# Intuition for the Yang-Baxter Equation (was: Giving relations via formal power series)

I'm reading a book (Yangians and Classical Lie Algebras by Molev) which regularly uses (what appear to me to be) clever tricks with formal power series to encapsulate lots of relations. For instance, if we let $S_n$ act on $(\mathbb{C}^N)^{\otimes n}$ by permuting tensor components, so that e.g. $P_{(1 2)} (a \otimes b \otimes c) = b \otimes a \otimes c$, then working in $({\rm End} \mathbb{C}^N)^{\otimes 3}(u, v)$ we have the identity

$\left(u - P_{(1 2)}\right)\left(u + v - P_{(1 3)}\right)\left(v - P_{(2 3)}\right) = \left(v - P_{(2 3)}\right)\left(u + v - P_{(1 3)}\right)\left(u - P_{(1 2)}\right)$

This is used to motivate the definition of an operator $R_{(j k)}(u) = 1 - P_{(j k)} u^{-1}$, the Yang R-matrix, which is then used to express an enormous family of relations on an algebra by multiplying by a matrix of formal power series.

Of course it's straightforward to verify that the above expression holds if we multiply out the terms. That said, it seems considerably less straightforward to me how one would start from $S_3$ and end up at the equation above. Is this just a marvelous ad-hoc construction, or does it belong to some class of examples?

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Well, that equation a special case of the so called parameter-dependent Yang–Baxter equation. It is supposedly natural in the context of statistical mechanics, where it originally shows up... After that, it was observed in nature in many, many contexts, specially in the equivalent form of the braid equation (which I find enormously more palatable) In the statistic mechanics context it was some time ago explained to me in terms of collisions of particles in a one dimensional system, and did appear quite natural. – Mariano Suárez-Alvarez Feb 2 '11 at 23:20
Care to recapitulate that explanation, or suggest where I can read about it? – Daniel McLaury Feb 3 '11 at 1:02
I would not be able to. This was a course by Marc Rosso at the École Polytechnique some 8ish years ago---maybe you can find notes online. – Mariano Suárez-Alvarez Feb 3 '11 at 1:25