# Examples of triality in mathematics

There are tons of interesting examples of duality in mathematics (Poincaré duality, Verdier duality, Stone duality, s-duality, Tannaka duality, Koszul duality, Spanier-Whitehead duality ... ). What examples are there of triality in mathematics?

Note: this is not a duplicate of the question about trichotomies in mathematics. A trichotomy is any sort of classification into three. A triality is a classification into three where the relationship between those three is some sort of equivalence relation (especially one emphasizing that though things appear opposite they are in some sense the same).

• I think triality more often than not refers to operations involving the exceptional degree-$3$ outer automorphism of $Spin(8)$. – W. Cadegan-Schlieper Sep 19 '17 at 6:07
• There are a number of references here, not all of them are solely on Spin(8) mathoverflow.net/q/116666/4177 – David Roberts Sep 19 '17 at 7:03
• Not a duplicate. A trichotomy is any sort of classification into three, a triality is a classification into three where the relationship between those three is some sort of equivalence relation (more specifically, some sort of equivalence relation emphasizing how things which appear opposite are actually the same). – Trent Sep 19 '17 at 16:21
• @Trent You may wish to edit some elaboration of that comment into your question. Perhaps you can give a more precise definition of "triality", or list some specific features of the dualities you'd like to see generalized. – j.c. Sep 19 '17 at 17:10
• I like the question, but I agree with @j.c.: I think it takes some considerable work (at least for me) even to understand what it is, aside from the presence of 3 arms in the Dynkin diagram, that makes triality deserve the name 'triality'; so I'm not sure that I'd recognise another example of the same phenomenon when I saw one, even with the clarification in your comment above (since edited into the question). – LSpice Sep 20 '17 at 20:20

There are uses of ternary structures in the sense of $Z_3$-graded structures as generalizations of super-geometry (or $Z_2$-graded structures) in theoretical physics.

See papers by Richard Kerner:

Ternary and non-associative structures, “International Journal of Geometric Methods in Physics”, (volume en l’honneur de M. Dubois-Violette), Vol. 5,No. 8, pp. 1265-1294 (2008)

A Z3-generalization of Pauli’s principle, quark algebra and Lorentz invariance, AIP Conference Proceedings (International School of Field Theory and Gravitation, Petropolis 2011, Brésil), 1483, pp. 144-168 (2012)

and others on his homepage.

There is also the use of non-degenerate 3-forms as a generalization of symplectic manifolds, see

• Ševera, Pavol; Weinstein, Alan Poisson geometry with a 3-form background. Noncommutative geometry and string theory (Yokohama, 2001). Progr. Theoret. Phys. Suppl. No. 144 (2001), 145–154.

An interesting duality is polarity with respect to a conic in the projective plan. A conic is determined by a bilinear form $q$ (the equation of the conic being $q(u,u)=0$) and one way to describe the polarity is to say that the polar of a point corresponding to a vector $v$ in the underlying $3$-space is the line whose equation is $q(v,\cdot)=0$.

Now, this yields an immediate equivalent for trialities: consider a cubic, it is determined by a trilinear form $t(\cdot,\cdot,\cdot)$ and two points (with corresponding vectors $u$, $v$) and a line are in triality if $t(u,v,\cdot)=0$ is an equation of the line.

• Being a relationship between “two points and a line”, this one rather seems to lose the symmetry of the original duality, so feels less compelling to me as a candidate for “triality”. – Peter LeFanu Lumsdaine Sep 21 '17 at 6:39
• @PeterLeFanuLumsdaine Well, in duality you usually have two objects of different nature, so this seems difficult to avoid. You can see the line as a point in the dual space if you prefer (so, given a cubic, every two point define a polar point in the dual space) or use a conic to be able to identify lines with points. – Benoît Kloeckner Sep 21 '17 at 7:10
• @BenoîtKloeckner Can't we generalize to this higher 'ality'? – VS. Mar 6 '20 at 21:58
• @VS. yes, you can simply use higher-degree homogeneous polynomials. – Benoît Kloeckner Mar 8 '20 at 10:35
• @BenoîtKloeckner Perhaps you may want to give a general answer. – VS. Mar 8 '20 at 12:13