Robert's Mistakes

What's a Moufang loop?

Jan 09, 2022

In 1935 Ruth Moufang published a paper "Zur Struktur von Alternativkörpern", where she intoduced an algebra very similar to groups, but nonassociative.

https://en.wikipedia.org/wiki/Moufang_loop

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.

Definition

Let's look at its definition. The Moufang identities are a generalization of associativity:

z(x(zy))=((zx)z)yx(z(yz))=((xz)y)z(zx)(yz)=(z(xy))z(zx)(yz)=z((xy)z) z(x(zy)) = ((zx)z)y \\ x(z(yz)) = ((xz)y)z \\ (zx)(yz) = (z(xy))z \\ (zx)(yz) = z((xy)z)

More precisely, they generalize alternative algebras:

Alternative Algebra

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:

x(xy)=(xx)y left alternative identityy(xx)=(yx)x right alternative identity x(xy) = (xx)y \textrm{ left alternative identity} \\ y(xx) = (yx)x \textrm{ right alternative identity}

There's also the flexible identity:

x(yx)=(xy)x x(yx) = (xy)x

but any two can prove the remaining one.

One consequence, which justifies the name, is that the associator

[a,b,c]=(ab)ca(bc) [a,b,c] = (ab)c - a(bc)

of any alternating algebra is totally skew symmetric:

[xσ(1),xσ(2),xσ(3)]=sgn(σ)[x1,x2,x3] [x_{σ(1)}, x_{σ(2)}, x_{σ(3)}] = sgn(σ)[x_1, x_2, x_3]

https://en.wikipedia.org/wiki/Alternative_algebra

But I digress...

Moufang's inverse property

A Moufang loop also requires the inverse property:

x1(xy)=y=(yx)x1 x^{-1}(xy) = y = (yx)x^{-1}

which implies that left and right inverses exist. Using that, we can reformulate the first two Moufang identites in a more practical form:

(xy)z=(xz1)(zyz)x(yz)=(xyx)(x1z) (xy)z = (xz^{-1})(zyz) \\ x(yz) = (xyx)(x^{-1}z)

Looking at it differently

There's also an operator version of of these, which might be easier to remember:

LzLzLz(y)=Lzzz(y)RzRyRz(x)=Rzyz(x) L_z L_z L_z (y) = L_{zzz} (y) \\ R_z R_y R_z (x) = R_{zyz} (x)

And another, equivalent to the remaining identities:

LzRz(xy)=Lz(x)Rz(y)=RzLz(xy) L_z R_z (xy) = L_z (x) R_z (y) = R_z L_z (xy)

In 1937, Gerrit Bol introduced another kind of loop:

a(b(ac))=(a(ba))c left Bol loop((ca)b)a=c((ab)a) right Bol loop a(b(ac)) = (a(ba))c \textrm{ left Bol loop} \\ ((ca)b)a = c((ab)a) \textrm{ right Bol loop}

If both hold, then the Moufang identities will also hold.

https://en.wikipedia.org/wiki/Bol_loop

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).

https://en.wikipedia.org/wiki/Pascal%27s_theorem

Let's get back to our story:

Little Desargues

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.

https://en.wikipedia.org/wiki/Little_Desargues_theorem#The_little_Desargues_theorem

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.

https://en.wikipedia.org/wiki/Moufang_plane

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?

translation planes

Those translation planes (a kind of non-Desarguian projective plane) in which little Desargues still holds for every line are Moufang planes!

https://en.wikipedia.org/wiki/Translation_plane

Planar ternary rings are related, and quasifields can be used to coordinatize translation planes, but I know even less about these things.

https://en.wikipedia.org/wiki/Planar_ternary_ring

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?

[twitter]

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:

(a,b,c)=a(bc)(ab)c (a,b,c) = a(bc)-(ab)c

But in the context of groups or quasigroups we only have one operation available, so here's a better way to define an associator:

(a,b,c)=(a(bc))((ab)c)1 (a,b,c) = (a(bc))((ab)c)^{-1}

With that out of the way, let's, for simplicity, look at commutative Moufang loops (CML). Here, it suffices to require either

(xy)(xz)=x2(yz) orx(y(xz))=(x2y)z (xy)(xz) = x^2(yz)\textrm{ or} \\ x(y(xz)) = (x^2y)z

to derive the rest. Supposedly, to see the similarity with groups, it's helpful to also stare at power associativity:

x(xy)=x2y x(xy) = x^2y

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 Z(E)Z(E) all elements for which associativity holds:

x,y,zZ(E):x(yz)=(xy)z \forall x,y,z \in Z(E) : x(yz) = (xy)z

That is similar to noncommutative groups, where we often study their commutative center Z(G)Z(G)

x,yZ(G):xy=yx \forall x,y \in Z(G) : xy = yx

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:

1E2fEgE11 \begin{CD} 1 @>>> E_2 @>f>> E @>g>> E_1 @>>> 1 \end{CD}

where, as usual, ff maps E2E_2 to the kernel of gg, and then say that ff extends E1E_1 by E2E_2. Or we could define EE using a quotient:

E1E/f(E2) E1 \simeq E/f(E2)

The quotient of a CML by its center E/Z(E)E/Z(E) is a CML of exponent 3. This means

x3=1 x^3 = 1

holds for any xx. This leads to two deep theorems:

Theorem: The center of a CML of exponent 3 is non-trivial.

and:

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 zz. Of course, this works best in projective space. Anyways, we get a 3-way relation CC which holds for any permutation of its arguments:

C(x,y,z):xy=z C(x,y,z): x \circ y = z

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 uu by defining a new composition like this

xy=u(xy) xy = u \circ (x \circ y)

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 cc from Z(E)Z(E), then

xy=c(x1y1) x \circ y = c(x^{-1}y^{-1})

turns EE into a CH-quasigroup. However, different choices for cc need not give isomorphic CH-quasigroups!

(to be continued)

Links

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:

[youtube]

For the full interview see here:

https://www.simonsfoundation.org/2012/01/27/yuri-manin/

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.