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?

I share the thoughts above on the Yang-Baxter equation. My viewpoint on this is perhaps more algebraic: in light of the quantum group theory, the Yang R-matrix, as well as its trigonometric and elliptic counterparts, are indeed somewhat miraculous objects. Since no classification of solution of the Yang-Baxter equation is known, it is not clear how to put these examples in perspective. The R-matrix form of the defining relations does bring new tools to work these algebras. As pointed out above, the whole variety of relations is written as a single matrix relation. This is a starting point for special matrix techniques.
