Moufang set
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In mathematics, a Moufang set is a particular kind of combinatorial system named after Ruth Moufang.
Definition
A Moufang set is a pair where X is a set and is a family of subgroups of the symmetric group indexed by the elements of X. The system satisfies the conditions
- fixes y and is simply transitive on ;
- Each normalises the family .
Examples
Let K be a field and X the projective line P1(K) over K. Let Ux be the stabiliser of each point x in the group PSL2(K). The Moufang set determines K up to isomorphism or anti-isomorphism: an application of Hua's identity.
A quadratic Jordan division algebra gives rise to a Moufang set structure. If U is the quadratic map on the unital algebra J, let τ denote the permutation of the additive group (J,+) defined by
Then τ defines a Moufang set structure on J. The Hua maps ha of the Moufang structure are just the quadratic Ua (De Medts & Weiss 2006). Note that the link is more natural in terms of J-structures.
References
- De Medts, Tom; Segev, Yoav (2008). "Identities in Moufang sets". Transactions of the American Mathematical Society. 360 (11): 5831–5852. doi:10.1090/S0002-9947-08-04414-0. Zbl 1179.20030.
- De Medts, Tom; Segev, Yoav (2009). "A course on Moufang sets". Innovations in Incidence Geometry. 9: 79–122. doi:10.2140/iig.2009.9.79. Zbl 1233.20028.
- De Medts, Tom; Weiss, Richard M. (2006). "Moufang sets and Jordan division algebras" (PDF). Mathematische Annalen. 335 (2): 415–433. doi:10.1007/s00208-006-0761-8. Zbl 1163.17031.
- Segev, Yoav (2009). "Proper Moufang sets with abelian root groups are special". Journal of the American Mathematical Society. 22 (3): 889–908. Bibcode:2009JAMS...22..889S. doi:10.1090/S0894-0347-09-00631-6. MR 2505304. Zbl 1248.20031.
- Tits, Jacques (1992). "Twin buildings and groups of Kac–Moody type". In Liebeck, Martin W.; Saxl, Jan (eds.). Groups, Combinatorics and Geometry. London Mathematical Society Lecture Note Series. Vol. 165. Cambridge University Press. pp. 249–286. ISBN 978-0-521-40685-7. ISSN 0076-0552. Zbl 0851.22023.