Moufang identities $$x(y⋅xz)=(xy⋅x)z,$$ $$(zx⋅y)x=z(x⋅yx),$$ $$xy⋅zx=x(yz⋅x)$$ are identities deeply related with alternativity (since setting $z=1$ one recovers left and right alternativity), while a Moufang plane is a strictly geometrical concept that encodes the fact that the group of automorphisms that fixes every point of a line acts transitively on the points of the plane, not on the line. I know it is a well established fact that Moufang identities are deeply related to the geometric properties of the Moufang plane, but I could not find any direct proof. Can someone explain the main lines of the proof or at least give me a reference that directly addresses the question?
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$\begingroup$ please do not give me references in german... $\endgroup$– Dac0Commented Jul 10, 2023 at 10:25
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$\begingroup$ apart from the listed alternative quasigroup identities for the alternative division rings, you also need the additive structure of an Abelian group, and the usual distributive laws. $\endgroup$– Dima PasechnikCommented Jul 13, 2023 at 12:23
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$\begingroup$ @DimaPasechnik yes, please assume those (I didn't specified it but I'm assuming an alternative algebra) $\endgroup$– Dac0Commented Jul 14, 2023 at 9:44
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3 Answers
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This is discussed and proved in detail in Hall, Marshall jun., The theory of groups, New York: The Macmillan Company. xiii, 434 p. (1959). ZBL0084.02202, specifically in chapter 20 "Group Theory and Projective Planes", and in there in section 20.5 "Moufang and Desarguesian Planes".
See also https://www.ams.org/notices/200710/tx071001294p.pdf for a quick overview of the idea, esp. the definition of "ternary rings" and how to use them to coordinatize projective planes.
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$\begingroup$ Thank you very much for the references. This is what I was looking for $\endgroup$– Dac0Commented Jul 20, 2023 at 4:10
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The book "Moufang polygons" by Tits and Weiss provides a complete account on the more general question of Moufang $n$-gons (for Moufang planes, $n=3$).
One can read some parts of it to get the full answer. In Chapter 19 one finds a proof that Moufang plane (Moufang triangle, in terminology of the book) arises from alternative division rings. The latter are classified by Bruck-Kleinfeld theorem.
Moufang condition is weaker than the $(p,L)$-transitivity condition for the collineation group of a projective plane (a condition that is equivalent to the plane being Desarguesian), see e.g. Thm 2.2 in P.J.Cameron's lecture notes. So the proof is longer, too.
answered Jul 15, 2023 at 10:24
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$\begingroup$ Pasenick. While I appreciate the reference thus I upvoted the question, the answer to my question doesn't seem to be addressed there. Can you point a specific Chapter where you found the answer to my question? Thank you $\endgroup$– Dac0Commented Jul 15, 2023 at 16:11
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$\begingroup$ I added few details. Already an analogous proof for the Desarguesian planes is a bit tricky, so it's a bit hard to explain in such a post in detail. $\endgroup$ Commented Jul 17, 2023 at 12:11
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A very accessible book for such connections between geometric and algebraic properties in general, is John Faulkner's "The Role of Nonassociative Algebra in Projective Geometry" (https://bookstore.ams.org/gsm-159) from 2014.
More specifically, Theorem 3.17 is probably what you are looking for, but you might also be interested in the "translation" into configuration properties (rather than transitivity properties) in Theorem 5.18.
answered Jul 19, 2023 at 9:34
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1$\begingroup$ Thank you for the reference, it seems a very nice work and probably I will read it in its entirety $\endgroup$– Dac0Commented Jul 20, 2023 at 4:13