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

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

Examples

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

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

Then τ defines a Moufang set structure on J. The Hua maps h_a of the Moufang structure are just the quadratic U_a Template:Harv. Note that the link is more natural in terms of J-structures.

References

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