Abstract: |
The Yang-Baxter equation is a cubic equation in a parameter dependent endomorphism R of the tensor square of some vector space V. If R is constant with respect to the parameter and invertible, then the Yang-Baxter equation is equivalent to the braid relations of the braid groups, thus defining a representation of the n-th braid group B_n on the n-th tensor power of V. If, in addition, R squares to 1, then the defining relations of the symmetric groups are satisfied and the n-th tensor power of V is furnished with a representation of the n-th symmetric group. While this is quite a drastic reduction of complexity from the most general form the of the Young-Baxter equation, the set of solutions exhibits a lot of interesting structure and up to a natural notion of equivalence all solutions can be classified. |