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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

Michael Kapovich  and
John J. Millson
Michael Kapovich
Affiliation:
Department of Mathematics University of Utah Salt Lake City, Utah 84112 USA, email: kapovich@math.utah.edu
John J. Millson
Affiliation:
Department of Mathematics University of Maryland College Park, Maryland 20742 USA, email: jjm@math.umd.edu
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Abstract

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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.

Keywords

53D3053D50
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 44 , Issue 1 , 01 March 2001 , pp. 36 - 60
Copyright
Copyright © Canadian Mathematical Society 2001

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