, so they don’t decrease the extent to which I understand.

Posted by: Tom Leinster on March 16, 2009 5:45 AM | Permalink | Reply to this

Re: The Okubo Algebra

Great idea: if we all avoid installing fonts that correspond to subjects we don’t understand, we all have an excuse not to follow posts on those subjects. There’s a calligraphic font I haven’t installed, which seems to show up only in heavy-duty topos theory…

This reminds me of the snide remark I made at the beginning of the ‘TeXnical Issues’ page:

Why do all the equations look like gobbledygook?

Maybe you don’t know math, so all equations look like gobbledygook.

Posted by: John Baez on March 17, 2009 4:56 AM | Permalink | Reply to this

Re: The Okubo Algebra

I use Firefox myself, and I was unable to read the fraktur fonts until I followed Jacques’ directions and installed the STIX fonts. With the Euclid Fraktur font installed, they come out quite nicely.

I suspect that most people don’t have these fonts installed yet… hence my question.

Posted by: John Baez on March 16, 2009 7:13 AM | Permalink | Reply to this

Re: The Okubo Algebra

I now have the STIX fonts installed and so I see the fraktur letters, but I still think we should avoid using them. I think in the tradeoff between (1) using fancy letters in our math and (2) making our math accessible to as many readers as possible, I would come down in favor of (2). This goes for the nLab as well, and also for script letters like $\mathcal{T}$; they also looked like garbage to me until I installed STIX.

Posted by: Mike Shulman on March 17, 2009 1:28 AM | Permalink | Reply to this

STIX

At least in Firefox 3.1β, not installing the STIX fonts will generally lead to a variety of other MathML rendering problems (particularly with stretchy characters).

To get a get a relatively bug-free MathML experience (at least with current Mozilla-based browsers), you pretty much have to install the STIX fonts.

At that point, you can think of the fraktur support as just an added bonus.

There are other fonts which may work in the future (bug 372351, bug 407439) but, right now, the STIX fonts are the only game in town.

Posted by: Jacques Distler on March 17, 2009 4:04 AM | Permalink | PGP Sig | Reply to this

Re: STIX

I didn’t notice a lot of bugs before I installed STIX; it was just script fonts that I didn’t see. Installing STIX did subtly change the way a lot of MathML looks, but I’m not sure I prefer it this way.

Posted by: Mike Shulman on March 18, 2009 3:10 AM | Permalink | Reply to this

Re: STIX

Will installing STIX get better (more) spacing around binary operators? I don't know why I get space in $x + y$ but not $x \vee y$, considering that the MathML is identical (except for one character, of course).

Posted by: Toby Bartels on March 18, 2009 8:22 AM | Permalink | Reply to this

Re: STIX

Well, I also see more space in $x+y$ than $x\vee y$, so probably not.

Posted by: Mike Shulman on March 18, 2009 5:56 PM | Permalink | Reply to this

Re: STIX

I didn’t notice a lot of bugs before I installed STIX; it was just script fonts that I didn’t see. Installing STIX did subtly change the way a lot of MathML looks, but I’m not sure I prefer it this way.

Indeed. This gives me an opportunity to once again shamelessly promote my Stylish ‘skin’ for the n-category café, where the preferred math fonts are set to be (in this order):

Georgia, DejaVu Serif, BitStream Vera Serif, Linux Libertine, Constantia, Cambria Math, STIXNonUnicode, STIXSize1, STIXGeneral, Symbol.

Moreover, you get a wider text area and a gimungous comment box thrown in for free.

By the way, if you are using Firefox, try out some of the fonts temporarily by Tools -> Options -> Content -> Advanced (under Fonts and Colours) and change the Serif font, unchecking the “Allow pages to choose their own fonts” checkbox.

Posted by: Bruce Bartlett on March 18, 2009 2:38 PM | Permalink | Reply to this

Re: STIX

I definitely prefer it before I installed the STIX fonts, and I prefer that to the Gerogia fonts in Bruce's skin too. (That is, normally; of course only the STIX fonts do all of the large symbols.)

Compare:

Maybe I should figure out what font I was using before.

Posted by: Toby Bartels on March 20, 2009 6:15 PM | Permalink | Reply to this

Re: STIX

Maybe I should figure out what font I was using before.

