Abstract
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 σ 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, σ 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.
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Communicated by N. Yu. Reshetikhin
Research partially supported by NSF Grant DMS-90-01089
Research partially supported by NSF Grant DMS 90-01956 and Research Foundation of University of Pennsylvania
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Weinstein, A., Xu, P. Classical solutions of the quantum Yang-Baxter equation. Commun.Math. Phys. 148, 309–343 (1992). https://doi.org/10.1007/BF02100863
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DOI: https://doi.org/10.1007/BF02100863