## Abstract

Understanding the exceptional Lie groups as the symmetry groups of simpler objects is a long-standing program in mathematics. Here, we explore one famous realization of the smallest exceptional Lie group, G_{2}.Its Lie algebra g_{2}acts locally as the symmetries of a ball rolling on a larger ball, but only when the ratio of radii is 1:3. Using the split octonions, we devise a similar, but more global, picture of G_{2}:it acts as the symmetries of a ‘spinorial ball rolling on a projective plane’, again when the ratio of radii is 1:3. We explain this ratio in simple terms, use the dot product and cross product of split octonions to describe the G_{2}incidence geometry, and show how a form of geometric quantization applied to this geometry lets us recover the imaginary split octonions and these operations.

Original language | English |
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Pages (from-to) | 5257-5293 |

Number of pages | 37 |

Journal | Transactions of the American Mathematical Society |

Volume | 366 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2014 |

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