Harman Patil (Editor)

Moufang set

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

Contents

Definition

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

  • U y fixes y and is simply transitive on X ∖ { y } ;
  • Each U y normalises the family { U x } x ∈ X .

    • 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

    x ↦ − x − 1 = − U x − 1 ( x )   .

    Then τ defines a Mounfang 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

    Moufang set Wikipedia
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