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Moufang Sets and Structurable Division Algebras
About this Title
Lien Boelaert, Ghent University, Department of Mathematics, Algebra and Geometry, Krijgslaan 281-S25, 9000 Gent, Belgium, Tom De Medts, Ghent University, Department of Mathematics, Algebra and Geometry, Krijgslaan 281-S25, 9000 Gent, Belgium and Anastasia Stavrova, St. Petersburg State University, Chebyshev Laboratory, 14th Line V.O. 29B, 199178 Saint Petersburg, Russia
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 259, Number 1245
ISBNs: 978-1-4704-3554-7 (print); 978-1-4704-5245-2 (online)
DOI: https://doi.org/10.1090/memo/1245
Published electronically: April 10, 2019
Keywords: Structurable algebra,
Jordan algebra,
Moufang set,
root group,
simple algebraic group,
5-graded Lie algebra
MSC: Primary 16W10, 20E42, 17A35, 17B60, 17B45, 17Cxx, 20G15, 20G41
Table of Contents
Chapters
- Introduction
- 1. Moufang sets
- 2. Structurable algebras
- 3. One-invertibility for structurable algebras
- 4. Simple structurable algebras and simple algebraic groups
- 5. Moufang sets and structurable division algebras
- 6. Examples
Abstract
A Moufang set is essentially a doubly transitive permutation group such that each point stabilizer contains a normal subgroup which is regular on the remaining vertices; these regular normal subgroups are called the root groups, and they are assumed to be conjugate and to generate the whole group.
It has been known for some time that every Jordan division algebra gives rise to a Moufang set with abelian root groups. We extend this result by showing that every structurable division algebra gives rise to a Moufang set, and conversely, we show that every Moufang set arising from a simple linear algebraic group of relative rank one over an arbitrary field $k$ of characteristic different from $2$ and $3$ arises from a structurable division algebra.
We also obtain explicit formulas for the root groups, the $\tau$-map and the Hua maps of these Moufang sets. This is particularly useful for the Moufang sets arising from exceptional linear algebraic groups.
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