Classifying Low Dimensional Solutions of the Yang--Baxter Equation What is the present situation with classifying solutions of the Yang--Baxter equation in low dimensions?
To make my question more specific, have all solutions for dimension $2$ and $3$ been classified?
 A: Let $(V,c)$ be a braided vector space, that is: $V$ is a vector space and  $c\colon V\otimes V\to V\otimes V$ is an invertible linear map that satisfies $c_{12}c_{23}c_{12}=c_{23}c_{12}c_{23}$, where $c_{12}=(c\otimes\mathrm{id})$ and $c_{23}=(\mathrm{id}\otimes c)$. 
As far as I know, the classification of  braided vector spaces is completed in the case where $\dim V=2$:


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*Hietarinta, Jarmo. All solutions to the constant quantum Yang-Baxter equation in two dimensions. Phys. Lett. A 165 (1992), no. 3, 245--251. MR1169634 (93d:16050). doi. 


Other related interesting results:


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*Dye, H. A. Unitary solutions to the Yang-Baxter equation in dimension four. Quantum Inf. Process. 2 (2002), no. 1-2, 117--151 (2003). MR2032002 (2004k:81168). doi

*Galindo, César; Rowell, Eric C. Braid representations from unitary braided vector spaces. J. Math. Phys. 55 (2014), no. 6, 061702, 13 pp. MR3390645. doi
Edit: In general, producing solutions of the Yang-Baxter equation is a very hard problem. In MO Question 201901, you will find some information on the so-called set-theoretic solutions.
