nLab Freudenthal magic square

Idea

A special square array of Lie algebras/Lie groups labeled by pairs of normed division algebras and including all the exceptional Lie groups except G₂. Three constructions give this square, namely, the Tits construction, the Vinberg construction and the triality construction (see BaezMagic).

Michael Atiyah (AtiyahMagic) has proposed using the magic square to explain the Kervaire invariant problem similarly to how the existence of the four normed division algebras explains the Hopf invariant one problem.

Extensions

• Several mathematicians have constructed a larger triangle of Lie groups, known as the magic triangle, which contains the magic square.

• Michael Duff and colleagues (ABDHN) have proposed a magic pyramid whose base is the magic square.

• Bruce Westbury (Westbury) has proposed an extension to a $5 \times 5$-square with the new row and column indexed by the sextonions.

Related concepts

• magic triangle

• magic pyramid

• magic supergravity

References

• Hans Freudenthal, Lie groups in the foundations of geometry, Advances in Mathematics, volume 1, (1965) pp. 145 - 190 (dspace)

• Tonny Springer, Ferdinand Veldkamp, chapter 8.5 of Octonions, Jordan Algebras, and Exceptional Groups, Springer Monographs in Mathematics, 2000

• John Baez, 4.3 The Magic Square, webpage

• Michael Atiyah, The Geometry and Topology of the Freudenthal Magic Square, videos

• Bruce Westbury, Sextonions and the magic square, (pdf)

See also

• Wikipedia, Freudenthal magic square

Discussion in relation to super Yang-Mills theory, supergravity and U-duality includes

• Leron Borsten, Michael Duff, L. J. Hughes, S. Nagy, A magic square from Yang-Mills squared (arXiv:1301.4176)

• A. Anastasiou, Leron Borsten, Michael Duff, L. J. Hughes, S. Nagy, A magic pyramid of supergravities, arXiv:1312.6523

See also at supersymmetry and division algebras.