• Corpus ID: 119096562

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}
}
  • Jonathan M. Evans
  • Published 9 October 2009
  • Mathematics
  • arXiv: High Energy Physics - Theory
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… 

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References

SHOWING 1-10 OF 23 REFERENCES

Magic squares and matrix models of Lie algebras

Magic squares of Lie algebras

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

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

Supersymmetric {Yang-Mills} Theories and Division Algebras

Division Algebras and Supersymmetry I

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

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

? 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

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

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

Super p-Branes