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Freudenthal magic square

Contents

Context

Algebra

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

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 5×55 \times 5-square with the new row and column indexed by the sextonions.

References

Named after Hans Freudenthal.

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

See also at supersymmetry and division algebras.

Last revised on September 20, 2016 at 00:37:11. See the history of this page for a list of all contributions to it.