dorsal/arxiv
View SchemaQuantization of the Algebra of Chord Diagrams
| Authors | Jørgen Ellegaard Andersen, Josef Mattes, Nicolai Reshetikhin |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9701018 |
| URL | https://arxiv.org/abs/q-alg/9701018 |
| DOI | 10.1017/S0305004198002813 |
Abstract
In this paper we define an algebra structure on the vector space $L(\Sigma)$ generated by links in the manifold $\Sigma \times [0,1]$ where $\Sigma $ is an oriented surface. This algebra has a filtration and the associated graded algebra $L_{Gr}(\Sigma)$ is naturally a Poisson algebra. There is a Poisson algebra homomorphism from the algebra of chord diagrams $ch(\Sigma)$ on $\Sigma $ to $L_{Gr}(\Sigma)$. We show that multiplication in $L(\Sigma)$ provides a geometric way to define a deformation quantization of the algebra of chord diagrams, provided there is a universal Vassiliev invariant for links in $\Sigma\times [0,1]$. The quantization descends to a quantization of the moduli space of flat connections on $\Sigma $ and it is universal with respect to group homomorphisms. If $\Sigma $ is compact with free fundamental group we construct a universal Vassiliev invariant.
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"abstract": "In this paper we define an algebra structure on the vector space $L(\\Sigma)$\ngenerated by links in the manifold $\\Sigma \\times [0,1]$ where $\\Sigma $ is an\noriented surface. This algebra has a filtration and the associated graded\nalgebra $L_{Gr}(\\Sigma)$ is naturally a Poisson algebra. There is a Poisson\nalgebra homomorphism from the algebra of chord diagrams $ch(\\Sigma)$ on $\\Sigma\n$ to $L_{Gr}(\\Sigma)$.\n We show that multiplication in $L(\\Sigma)$ provides a geometric way to define\na deformation quantization of the algebra of chord diagrams, provided there is\na universal Vassiliev invariant for links in $\\Sigma\\times [0,1]$. The\nquantization descends to a quantization of the moduli space of flat connections\non $\\Sigma $ and it is universal with respect to group homomorphisms. If\n$\\Sigma $ is compact with free fundamental group we construct a universal\nVassiliev invariant.",
"arxiv_id": "q-alg/9701018",
"authors": [
"J\u00f8rgen Ellegaard Andersen",
"Josef Mattes",
"Nicolai Reshetikhin"
],
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"q-alg",
"math.QA"
],
"doi": "10.1017/S0305004198002813",
"title": "Quantization of the Algebra of Chord Diagrams",
"url": "https://arxiv.org/abs/q-alg/9701018"
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