TITLE: Solutions of the Quantum Yang-Baxter Equation
QUESTION [9 upvotes]: I am interested in finding non-constant solutions to the following Yang Baxter equation
$$R_{12}(x/y) R_{13}(x/z) R_{23}(y/z) = R_{23}(y/z)
R_{13}(x/z) R_{12}(x/y)$$
where $R(x)$ is an endomorphism of $V\otimes V$. Personally I happen to be interested in finding explicit solutions in the case where $V$ is of dimension 2.
I would like to know what families of solutions are known. For example, there is an example at the end of Ch 12 of Chari and Pressley related to quantum affine sl2. I would prefer answers that are as explicit as possible (eg writing $R$ as 4x4 matrix) and some context as to where they come from.

REPLY [3 votes]: The general theory (due to Jimbo) is that each irreducible finite dimensional representation
of the quantised enveloping algebra of a Kac-Moody algebra (not of finite type) gives a trigonometric R-matrix. There is substantial information on these representations but the R-matrices are not explicit. There is a special case which is explicit and is given by the
"tensor product graph" method (this was worked out by Niall MacKay and Gustav Delius).
I used this in my paper:
R-matrices and the magic square. J. Phys. A, 36(7):1947–1959, 2003.
and you can find the references there.
If you want to go beyond this special case and be explicit then you can use
"cabling" a.k.a "fusion".
The only papers which deal with R-matrices not covered by the tensor product graph method that I know of are
Vyjayanthi Chari and Andrew Pressley. Fundamental representations
of Yangians and singularities of R-matrices. J. Reine Angew. Math.,
417:87–128, 1991.
G´abor Tak´acs. The R-matrix of the Uq(d(3)4 ) algebra and g(1)2 affine
Toda field theory. Nuclear Phys. B, 501(3):711–727, 1997.
Bruce W. Westbury. An R-matrix for D(3) 4 . J. Phys. A, 38(2):L31–L34, 2005
Deepak Parashar, Bruce W. Westbury
R-matrices for the adjoint representations of Uq(so(n))
arXiv:0906.3419
The Chari & Pressley paper deals with rational R-matrices.
The last preprint was an incomplete attempt to try and find the trigonometric analogues of these R-matrices.