School Seminars and Colloquia

G2 and the Rolling Ball

Algebra/Geometry/Topology Seminar

by John Huerta


Institution: Australian National University
Date: Fri 12th October 2012
Time: 3:15 PM
Location: 213 Richard Berry

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 \(\mathfrak{g}_2\) 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 \(G_2\): it acts as the symmetries of a
`spinorial ball rolling on a projective plane', again when the ratio
of radii is 1:3. We describe the incidence geometry of both systems,
and use it to explain the mysterious 1:3 ratio in simple, geometric
terms.