Classical solutions of the quantum Yang-Baxter equation

The classical analogue is developed here for part of the construction in which knot and link invariants are produced from representations of quantum groups. Whereas previous work begins with a quantum group obtained by deforming the multiplication of functions on a Poisson Lie group, we work directly with a Poisson Lie group G and its associated symplectic groupoid. The classical analog of the quantum R-matrix is a lagrangian submanifold[Figure not available: see fulltext.] in the cartesian square of the symplectic groupoid. For any symplectic leaf S in G,[Figure not available: see fulltext.] induces a symplectic automorphism σ of S×S which satisfies the set-theoretic Yang-Baxter equation. When combined with the "flip" map exchanging components and suitably implanted in each cartesian power Sⁿ, σ generates a symplectic action of the braid group B_n on Sⁿ. Application of a symplectic trace formula to the fixed point set of the action of braids should lead to link invariants, but work on this last step is still in progress.