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Tevian Dray
Using Division Algebras to Describe Symmetry, with Applications to Physics
Abstract: Quaternions are often used to describe rotations in 3 (Euclidean) dimensions. Several generalizations of this fundamental idea will be discussed, notably the extension to the octonions and the inclusion of spinor transformations as well as vector rotations. The symmetry groups described by the resulting framework include the Lorentz group in 3, 4, 6, and 10 (spacetime) dimensions, which are precisely the dimensions in which classical supersymmetry holds. This framework culminates in the well-known Tits-Freudenthal magic square of Lie algebras, providing a unified treatment of the exceptional Lie groups. Some applications of particle physics will be briefly mentioned if time permits.
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