What's a Moufang loop?
In 1935 Ruth Moufang published a paper "Zur Struktur von Alternativkörpern", where she intoduced an algebra very similar to groups, but nonassociative.
Back then people said quasigroup instead of loop. A loop is simply a complete operation with inverses and an identity element. Today, a quasigroup need not have an identity.
Let's look at its definition. The Moufang identities are a generalization of associativity:
More precisely, they generalize alternative algebras:
By setting various variables to the identity, we can derive the laws of alternative algebra, which really do look a lot like the associative law:
There's also the flexible identity:
but any two can prove the remaining one.
One consequence, which justifies the name, is that the associator
of any alternating algebra is totally skew symmetric:
But I digress...
Moufang's inverse property
A Moufang loop also requires the inverse property:
which implies that left and right inverses exist. Using that, we can reformulate the first two Moufang identites in a more practical form:
Looking at it differently
There's also an operator version of of these, which might be easier to remember:
And another, equivalent to the remaining identities:
In 1937, Gerrit Bol introduced another kind of loop:
If both hold, then the Moufang identities will also hold.
They are useful in general relativity!
A first example: the octonions
Octonion multiplication without zero (it has no inverse) forms a Moufang loop! Actually, they are the prime example, easily remembered, and Ruth Moufang used them as example in her paper. But I'll talk about other examples below. Read on!
Reading Moufang's paper
As her paper is in german, let me give you a short synopsis of what it's about:
In a quasigroup, any two elements generate a group! As do three elements which are associative among themselves.
Given that, she introduces the inverse property, and says it's easy to get a skew ring from there. She then explains what an alternative ring is, introducing the alternative identities. And goes on to derive what we know as Moufang identities today. .
In her arguments she references Cayleys number system, the octonions! A notable trick is applied: the flexible identity allows us to also read any identity from right to left!
And then she proves her claims.
I like the style of her paper. Almost all claims and all curious facts are mentioned right at the beginning, and only then come the technicalities, which are still worth looking at, because some of the tricks are too nice to skim.
The paper ends inspecting the geometrical meaning, looking at Desargue's little theorem, and pascal's configuration (his hexagrammum mysticum theorem, from a time when people gave names to their ideas).
Let's get back to our story:
Desargues' little theorem is just Desargues' theorem with the additional restriction that the center of perspectivity has to lie on the axis of perspectivity. So it's less general.
Did you know that Desargues' theorem itself is equivalent to associativity? So while the Moufang identities look like a weird phantasm, a monstrous mutation of the law of associativity, it's actually a very reasonable, even conservative generalization!
A projective geometry is said to be Desarguesian if whenever two triangles are perspective from a point, they are perspective from a line, and vice versa. If this property fails, it is said to be non-Desarguesian. – Charles Weibel
Here is another nice characterization of Moufang planes I found on Wikipedia:
The group of automorphisms fixing all points of any given line acts transitively on the points not on the line.
Or, closer to the algebra, we can say that any Moufang planes arises as projective plane over an alternative division ring. Or vice versa, any isomorphism class of alternative division rings corresponds to a Moufang plane!
According to Artin-Zorn's theorem, every finite alternative division ring is a field, and that means that all finite Moufang planes are Desarguesian. But the infinite plane over the octonions is a non-Desarguesian example, so they exist! Are there more? What are they?
Those translation planes (a kind of non-Desarguian projective plane) in which little Desargues still holds for every line are Moufang planes!
Planar ternary rings are related, and quasifields can be used to coordinatize translation planes, but I know even less about these things.
If that doesn't help, Charles Weibel's wrote a nice "Survey of Non-Desarguesian Planes", p1294 ff here: https://www.ams.org/notices/200710/200710FullIssue.pdf
So let's talk about some of these... later.
Michael Kinyon wrote a nice thread about the Larange's theorem about groups, but for Moufang loops: Does the order of any subloop divide the order of the larger loop?
How are Moufang loops like groups?
Last time, I defined the associator as a difference, which makes good sense as long as we are talking about octonions or matrix groups:
But in the context of groups or quasigroups we only have one operation available, so here's a better way to define an associator:
With that out of the way, let's, for simplicity, look at commutative Moufang loops (CML). Here, it suffices to require either
to derive the rest. Supposedly, to see the similarity with groups, it's helpful to also stare at power associativity:
However that didn't click with me. What seems much more familiar, is the idea to think about the associative center of a CML. That is the set all elements for which associativity holds:
That is similar to noncommutative groups, where we often study their commutative center
Hence we can say that a CML arises from a commutative group by extending it with nonassociative elements. We can express this using a short exact sequence:
where, as usual, maps to the kernel of , and then say that extends by . Or we could define using a quotient:
The quotient of a CML by its center is a CML of exponent 3. This means
holds for any . This leads to two deep theorems:
Theorem: The center of a CML of exponent 3 is non-trivial.
Theorem: Any CML of exponent 3 with finitely many generators is finite.
Cubic hypersurface quasigroups
Remember groups derived from cubic curves? Given any two points x,y on the curve, the line through these meets the curve once more in . Of course, this works best in projective space. Anyways, we get a 3-way relation which holds for any permutation of its arguments:
At first I thought that just means that every element is its own inverse, which is certainly true for groups coming from cubic curves, but it seems that's not always true.
Generally, if we were to forget the identity element of a group, what remains is a quasigroup. If the above holds, then we may call it a CH (cubic hypersurface) quasigroup.
CH-quasigroups were introduced by Yuri Ivanovich Manin. Note that his notion is not limited to 1d curves, but the analogy with cubic curves doesn't work in higher dimensions.
If we then introduce a new identity element by defining a new composition like this
what we end up with is not a group, but a CML! Different choices for u yield isomorphic variants. We can also go the other way around. Pick any from , then
turns into a CH-quasigroup. However, different choices for need not give isomorphic CH-quasigroups!
(to be continued)
I learned all this stuff from Yuri Manin's book, which is about number theory. You know, the kind which is concerned with solving Diophantine equations, considers elliptic curves and ends up doing lots of algebraic geometry.
Its first chapter introduces quasigroups much better than I might, and after that goes on to prove much of stuff I summarized. If you're serious about Moufang loops, you might want to get a copy.
Yuri Ivanovich Manin, "Cubic Forms: Algebra, Geometry, Arithmetic"
Here's an Interview with Manin, where he speaks about his book:
For the full interview see here:
What's Mal'cev algebra?
There's even a smooth version, similar to a Lie algebra, introduced by Quillen following the work of Anatoly Ivanovich Mal'cev.