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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?

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  • $\begingroup$ Could you clarify exactly what you want. Is there spectral parameter? Is it dynamical version? Is this dimension of the representation not some dimension associated with the algebra? $\endgroup$
    – AHusain
    Commented Feb 7, 2016 at 22:02

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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 $\dim V=2$:

  • 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:

  • 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.

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