𝐺₂ and the rolling ball

Baez, John; Huerta, John

doi:10.1090/S0002-9947-2014-05977-1

$\mathrm {G}_2$ and the rolling ball
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by John C. Baez and John Huerta PDF

Trans. Amer. Math. Soc. 366 (2014), 5257-5293

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, $\mathrm {G}_2$. Its Lie algebra $\mathfrak {g}_2$ 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 $\mathrm {G}_2$: 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 $\mathrm {G}_2$ incidence geometry, and show how a form of geometric quantization applied to this geometry lets us recover the imaginary split octonions and these operations.

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