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 groupG and its associated symplectic groupoid. The classical analog of the quantumR-matrix is a lagrangian submanifold in the cartesian square of the symplectic groupoid. For any symplectic leafS inG, induces a symplectic automorphism ofS×S which satisfies the set-theoretic Yang-Baxter equation. When combined with the flip map exchanging components and suitably implanted in each cartesian powerS ⁿ , generates a symplectic action of the braid groupB _n onS ⁿ . 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.Research partially supported by NSF Grant DMS-90-01089Research partially supported by NSF Grant DMS 90-01956 and Research Foundation of University of Pennsylvania