Moufang set

In mathematics, a Moufang set is a particular kind of combinatorial system named after Ruth Moufang.

Definition

A Moufang set is a pair $\left({X;\{U_{x}\}_{x\in X}}\right)$ where X is a set and $\{U_{x}\}_{x\in X}$ is a family of subgroups of the symmetric group $\Sigma _{X}$ indexed by the elements of X. The system satisfies the conditions

$U_{y}$ fixes y and is simply transitive on $X\setminus \{y\}$ ;
Each $U_{y}$ normalises the family $\{U_{x}\}_{x\in X}$ .

Examples

Let K be a field and X the projective line P¹(K) over K. Let U_x 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

x\mapsto -x^{-1}=-U_{x}^{-1}(x)\ .

Then τ defines a Moufang set structure on J. The Hua maps h_a of the Moufang structure are just the quadratic U_a (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" (PDF). 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.