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).
 A: 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.

A: 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.
