Talk:Triality

Needing citations

There are no citations in this article, and just two blanketing references listed at the end. This needs to be done better! — Preceding unsigned comment added by 2600:1010:B05B:DDC5:8881:5F53:4BED:E5E9 (talk) 17:05, 8 May 2018 (UTC)

Article misrepresents the thrust of "triality"

The version of the "triality principle" alluded to here is described more accurately (with detailed references) in Freudenthal_magic_square#Triality. (Personally, I found Landsberg and Manivel's exposition more readable that the Barton and Sudbery version.) It's not very interesting (or encylopedic) to wave around the exceptional symmetry of the Dynkin diagram of $D_{4}$. The exceptional Jordan algebra is much more central to the original Tits construction, and there is no reference to the graph of $D_{4}$ in the paper of Ramond in which the "triality principle" first appears. (Of course there are references to SO(8), because octonions; but the emphasis on the $D_{4}$ diagram here and in SO(8)#Triality is pure retcon.)

While it is true that the only Dynkin diagram of a classical Lie algebra that has an $S_{3}$ graph automorphism is $D_{4}$, similar symmetries exist in the Dynkin diagrams of some Kac-Moody algebras, notably the affine Kac-Moody algebra ${\tilde {E_{6}}}$. Pace John Baez, any claim that triality is essentially "about" this graph automorphism is mere hand-waving without A) historical support, B) an exposition of Ramond's triality construction as an automorphism of the $D_{4}$ root system, and C) an extension of this idea to ${\tilde {E_{6}}}$ et al. (Which, as far as I can tell, would be in a big way.)

Michael K. Edwards (talk) 04:22, 9 July 2018 (UTC)

Mmm, I stated that badly. The symmetry of $D_{4}$ is interesting, just not for the reasons this article seems to be hinting at. At least as a mathematical amateur and a poor reader of French, I'm unable to find any very meaningful commonality between "triality" in the sense of Tits 1959 (and prior work by Cartan and Chevalley) and the "triality principle" of Ramond 1977 (and apparently Tits 1966). I'm not saying there's no connection between graph automorphisms of $D_{4}$ and automorphisms of ${\mathfrak {so}}(8)$. You can go farther back (to Cartan 1925 and the other papers cited in Robert Bryant's answer to a related question) or find a nice modern exposition in Mikosz and Weber 2013. But that's not the "triality" that relates the exceptional Lie algebras in the Barton-Sudbery construction, which is what this article seems mostly to be hinting at.

Ramond's paper is hardly the first to invoke a "triality principle", just the first (that I know of) to emphasize a modified distributive law for the octonions as a form of "triality". Ramond reframed the construction of the exceptional Lie algebras in these terms, which was the supposed source of inspiration for the Barton and Sudbery approach. The "triality principle" in this line of research doesn't seem all that closely related to the triality automorphism used in the construction of 3D4. (Or if it is, I cannot find that relationship in any of the papers cited.) It's a lot more closely related to the idea of the general derivation on an algebra, generalizing that idea to permit the terms on the RHS of the "transformation law" to contain, not the same transformation as the LHS, but relatives of that transformation.

The triality principle for the octonions can be derived, as in Rubenthaler 2008, from an embedding of the octonions into $E_{6}$. Yes, there's a $D_{4}$ inside $E_{6}$, and in the Rubenthaler construction it's represented differently on the six 8-dimensional "blocks" involved in the octonion embedding. If you squint hard you can see a discrete automorphism of this construction (exchanging the two "spinor representation" blocks by flipping the $A_{2}$-grading) associated with the (2-fold) graph automorphism of $E_{6}$. But nowhere in this construction, or anywhere else that I can find, does the general outer automorphism of $D_{4}$ enter in. And I really wouldn't expect it to, since the third "positive root" block is the "vector representation" block, which is related to the Lie bracket of the two "half-spin" generators. There are automorphisms that permute the roles of the six blocks, but they don't have anything to do with the third branch of $D_{4}$; they're automorphisms of the $A_{2}$-grading, and with the exception of the aforementioned $\alpha _{1}$⇔$\alpha _{2}$ flip, they alter the hyperplane in the root system that separates "positive" from "negative" roots. (They also change the formula for the octonion multiplication in terms of Lie brackets in $E_{6}$.

So I think somebody who knows this area of mathematics well (which is not me) should separate out the "trialities" which originate from outer automorphisms of $D_{4}$ from the ones that don't. The ones that do, probably belong in an article about $D_{4}$ (and ${\mathfrak {so}}(8)$ and SO(8) and Spin(8) and 3D4). The ones that don't, probably belong in an expansion of Freudenthal_magic_square#Triality that captures the nuances of Rubenthaler's $A_{2}$-grading of $E_{6}$. It's not clear to me that this separate Triality article is needed at all.

Michael K. Edwards (talk) 09:09, 9 July 2018 (UTC)

Elaboration of briefly described phenomena would greatly improve this article

The section General formulation ends with this passage:

"By choosing vectors e_i in each V_i on which the trilinear form evaluates to 1, we find that the three vector spaces are all isomorphic to each other, and to their duals. Denoting this common vector space by V, the triality may be re-expressed as a bilinear multiplication

$V\times V\to V$

where each e_i corresponds to the identity element in V. The non-degeneracy condition now implies that V is a composition algebra. It follows that V has dimension 1, 2, 4 or 8. If further F = R and the form used to identify V with its dual is positively definite, then V is a Euclidean Hurwitz algebra, and is therefore isomorphic to R, C, H or O.

Conversely, composition algebras immediately give rise to trialities by taking each V_i equal to the algebra, and contracting the multiplication with the inner product on the algebra to make a trilinear form.

An alternative construction of trialities uses spinors in dimensions 1, 2, 4 and 8. The eight-dimensional case corresponds to the triality property of Spin(8)."

The article would be greatly improved if at least the last two paragraphs quoted were elaborated on clearly.

As they are now, they are much too brief to convey a clear understanding of what they mean.

I hope someone knowledgeable about this subject can fill in this missing information. — Preceding unsigned comment added by 98.36.148.11 (talk) 17:48, November 18, 2024 (UTC)