Combinatorial Yang-Baxter, 2

It is five years since Tatiana Gateva-Ivanova introduced me to the combinatorial Yang–Baxter equation. I may be slow; I have just understood why this is a good thing for a mathematician to study, aside from its applications in physics (whatever they may be).

Let me start with an analogy. A binary operation on a set A is a function b from X×X to X, where X is a set. Now given a binary operation, we can define a function fx (left translation) on X, for any xX, by

fx(y) = b(x,y).

Then the associative law is equivalent to the assertion that

If b(x,y) = z, then fxfy = fz,

in other words, left translation gives a homomorphism to a transformation semigroup.

Now let r be a map from X×X to itself. Considering X×X×X, let r12 denote the map which acts on the first two coordinates as r and fixes the third, and r23 be the map fixing the first coordinate and acting on the second and third as r. The combinatorial Yang–Baxter equation asserts:

r12r23r12 = r23r12r23.

Now, for each xX, let fx be the map on X given by

r(x,y)=(fx(y),?).

Then, if r satisfies the CYB, we have:

If r(x,y) = (u,v), then fxfy = fufv.

In other words, there is a sense in which solutions to the CYB are “2-dimensional semigroups”.

So it is OK to study them!

The proof is simple. Let r(x,y) = (u,v). Then r23r12r23 maps

(x,y,z) → (x,fy(z),?) → (fx(fy(z)),?,?) → (fx(fy(z)),?,?),

while r12r23r12 maps

(x,y,z) → (u,v,z) → (u,fv(z),?) → (fu(fv(z)),?,?).