Moufang set

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

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

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

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

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.

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