For the record, it looks like it was DejaVu Serif.

Posted by: Toby Bartels on March 20, 2009 9:26 PM | Permalink | Reply to this

Re: The Okubo Algebra

Mike wrote:

I think in the tradeoff between (1) using fancy letters in our math and (2) making our math accessible to as many readers as possible, I would come down in favor of (2).

In general I agree with you, which is why — for example — This Week’s Finds is still available in ASCII. But I find it very painful to be writing in a version of TeX while still not using Gothic letters for Lie algebras. Lie algebras that aren’t in Gothic font make me feel sad.

And besides, the whole point of Gothic letters is to diminish readership. That’s how the Goths destroyed literacy and overthrew the Roman empire in the first place!

But, I’m weighing the issue judiciously.

Posted by: John Baez on March 17, 2009 6:34 AM | Permalink | Reply to this

Re: The Okubo Algebra

looks like meaningless junk to me

Posted by: jim stasheff on March 16, 2009 2:48 PM | Permalink | Reply to this

Re: The Okubo Algebra

The Fraktur fonts look great after installing STIX (which only takes a moment). It’s nice to finally be seeing these symbols correctly - don’t stop using them now! :-)

Posted by: Charles G Waldman on March 16, 2009 9:55 AM | Permalink | Reply to this

Re: The Okubo Algebra

Yeah, I never got it working with Firefox 2 and the old version of OSX I used to have, but now everything’s been working great for more than a year and I don’t want to go back. No junk for me! Jacques’ instructions were great.

Posted by: Tim Silverman on March 16, 2009 3:40 PM | Permalink | Reply to this

Re: The Okubo Algebra

Just installed STIX fonts. Easy to do; looks great! Thanks!

Posted by: Charlie C on March 17, 2009 12:10 PM | Permalink | Reply to this

Hadronic Journal (was: The Okubo Algebra)

The Hadronic Journal is a bit of a mystery to me. I’ve seen a lot of quirky papers on nonassociative algebras and physics in this journal.

The Hadronic Journal is the creation of Ruggero Santilli. Because of his interest in Lie-admissible algebras back in the 1980s, his journal attracted people in the nonassociative algebra community. After all, who would not be pleased that that one’s research, generally considered esoteric, was being looked at by physicist(s)?

However, my understanding is that the predictions of Santilli’s theories at the time violate SR. Based on his unsuccessful attempts to get funding as well as other events, he made various accusations against “followers” of Einstein. As his views became known, nonassociative algebra folks moved elsewhere. For instance, the 5th (and last) International Conference on Hadronic Mechanics and Nonpotential Interactions in Ceder Falls, Iowa in August 1990 essentially turned into a nonassociative algebra conference; there were a few physicists, but Santilli didn’t attend. (There is more to the story, but the rest lacks documentation, so I’ll leave it at that.)

I’ll bet most of you didn’t know that Santilli won the 2009 Gold Prize for Scientific Discoveries from the Fondazione Mediterraneo. I have been unable to find this at the Fondazione’s own site.

Posted by: Michael Kinyon on March 20, 2009 12:51 PM | Permalink | Reply to this

Re: Hadronic Journal (was: The Okubo Algebra)

I’d never heard of this fellow Santilli, but the front page of his tome of accusations against Einstein’s followers is awesome:

IL GRANDE GRIDO: ETHICAL PROBE ON EINSTEIN FOLLOWER’S
IN USA

by

Ruggero Maria Santilli da Capracotta
The greatest scientist of all times

This work represent to us, the most important historical manuscript of the 20th century science, and the most influential book ever written:

Posted by: Todd Trimble on March 20, 2009 3:56 PM | Permalink | Reply to this

Re: Hadronic Journal (was: The Okubo Algebra)

So that explains why

To the Foundation’s best knowledge, during the three decades following the inception of hadronic mechanics in 1978, Prof. Ruggero Maria Santilli has received hundreds of nominations for the Nobel Prize in Physics…

Posted by: David Corfield on March 20, 2009 5:49 PM | Permalink | Reply to this

Re: The Okubo Algebra

Thanks for all the information! What a nice coincidence, that you went to a talk on the Okubo algebra just last week.

Posted by: Matew on March 22, 2009 10:17 PM | Permalink | Reply to this

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