Duality
Understand the difference between triality and duality and what that means for Dynkin diagrams.
D4 looks like Kirby's coordinate system for the tetrahedron.
- The Dynkin diagrams describe how to relate the reflections that generate a group. These reflections are highly relevant in thinking about the Platonic solids, for example. These relations between the reflections are highly constrained, especially as the dimension of the space grows large. Basically they are chained together in a long line with at most one short "leg" sticking out at or near the beginning. But in the particular Dynkin diagram for D4 the line is so short that it equals that leg as in the picture. (F4 is another exception). See Wikipedia: Dynkin diagram.
Notes
- Baez's review of Conway and Smith: The octonions can be described not only as the vector representation of {$\text{Spin}(8)$}, but also the left-handed spinor representation and the right-handed spinor representation. This fact is called 'triality'. It has many amazing spinoffs, including structures like the exceptional Lie groups and the exceptional Jordan algebra, and the fact that supersymmetric string theory works best in 10-dimensional spacetime — fundamentally because {$8 + 2 = 10$}.
- The triality suggests that the Father and the Son are oppositely handed spinor representations and the vector representation is the Spirit? Or the unconscious and conscious are oppositely handed and the vector representation is consciousness?
- Yu. I. Manin. Cubic Forms: Algebra, Geometry, Arithmetic. Related to ternary operations and triality.
- Triality: C at the center, three legs: quaternions, even-dimensional reals, odd-dimensional reals. Fold, fuse, link.
- John Baez. Triality of vector spaces forces their dimension to be 1, 2, 4 or 8.
- Triality - the coming together of three conceptions - is divinely miraculous.
- Consider triality as variously manifesting all the forms of duality.
- Jacques Distler. G2 and Spin(8) Triality.
- SO(8) has triality. The generators become spinors.
- Math Overflow: Triality of Spin(8)
- SO(8) and triality
- Rodolfo De Sapio. On Spin(8) and Triality: A Topological Approach.
- We can consider triality as given by Dynkin diagrams with a threefold branching, the D-series and the E-series. Here we may have three different lowering operators but one raising operator. At the root we may have three lowering operators and three raising operators, thus two trinities, a sixfold structure.
- https://johncarlosbaez.wordpress.com/2021/12/24/the-binary-octahedral-group-part-2/
- Baez: I don’t know the deep meaning of this fact. I know that the vertices of the 24-cell correspond to the 24 roots of the Lie algebra \mathfrak{so}(8). I know that the famous ‘triality’ symmetry of \mathfrak{so}(8) permutes the three cross-polytopes in the 24-cell, which are in some rather sneaky way related to the three 8-dimensional irreducible representations of \mathfrak{so}(8). I also know that if we take the two 24-cells in the binary octahedral group, and expand one by a factor of \sqrt{2}, so the vertices of other lie exactly at the center of its faces, we get the 48 roots of the Lie algebra \mathfrak{f}_4. But I don’t know how to extend this story to get a nice story about the six cross-polytopes in the binary octahedral group.
- Could {$D_4$} be of special significance? {$D_4$} has triality - thus this occurs within the classical root systems. And {$D_4$} has 24 root vectors. {$24 = 2\times 4\times 3 = 2n(n-1)$}. The Weyl group has order {$2^{(n-1)}n!= 48$}.
- In 8 Euclidean dimensions, there are two Weyl–Majorana real 8-dimensional representations that are related to the 8-dimensional real vector representation by a special property of Spin(8) called triality.
- Mia Hughes's thesis explains triality