G2 and the Rolling Ball

• Published in 2012
• Added on 2012-05-14

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 using the incidence geometry of G2, and show how a form of geometric quantization applied to this system gives the imaginary split octonions.

Links

• http://arxiv.org/abs/1205.2447
• http://arxiv.org/pdf/1205.2447v4

Other information

BibTeX entry

@article{Baez2012,
	key = {Baez2012},
	type = {article},
	title = {G2 and the Rolling Ball},
	author = {Baez, John C and Huerta, John},
	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 using the incidence geometry of G2, and show how a form of geometric quantization applied to this system gives the imaginary split octonions.},
	comment = {},
	date_added = {2012-05-14},
	date_published = {2012-05-01},
	urls = {http://arxiv.org/abs/1205.2447,http://arxiv.org/pdf/1205.2447v4},
	collections = {Basically physics},
	archivePrefix = {arXiv},
	arxivId = {1205.2447},
	eprint = {1205.2447},
	month = {may},
	pages = 28,
	url = {http://arxiv.org/abs/1205.2447 http://arxiv.org/pdf/1205.2447v4},
	year = 2012,
	primaryClass = {math.DG},
	urldate = {2012-05-14}
}