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 [math]\displaystyle{ \left({ X; \{U_x\}_{x \in X} }\right) }[/math] where X is a set and [math]\displaystyle{ \{U_x\}_{x \in X} }[/math] is a family of subgroups of the symmetric group [math]\displaystyle{ \Sigma_X }[/math] indexed by the elements of X. The system satisfies the conditions

• [math]\displaystyle{ U_y }[/math] fixes y and is simply transitive on [math]\displaystyle{ X \setminus \{y\} }[/math];
• Each [math]\displaystyle{ U_y }[/math] normalises the family [math]\displaystyle{ \{U_x\}_{x \in X} }[/math].

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

[math]\displaystyle{ x \mapsto -x^{-1} = - U_x^{-1}(x) \ . }[/math]

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). 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. 
• 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. https://projecteuclid.org/journals/innovations-in-incidence-geometry/volume-9/issue-none/A-course-on-Moufang-sets/10.2140/iig.2009.9.79.pdf. 
• De Medts, Tom; Weiss, Richard M. (2006). "Moufang sets and Jordan division algebras". Mathematische Annalen 335 (2): 415–433. doi:10.1007/s00208-006-0761-8. http://cage.ugent.be/~tdemedts/preprints/moufsets.pdf. 
• Segev, Yoav (2009). "Proper Moufang sets with abelian root groups are special". Journal of the American Mathematical Society 22 (3): 889–908. doi:10.1090/S0894-0347-09-00631-6. Bibcode: 2009JAMS...22..889S. 
• Tits, Jacques (1992). "Twin buildings and groups of Kac–Moody type". in Liebeck, Martin W.; Saxl, Jan. Groups, Combinatorics and Geometry. London Mathematical Society Lecture Note Series. 165. Cambridge University Press. pp. 249–286. ISBN 978-0-521-40685-7.