Moufang set explained
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
;
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 . Note that the link is more natural in terms of
J-structures.
References
- De Medts . Tom . Segev . Yoav . Identities in Moufang sets . 1179.20030 . Transactions of the American Mathematical Society. 360 . 11 . 5831–5852 . 2008 . 10.1090/S0002-9947-08-04414-0 . free .
- De Medts . Tom . Segev . Yoav . A course on Moufang sets . 1233.20028 . Innovations in Incidence Geometry . 9 . 79–122 . 2009 . 10.2140/iig.2009.9.79 . free .
- De Medts . Tom . Weiss . Richard M. . Moufang sets and Jordan division algebras . 1163.17031 . Mathematische Annalen. 335 . 2 . 415–433 . 2006 . 10.1007/s00208-006-0761-8 .
- Proper Moufang sets with abelian root groups are special . Yoav . Segev . Journal of the American Mathematical Society. 22 . 3 . 2009 . 889–908 . 2505304 . 1248.20031 . 10.1090/S0894-0347-09-00631-6 . 2009JAMS...22..889S . free .
- Book: Tits, Jacques . Jacques Tits
. Jacques Tits . Twin buildings and groups of Kac–Moody type . 0851.22023. 249–286 . Groups, Combinatorics and Geometry . 165 . London Mathematical Society Lecture Note Series . 0076-0552 . Martin Liebeck . Martin W. . Liebeck . Jan . Saxl . . 1992 . 978-0-521-40685-7 .