• Corpus ID: 50818244

G2 and the Rolling Ball

@article{Baez2012G2AT,
  title={G2 and the Rolling Ball},
  author={John C. Baez and John Huerta},
  journal={arXiv: Differential Geometry},
  year={2012}
}
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… 
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References

SHOWING 1-10 OF 31 REFERENCES
G_2 and the "Rolling Distribution"
Associated to the problem of rolling one surface along another there is a five-manifold M with a rank two distribution. If the two surfaces are spheres then M is the product of the rotation group
Quaternions and octonions in Mechanics
In fact, this group is Spin(3), the 2-fold cover of SO(3), the group of rotations of R. This has been known for quite some time and is perhaps the simplest realization of Hamilton’s expectations
Buildings and shadows
This is an expository paper describing geometries associated to the groups of Lie type, following Jacques Tits' theory of buildings and his earlier theory of shadows. The geometry of shadows in the
Split octonions and generic rank two distributions in dimension five
In his famous five variables paper Elie Cartan showed that one can canonically associate to a generic rank 2 distribution on a 5 dimensional manifold a Cartan geometry modeled on the homogeneous
On variational approach to differential invariants of rank two distributions
The Octonions
The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics.
Buildings of Spherical Type and Finite BN-Pairs
These notes are a slightly revised and extended version of mim- graphed notes written on the occasion of a seminar on buildings and BN-pairs held at Oberwolfach in April 1968. Their main purpose is
Élie Cartan and geometric duality
It is a great honor for me to be asked to give a lecture about the work of Élie Cartan at the institute that was founded in his name. When I was asked to do this, I was immediately beset by doubts as
Geometric Quantization
We review the definition of geometric quantization, which begins with defining a mathematical framework for the algebra of observables that holds equally well for classical and quantum mechanics. We
Buildings: Theory and Applications
TLDR
This book treats Jacques Tits's beautiful theory of buildings, making that theory accessible to readers with minimal background, and focuses on all three approachs to buildings, “old-fashioned," combinatorial (chamber systems), and metric so that the reader can learn all three or focus on only one.
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