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