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Quantization of Bending Deformations of Polygons In
, Hypergeometric Integrals and the Gassner Representation
Published online by Cambridge University Press: 20 November 2018
Abstract
The Hamiltonian potentials of the bending deformations of $n$-gons in
${{\mathbb{E}}^{3}}$ studied in
$\left[ \text{KM} \right]$ and [Kly] give rise to a Hamiltonian action of the Malcev Lie algebra
${{P}_{n}}$ of the pure braid group
${{P}_{n}}$ on the moduli space
${{M}_{r}}$ of
$n$-gon linkages with the side-lengths
$r\,=\,\left( {{r}_{1}},\ldots ,{{r}_{n}} \right)$ in
${{\mathbb{E}}^{3}}$. If
$e\,\in \,{{M}_{r}}$ is a singular point we may linearize the vector fields in
${{P}_{n}}$ at
$e$. This linearization yields a flat connection
$\nabla$ on the space
$\mathbb{C}_{*}^{n}$ of
$n$ distinct points on
$\mathbb{C}$. We show that the monodromy of
$\nabla$ is the dual of a quotient of a specialized reduced Gassner representation.
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- Research Article
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- Copyright © Canadian Mathematical Society 2001
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