# Trialities and Exceptional Lie Algebras: DECONSTRUCTING the Magic Square

@article{Evans2009TrialitiesAE, title={Trialities and Exceptional Lie Algebras: DECONSTRUCTING the Magic Square}, author={Jonathan M. Evans}, journal={arXiv: High Energy Physics - Theory}, year={2009} }

A construction of the magic square, and hence of exceptional Lie algebras, is carried out using trialities rather than division algebras. By way of preparation, a comprehensive discussion of trialities is given, incorporating a number of novel results and proofs. Many of the techniques are closely related to, or derived from, ideas which are commonplace in theoretical physics. The importance of symmetric spaces in the magic square construction is clarified, allowing the Jacobi property to be…

## 8 Citations

### Squaring the Magic

- Mathematics
- 2012

We construct and classify all possible Magic Squares (MS's) related to Euclidean or Lorentzian rank-3 simple Jordan algebras, both on normed division algebras and split composition algebras. Besides…

### Octonions in Particle Physics through Structures of Generalised Proper Time

- Physics, Mathematics
- 2019

In considering the nature of the basic mathematical structures appropriate for describing the fundamental elements of particle physics a significant role for the octonions, as an extension from the…

### On the Origin of the Mass without Higgs Bosons

- Physics
- 2010

This note presents an alternative to the Higgs mechanism accounting for mass. Instead mass derives from the entropy of the weak interaction about 10 picoseconds after the Big Bang, when particles…

### Generalised Proper Time as a Unifying Basis for Models with Two Right-Handed Neutrinos

- Physics
- 2019

Models with two right-handed neutrinos are able to accommodate solar and atmospheric neutrino oscillation observations as well as a mechanism for the baryon asymmetry of the universe. While…

### Superconformal Yang-Mills quantum mechanics and Calogero model with $$ {\text{OSp}}\left( {\mathcal{N}|{2},\mathbb{R}} \right) $$ symmetry

- Physics, Mathematics
- 2012

A bstractIn spacetime dimension two, pure Yang-Mills possesses no physical degrees of freedom, and consequently it admits a supersymmetric extension to couple to an arbitrary number, $ \mathcal{N} $…

### Octonions in Particle Physics through Structures of Generalised Proper Time

- Physics, Mathematics
- 2019

In considering the nature of the basic mathematical structures appropriate for describing the fundamental elements of particle physics a significant role for the octonions, as an extension from the…

### A kind of magic

- Mathematics
- 2017

We introduce the extended Freudenthal–Rosenfeld–Tits magic square based on six algebras: the reals R, complexes C, ternions T, quaternions H, sextonions S and octonions O. The sextonionic row/column…

## References

SHOWING 1-10 OF 23 REFERENCES

### Magic squares of Lie algebras

- Mathematics
- 2000

This paper is an investigation of the relation between Tit's magic square of Lie algebras and certain Lie algebras of 3 ×3 and 6 × 6 matrices with entries in alternative algebras. By refor- mulating…

### Cli ord Algebras and the Classical Groups

- Mathematics
- 1995

The Clifford algebras of real quadratic forms and their complexifications are studied here in detail, and those parts which are immediately relevant to theoretical physics are seen in the proper…

### Division Algebras and Supersymmetry I

- Mathematics
- 2009

Supersymmetry is deeply related to division algebras. Nonabelian Yang-Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or 10.…

### The Octonions

- Mathematics
- 2001

The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics.…

### On division algebras

- Mathematics
- 1921

? 1. The object of this paper is to develop some of the simpler properties of division algebras, that is to say, linear associative algebras in which division is possible by any element except zero.…

### Division algebras, (pseudo)orthogonal groups and spinors

- Mathematics
- 1984

The groups SO( nu -1), SO( nu ), SO( nu +1), SO( nu +1, 1) and SO( nu +2, 2) ( nu =1, 2, 4, 8) and their spin representations are described in terms of the division algebras R, C, H and O.

### The Berry Phase of D0-Branes

- Physics
- 2008

We study SU(2) Yang-Mills quantum mechanics with N = 2, 4, 8 and 16 supercharges. This describes the non-relativistic dynamics of a pair of D0-branes moving in d = 3, 4, 6 and 10 spacetime dimensions…