2012/05/11 by John C. Baez, John Huerta · 2 voices
Mathematics · Physics and Astronomy · #math-ph #math.DG #math.MP #msc:17A35 #msc:17A75 #msc:17B25 #msc:20G41 #msc:51A45 #msc:53D50
published as Trans. Amer. Math. Soc., Vol. 366 No. 10 (2014), 5257-5293 · 35 pages, 2 png figures, many typos corrected
arxiv published 2012/05/11 · arxiv created 2015/02/11 · arxiv updated 2017/08/22
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, G2. Its Lie algebra 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 G2: 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 G2 incidence geometry, and show how a form of geometric quantization applied to this geometry lets us recover the imaginary split octonions and these operations.