## 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 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 S^{n}, σ generates a symplectic action of the braid group B_{n} on S^{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.

Original language | English (US) |
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Pages (from-to) | 309-343 |

Number of pages | 35 |

Journal | Communications In Mathematical Physics |

Volume | 148 |

Issue number | 2 |

DOIs | |

State | Published - Aug 1 1992 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics