Moufang set

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

• $U\_y$ fixes y and is simply transitive on $X\; \backslash setminus\; \backslash \{y\backslash \}$;
• Each $U\_y$ normalises the family $\backslash \{U\_x\backslash \}\_\{x\; \backslash in\; X\}$.

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

:$x\; \backslash mapsto\; -x^\{-1\}\; =\; -\; U\_x^\{-1\}(x)\; \backslash \; .$

Then τ defines a Moufang set structure on J. The Hua maps h_a of the Moufang structure are just the quadratic U_a {{harv|De Medts|Weiss|2006}}. Note that the link is more natural in terms of J-structures.

• {{cite journal | last1=De Medts | first1=Tom | last2=Segev | first2=Yoav | title=Identities in Moufang sets | zbl=1179.20030 | journal= Transactions of the American Mathematical Society| volume=360 | number=11 | pages=5831–5852 | year=2008 | doi=10.1090/S0002-9947-08-04414-0 | doi-access=free }}
• {{cite journal | last1=De Medts | first1=Tom | last2=Segev | first2=Yoav | title=A course on Moufang sets | zbl=1233.20028 | journal= Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial| volume=9 | pages=79–122 | year=2009 | doi=10.2140/iig.2009.9.79 | url=https://projecteuclid.org/journals/innovations-in-incidence-geometry/volume-9/issue-none/A-course-on-Moufang-sets/10.2140/iig.2009.9.79.pdf | doi-access=free }}
• {{cite journal | last1=De Medts | first1=Tom | last2=Weiss | first2=Richard M. | title=Moufang sets and Jordan division algebras | zbl=1163.17031 | journal= Mathematische Annalen| volume=335 | number=2 | pages=415–433 | year=2006 | url=http://cage.ugent.be/~tdemedts/preprints/moufsets.pdf | doi=10.1007/s00208-006-0761-8 }}
• {{cite journal | title=Proper Moufang sets with abelian root groups are special | first=Yoav | last=Segev | journal= Journal of the American Mathematical Society| volume=22 | issue=3 | year=2009 | pages=889–908 | mr=2505304 | zbl=1248.20031 | doi=10.1090/S0894-0347-09-00631-6 | bibcode=2009JAMS...22..889S | doi-access=free }}
• {{cite book | last=Tits | first=Jacques | authorlink=Jacques Tits | chapter=Twin buildings and groups of Kac–Moody type | zbl=0851.22023| pages=249–286 | title=Groups, Combinatorics and Geometry | volume=165 | series=London Mathematical Society Lecture Note Series | issn=0076-0552 |editor-link1=Martin Liebeck | editor1-first=Martin W. | editor1-last=Liebeck | editor2-first=Jan | editor2-last=Saxl | publisher=Cambridge University Press | year=1992 | isbn=978-0-521-40685-7 }}

Category:Group theory