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Moufang set

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

Definition

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

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

Examples

Let K be a field and X the projective line P(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

x ↦ − x − 1 = − U x − 1 ( x )   . {\displaystyle x\mapsto -x^{-1}=-U_{x}^{-1}(x)\ .}

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 2006). Note that the link is more natural in terms of J-structures.

References

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