FROM REPRESENTATIONS OF THE BRAID GROUP TO SOLUTIONS OF THE YANG-BAXTER EQUATION

A systematic method is developed for constructing solutions of the Yang-Baxter equation from given braid group representations, arising from such finite dimensional irreps of quantum groups that any irrep can be affinized and the tensor product of the irrep with itself is multiplicity-free. The main tool used in the construction is a tensor product graph, whose circuits give rise to consistency conditions. A maximal tree of this graph leads to an explicit formula for the quantum R-matrix when the consistency conditions are satisfied. As examples, new solutions of the Yang-Baxter equation are found, corresponding to braid group generators associated with the symmetric and antisymmetric tensor irreps of U(q)[gl(m)], a spinor irrep of U(q)[so(2n)], and the minimal irreps of U(q)[E6] and U(q)[E7].