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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

$ \mathrm{G}_2$ and the rolling ball


Authors: John C. Baez and John Huerta
Journal: Trans. Amer. Math. Soc. 366 (2014), 5257-5293
MSC (2010): Primary 20G41, 17A75; Secondary 57S25, 51A45, 20E42
Published electronically: May 15, 2014
MathSciNet review: 3240924
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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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Additional Information

John C. Baez
Affiliation: Department of Mathematics, University of California, Riverside, California 92521 — and — Centre for Quantum Technologies, National University of Singapore, Singapore 117543
Email: baez@math.ucr.edu

John Huerta
Affiliation: Department of Theoretical Physics, Research School of Physics and Engineering — and — Mathematical Sciences Institute, The Australian National University, Canberra, ACT 0200, Australia
Email: john.huerta@anu.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-05977-1
PII: S 0002-9947(2014)05977-1
Keywords: Split $G_2$, split octonions, rolling ball, (2, 3, 5) distributions, buildings
Received by editor(s): August 16, 2012
Received by editor(s) in revised form: September 28, 2012, and October 4, 2012
Published electronically: May 15, 2014
Article copyright: © Copyright 2014 John C. Baez and John Huerta