# nLab Freudenthal magic square

Contents

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

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