G2 and the Rolling Ball
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, 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2012
- DOI:
- 10.48550/arXiv.1205.2447
- arXiv:
- arXiv:1205.2447
- Bibcode:
- 2012arXiv1205.2447B
- Keywords:
-
- Mathematics - Differential Geometry;
- Mathematical Physics;
- 17B25;
- 20G41;
- 17A35;
- 17A75;
- 53D50;
- 51A45
- E-Print:
- 35 pages, 2 png figures, many typos corrected