Division Algebras, Magic Squares and Supersymmetry
Of the four exclusive normed division algebras, only the real and complex numbers prevail in both mathematics and physics. The noncommutative quaternions and the nonassociative octonions have found limited physical applications. In mathematics, division algebras unify both classical and exceptional Lie algebras with the exceptional ones appearing in a table known as the magic square generated by tensor products of division algebras. This work reviews the normed division algebras and the magic square as well as necessary preliminaries for its construction. Space-time transformations, pure super Yang-Mills theories in space-time dimensions D = 3, 4, 6, 10, dimensional reduction and truncation of supersymmetry are also described here by the four division algebras. Supergravity theories, seen as tensor products of super Yang-Mills theories, are described as tensor products of division algebras leading to the identi cation of a magic square of supergravity theories with their U-duality groups as the magic square entries, providing applications of all division algebras to physics and suggesting division algebraic underpinnings of supersymmetry. Other curious uses of octonions are also mentioned.